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Logic and Its Philosophy

by Jan Woleński (Author)
Monographs 264 Pages

Table Of Content


Preface

This book collects 20 of my papers published in the years 2011–2016. All of them, with one exception, are devoted to logic and its philosophy. The essay XX is the only exception, although it also alludes to some logical questions. However, I included considerations on Keret’s House for personal reasons. My roots are Jewish. I spent the years 1941–1944 in Warsaw, not in ghetto, but in relatively normal (if anything was normal at the time) circumstances. Although my family survived, but… (let me not finish). Hence, I am very sensitive to philosophical (and other) problems related to the Holocaust.

The rest of the book is a sequel to my Essays on Logic and its Applications in Philiosophy, published by Peter Lang in 2011. But this collection is more compact because all chapters belong to systematic philosophy (I did not include historical studies). However, I continue topics considered in the mentioned book of 2011 and use similar analytic tools taken from formal logic, especially the logical square and its generalization. Generally speaking, both collections can be viewed as contributions to so-called philosophical logic.

The papers included in this collection are reprinted here with changes introduced for avoiding repetitions. Yet, particular chapters overlap at some points; in particular, the diagrams re-appear in few places. Although I tried to unify symbolism to some extent, there are some differences caused by the fact that specific letters and signs play various roles. However, the context always explains what is going on at a given place. The numeration of formulas and statements is separate for each chapter.

The book is financially supported by the University of Information, Technology and Management in Rzeszów, Poland. I am grateful to its authorities, especially Dr. Wirgiliusz Gołąbek for this help. I would also like to thank Mr. Łukasz Gałecki for his initiative that this collection could be published by Peter Land. I am also grateful the text-editors for thei efforts toward preparing the final version of the book. All papers are reprinted with permission of particular publishers.

Jan Woleński

← 7 | 8 → ← 8 | 9 →

Source note

Semantic Loops, in Philosophy in Science. Methods and Applications, ed. by B. Brożek, W. P. Grygiel and J. Mączka, Copernicus Center Press, Kraków 2011, 256–271.

Logic as Calculus Versus Logic as Universal Medium, and Syntax Versus Semantics, Logica Universalis 6(3)(2012), 587–596.

Do We Need to Reform the Old T-Scheme, Discusiones Filozóficas (20)(13) (2012), 73–85.

Truth is Eternal if and only if it is Sempiternal, in: Studies in the Philosophy of Herbert Hochberg, ed. by E. Tegtmeier, Ontos, Frankfurt am Main 2012, 223-230.

Is Identity a Logical Constant and are there Accidental Identities?, Studia Humana 3-4(2012), studia humana.com/current-issue.html.com.

Naturalism and the Genesis of Logic, in Papers on Logic and Rationality. Festschrift in Honor of Andrzej Grzegorczyk (Studies in Logic, Grammar and Rhetoric 27(40)); together with K. Trzęsicki and S. Krajewski, University of Bialystok, Bialystok 2012, 223–240.

Some Analogies between Normative and Epistemic Discourse, in: The Many Faces of Normativity, ed, by J. Stelmach, B. Brożek and M. Hohol, Copernicus Center, Kraków 2013, 51–71.

Theology and Logic, in: Logic in Theology, ed. by B. Brożek, A. Olszewski and M. Hohol, Copernicus Center Press, Kraków 2013, 11–38.

Ens et Verum Concertuntur (Are Being and Truth Convertible)? A Contermporary Pesrpective, in: Truth, ed. by M. Dumitriu and G. Sandu, Editura Universitãtii din Bucureşti, Bucureşsti 2013, 75–83.

An Abstract Approach to Bivalance, Logic and Logical Philosophy 23(1)(2014), 3–14.

Philosophical Reflections on Logic, Proof and Truth, in: Trends in Logic XIII. Gentzen’s and Jaskowski’s Heritage. 80 Years of Natural Deduction and Sequent Calculi, ed. by A. Indrzejcak, J. Kaczmarek and M. Zawidzki, Wydawnictwo Uniwersytetu Łódzkiego, Lódź 2014, 27–39.

Constructivism and Metamathematics, in The Road to Universal Logic. Festschrift for 50th Birthday of Jean-Yves Béziau, v. I, ed. by A. Koslow and A. Buchsbaum, Birkhäuser, Basel 2015, 513-520. ← 9 | 10 →

Rule-Following and Logic, in Problems of Normativity, Rules and Rule-Following, ed. by M. Araszkiewicz, P. Banas, T. Gizbert-Studnicki and Krzysztof Płeszka, Springer, Heidelberg 2014, 395–402.

Truth-Makers and Convention T, in Philosophical Papers for Kevin Mulligan, ed. by A. Reboul, in Mind, Values, and Metaphysics. Philosophical Essays in Honor of Kevin Mulligan, v. 1, Springer, Dordrecht 2014, 79–84.

Constructivism and Metamathematics, in The Road to Universal Logic. Festschrift for 50th Birthday of Jean-Yves Béziau, v. I, ed. by A. Koslow and A. Buchsbaum, Birkhäuser, Basel 2015, 513–520.

An Analysis of Logical Determinism, in Themes from Ontology, Mind and Logic. Present and Past. Essays in Honour of Peter Simons, ed. by S. Lapointe, Brill, Leiden 2015, 423–442.

Normativity of Logic, in The Normative Mind, ed. by J. Stelmach, B. Brożek and Ł. Kwiatek, Copernicus Center 2016, 169–195.

Formal and Informal Aspects of the Semantic Theory of Truth, in: Uncovering Facts and Values. Studies in Contemporary Epistemology and Political Philosophy, ed. by A. Kuźniar and J. Odrowąż-Sypniewska, Brill/Rodopi, Leiden 2016, 56–66.

The Paradox of Analyticity and Related Issues, in Modern Logic 1850–1950. East and West, ed. by F. Abeles and M. E. Fuller, Birkhäuser, Basel 2016, 135–138.

Some Liar-Like Paradoxes, Logic-Philosophical Studies. Yearbook of St. Perterburg Logic Association (14(2016), 70–75.

Nothingness, Philosophy and Keret’s House, in: Przestrzenie pustki/Void Spaces, Fundacja Polskiej Sztuki Nowoczesnej/Foundation of Polish Modern Art., Warszawa 2013, 23–39. ← 10 | 11 →

I
Semantic Loops

Michael Heller introduced (see Heller 1999, pp. 90, 99–100) the idea of semantic loops. They are essentially involved in interactions between languages and their metalanguages (I will use the letter L as referring to a language and the symbol ML as denoting a metalanguage of L). As it is well-known, such interactions lead to semantic antinomies related to the self-referential use of expressions. The famous Liar antinomy (LA for brevity) is perhaps the most paradigmatic example. Consider the sentence:

   (λ) the sentence denoted by (λ) is false.

A simple inspection shows that (λ) is true if and only if it is false (it will be demonstrated below). The situation illustrated by (λ) can be metaphorically characterized as a closed semantic loop, because we pass from truth to falsehood and back without any possibility of leaving the loop (or a circle, if you prefer this, more popular, figure of speech; see also a historico-terminological digression below). On the other hand, the language/metalanguage distinction cannot be liquidated, because we need to speak about languages and their various properties.

Although ML can be effectively reduced to L in the case of syntax (for example, via the method of Gödel numbering), this is impossible in the case of semantics; Tarski showed that doing semantics of L in ML requires that the latter is essentially richer than the former. Thus, closed semantic loops operate somehow between L and ML. As Heller indirectly (as he speaks about loops in physics) suggests, a solution of a semantic paradox assumes that loops related to such antinomies are not quite closed (for stylistic reasons, I use the labels ‘antinomy’ and ‘paradox’ as synonyms). In order to have a convenient façon de parler, I will contrast closed loops resulting in paradoxes and open loops, which save us from falling into antinomies as contradictions. The Gödelian sentence

   (G) (G) is not provable,

provides an example of an open loop, because it is not paradoxical, but generates undecidable sentences or shows that arithmetic is incomplete (of course, after suitable formulation in terms of number theory and assuming its consistency); in fact, (G) is purely syntactic and does not involves semantic notions. My main task is to outline a formal approach to the problem of semantic loops. ← 11 | 12 → Yet, I will also make some general and special philosophical suggestions at the end of this paper.

Let me start with a diagnosis of LA paradox outlined by Leśniewski and Tarski (see Tarski 1933, Tarski 1935, and especially Tarski 1944, pp. 671–674). They argued (Leśniewski never published his results) that LA is generated by the following facts (my formulation differs literally from that of Tarski):

This diagnosis opens three ways out. Firstly, we can exclude self-reference from semantics. Secondly, we drop the T-scheme. Thirdly, we decide to modify logic. In each case, a solution of LA costs something in the sense that we must sacrifice self-reference or the T-scheme or classical logic. Nothing is free of charge here and eliminating semantic antinomies requires a sacrifice of something for something. For Tarski (and the same concerns Leśniewski), playing with (A) was the most natural way (or cost less than rejecting the T-scheme) or changing logic. I do not know of any solutions consisting in rejecting the T-scheme. However, this remark must be properly understood. If we exclude self-reference, some T-equivalences resulting from the T-scheme are not admissible, for instance ‘λ is true if and only if λ’, because it leads to LA (see Thomason 1986). Similarly, we block analogous statements having the form of equivalences and related to other semantic paradoxes, for instance, Grelling’s antinomy of heterological adjectives (see below).

It is important to see the role of the T-scheme in producing LA. Not every case of self-reference is harmful (in the sense of leading to an antinomy). In fact, we have ‘innocent’ self-referential sentences (for a general concise treatment of cases of this phenomenon, not only linguistic, see Smith 1986; Bartlett, Suber 1987 and Bartlett 1992 are useful collections) in which self-reference occurs. Consider, the phrase

Clearly, (*) is self-referential and false. However, it does not produce any antinomy. (T) serves as a device of abbreviation. Consider LA once again. The main step consists in observing that (a) λ = ‘λ is false’. Hence, having (b) ‘λ is true if and only if λ’, and using (a), we arrive at: ‘λ is true if and only if λ is false’. Similarly, define an autological adjective as such which possesses the property ← 12 | 13 → expressed by it; an adjective is heterological in the opposite case. For instance, the word ‘short’ is short and thereby autological, but the expression ‘long’ is just heterological. Now we ask: is ‘hetelogical’ heterological or autological? Denote ‘heterological’ by the symbol Ht and ‘autological’ by At. Assume that Ht is Ht, that is Ht(Ht) holds. If so, Ht is not-Ht, but At. In other words, if Ht(Ht), then At(Ht). Assume, then At(Ht). If so, ‘heterological’ possesses the property of being heterological, but this means that Ht(Ht) holds. Thus, Ht(Ht) if and only if At(Ht). Generally speaking, we have the equivalence (**) ‘At(E) if and only if E(E)’, which plays a similar role as the T-equivalence. Roughly speaking, semantic equivalences, like instances of (T) or (**), omit all terms expressing semantic properties, like truth or reference. Using the introduced terminology, semantic equivalences close loops.

Changing logic is a popular strategy in fighting against semantic antinomies. There are several attempts of this kind (see papers in Martin 1978 and Martin 1984, Sheard 1994, Antonelli 2000, Bromand 2001, Bolander, Hendricks, Pedersen 2006; this is only a sample of a very rich literature). Without entering into details, one may mention the transition from classical to many-valued logic (paradoxical sentences have another logical value than truth or falsehood; the most popular version assumes three-valued logic), gappy semantics (paradoxical sentences have no logical value) or paraconsistent logic (paradoxical sentences are so-called dialetheias, statements which are simultaneously true and false; λ is an example here). Other solutions propose to employ partial truthdefinitions (it is a response to the undefinability truth-theorem; see below) or so-called circular concepts (see Gupta 2000), that is, concepts which are regulated by special revision processes. All mentioned ways out open semantic loops. I leave these proposals without further analysis, because I would like to show that the standard Tarski’s recipe against antinomies or semantic closed loops is the simplest and most elegant solution.

I shall make one additional remark here. As I have already mentioned, any solution of LA and similar paradoxes comes at the expense of something. It is difficult to measure the amount which must or should be paid. Advocates of changing logic, partial truth-definitions or using circular concepts argue that we can define truth for L in L itself. They also point out that Tarski’s solution via the hierarchy of languages is artificial and at odds with the spirit of natural language. However, it is difficult to understand what is behind such claims. Why should we prefer changing logic over a stratification of our language into levels? Although no conclusive or ultimate answer is available, the ordinary parlance ← 13 | 14 → suggests, for instance, in making citations, that we clearly distinguish between the language about things and the language about words.

Let me make a historico-terminological digression. The problem of circularity goes back to Aristotle. He pointed out that circular arguments and circular (or idem per idem) definitions are incorrect and should be eliminated. These issues belonged to elementary school logic for a long time. A new and important impulse came from mathematical logic. When Russell discovered his famous paradox of sets which are not their own elements, logicians began to propose various ways out. The vicious circle principle (VCP) was proposed as one of devices. It says that if a set X is defined, its elements cannot be characterized by belonging to X. Otherwise speaking, impredicative constructions should be abandoned. Unfortunately, this rule is inconsistent with some accepted definitions in the arithmetic of real numbers. Russell tried to harmonize VCP with mathematical practice via the ramified theory of logical types. However, this solution became contested, as it is enormously complicated. On the other hand, the role of circularity in the sense of VCP suggests that not every circular construction is bad. Perhaps, we can say that open semantic loops tolerate good circularity, but closed semantic loops generate paradoxes and must be broken. As far as I know, the terms ‘loop’ and ‘closed loop’ were proposed by Haim Gaifman (see Gaifman 1992, pp. 230–231, 253, Gaifman 2000, pp. 88, 109; see also Koons 2000, p. 192). According to Gaifman, loops are sets of unevaluated pointers, that is, items which point to sentences and determinate their semantics. Thus, loops in Gaifman’s sense are something different than loops as understood in this paper, although the former are closely related to self-referential statements which generate LA and other paradoxes. Loops in Heller’s (and mine) meaning are comparable with Gaifman’s idea of black holes, that is, semantically gappy sentences. It is perhaps remarkable that various physical metaphors (loops, holes) became fairly popular in considerations about semantic difficulties stemming from self-referentiality. Let me add that Heller himself (see Heller 1999, p. 99) proposes a logic of loops as a weapon against closed loops and antinomies. According to him, such a logic should be non-linear, where non-linearity is understood analogically as in physics. As Heller points out to us, it is only a project. The name ‘non-linear logic’ can be slightly misleading, because the label ‘linear logic’ refers, in the jargon of contemporary logicians, to the logical theory combining classical and intuitionist logic in proof theory. Let me repeat, once again, that my further considerations are firmly set in the framework of classical logic and its metatheory. ← 14 | 15 →

As it is implicit in my previous remarks, Tarski’s semantic paradigm plays a crucial role in this paper. His truth definition (see Tarski 1933, Tarski 1935, Tarski 1936) is the most spectacular example of the paradigm in question. The concept has been used earlier, but rather informally, due to LA and other paradoxes, which plagued logic in the first decades of the 20th century. Tarski defined truth formally. In the contemporary setting (I will employ it, not Tarski’s original construction, except his remarks about the entire hierarchy of languages), truth is doubly relativized, namely to a language and a model. Thus, the fully defined context is ‘true in a language L and a model M’. Let us assume that A is a sentence of a language L. We say that A is true if and only if it is satisfied by all sequences of objects taken from the carrier of M (equivalently: it is satisfied by one sequence; it is satisfied by the empty sentence). This definition obeys the convention T, that is, the condition:

Here we have a new formulation of the T–scheme. As an example of a concrete T–equivalence, we have the sentence ‘n(snow is white) is true if and only if snow is white’ for the sentence ‘snow is white’. We assume that the expression ‘n(snow is white)’ is a name of the sentence ‘snow is white’, and that the last expression is a self-translation into English as a metalanguage. Instead of writing ‘n(snow is white)’ we could use quotes and say that the expression ‘snow is white’ is a name of the sentence occurring within the quotation marks. Also, instead of using English as our language and metalanguage simultaneously, we could say, for example, ‘n(Schnee is weiss)’ is true in German if and only if snow is white, where German is our object-language and English serves as the metalanguage in which the mentioned equivalence is formulated. Remarks about using ‘n(…)’ and quoting show that it is not easy to distinguish between naming sentences and using them (quotes indicate both these elements). In general, quotes (single ‘…’, double “…” or any other kind, for example, ‹…›) are only conventional devices and can be replaced any other notation (this remark applies for others papers included in this book). Typically, labels for formulas are made in metamathematics by ascribing numerals for them; an example will be given below.

Perhaps, the most important feature of Tarski’s (or semantic) truth-definition is its explicit metalinguistic character. Tarski originally worked in a version of Russell’s theory of types. This framework forces an infinite hierarchy of lan ← 15 | 16 → guages H = L0, L1, L2, … such that L Ln+1 (n = 0, 1, 2., …) is MLn, that is, a metalanguage for the language Ln. Consequently, we need a separate truthpredicate for every level. In fact, this variety is considerably limited, because it is cumulative: every next level contains former ones. The contemporary setting related to the distinction between first-order and higher-order languages assumes that it is enough to give the truth-definition for first-order sentences. If L is a first-order language, the truth-definition for its sentences is formulated in a second-order language as its metalanguage M; of course, we can always extend the construction for higher-order languages. Tarski showed that ML must be essentially richer than L in the sense that the former has to contain expressions of higher semantic categories than the latter and, moreover, the metalanguage has the translations of all expressions of the object language. Let us assume that L and ML have the same order. Then, we can construct a sentence, for example, λ, which is belongs to L and ML simultaneously and) leads to LA. This suggests that the set of true sentences in L cannot be defined in L itself. The last statement is a rough formulation of the Tarski undefinability theorem, one of the most important metalogical results. As far as the matter concerns the entire H, it is impossible to give a correct truth-definition for it. The Tarski undefinability theorem belongs to the family of limitative theorems, called in this way because they tell us something about limitations of formal systems. The Gödel incompleteness theorems, the Löwenheim-Skolem theorem about the size of models of first-order theories and the Church theorem about the undecidability of first-order logic are other members of this family. The first Gödel incompleteness theorem has two versions: syntactic and semantic. The former states that if Peano arithmetic (PA) is consistent, it is incomplete, because there are arithmetical sentences A and ¬A, such that both are not provable in PA, although the latter says that if PA is consistent, there are true but not provable arithmetical sentences.

The semantic version of the first Gödel theorem immediately suggests that there is a very deep connection between incompleteness and unprovability of truth in formal arithmetic. In my further remarks, I closely follow (as I did in Woleński 2004) the account given in Tarski, Mostowski, Robinson 1953, pp. 44–48 (I will also reproduce some proofs in order to demonstrate how formal devices work). Let TH be a first-order consistent formal theory with an infinite sequence of constant (that is, without free variables) terms t1, t2, t3, …. We say that a set of natural numbers KN is definable in TH (relatively to t1, t2, t3, …) if and only if there is a formula ATH such that A(tn) is valid in TH, provided that nK and ¬A(tn) is valid in TH, provided that nK. This definition is very ← 16 | 17 → easily generalized for arbitrary functions and relations. In particular, a function F is definable in TH if and only if for any nN, the sentence ∀v(A(tn, u) ⇔ v = tFn) is valid in TH and for any m, nN such that Fmn, the sentence ¬A(tm, tn) is valid in TH. Let us assume that we have any fixed one-to-one correspondence between natural numbers and expressions of TH; this strategy provides numerals as names of formulas. Without any loss of generality, we can concentrate on formulas of sentential category, that is, sentences or sentential open formulas. Thus, An is a formula corresponding with the number n and Nr(A) refers to the correlated number of A. Now, we define d (that is, the diagonal function) by the equality dn = Nr(An(tn)). Assume that v is the only free variable in An. According to the definition of the diagonal function, DFn is the correlated number of the formula obtainable from the formula An(v) by replacing the variable v by the term tn.

Let V be set of all natural numbers n such that An is a sentence true in TH. The following theorem holds:

Proof: Assume that DF and V are both definable in TH. This entails that there are formulas B and C such that for any natural number n, (i) the sentence ∀v(B(tn, v) ⇔ v = tDFn) is valid in TH (the definability of d); (ii) nV if and only if C(tn) is valid in TH (by the definability of V). Since the variable v has no free occurrence in C, we can assume that it does not occur in C at all. Let m = Nr(∀v(B(v, w) ⇒ ¬C(v)). This entails Am = ∀v(B(w, v) ⇒ ¬C(v)) and, furthermore, (iii) Am (tm) = ∀v(B(v, w) ⇒ ¬C(v)). If An(tn) is true, it is also true (by (i)) for the case n = m. This means that the sentence ¬C(tdm) is also true. At the same time, however, provided that the sentence Am(tdm) is not true, then Nr(Am(tDFm) ∉ V. By the definition of the diagonal function, we have (iv) dm = Nr(Am(tdm). This implies (by the contraposition of (ii)) that ¬C(tm) is true in TH. Consequently, (v) the sentence ¬C(tdm) is true in TH. The last result and (i) give (vi) ∀v(B(tm, v) ⇒ ¬C(v)) is true in TH. Finally, using (i) and (vi), we obtain (vii) C(tdm) is true in TH. Thus, we arrived at a contradiction, because C(tdm) and ¬C(tdm) are true in TH. Since we assumed that TH is consistent, the proof is concluded.

This result is followed by the following comment (p. 47): [This] theorem and its proof represent a metamathematical reconstruction and generalization of arguments involved in various semantical antinomies and, in particular, in the antinomy of the liar. The idea of this reconstruction and the realization of its far reaching implications is due to Gödel […]. The present version of this reconstruction is distinguished by its generality and simplicity. ← 17 | 18 →

In fact, the set V is intended to cover the truths of arithmetic. Hence, the expression ‘∈ V’ can read ‘is true’; this justifies calling V ‘the truth-predicate’. However, in order to avoid a confusion, let the symbol Tr stands for this predicate. Furthermore, the expression ‘nV’, as applied to sentences and due to the correspondence between natural names and sentences, can be read ‘the sentence named by the number n is true’. Thus, the method of arithmetization provides a convenient system of naming sentences. If we look at (i) as applied to V, we obtain that

it means that TH (assuming that it is arithmetized theory) satisfies the fixed-point lemma (or the diagonalization lemma; see below) with respect to the predicate Tr; the expression ADFn is the fix-point of Tr(An). Consider now the sentence Tm = ¬Tr(Tm). By (CT’), it is the fix-point of Tr(Tm). Thus, Tm is true, but on the other hand it is false by the assumption. This reasoning shows that the Liar paradox is a special case of the general proof of (UT). To see it, observe that (i) implies (CT’) and the latter entails that we have the appropriate fix-points, depending on whether the correlated number, let say k, of the formula ¬Tr(Tm) belongs to V or not; thus, we consider the case when k = m. Thus, the formula (CT’) ⇒ ¬Tr(Tm) (with Tm defined as above) becomes an instantiation of the formula ∀v(B(v, w) ⇒ ¬C(v)).

I will also give two other accounts of (UT). The first follows Bell 1999, pp. 219-222. Assume that Am(v) is a formula with the code number (the value of the diagonal function) m. Thus, for any n, it is possible to compute the code number of the sentence Am(n), where n is a numeral corresponding to n; the sentence Am(n) obtains from the open formula Am(v) by substituting the numeral n for the free variable v. This fact means that the computation in question is arithmetically representable and we are able to effectively define an arithmetical term s(u, v) such that for any natural numbers m, n, p, the sentence s(m, n) = p is true if and only if p is the code number of the sentence Am(n). Under these preliminaries, we have:

(UT’) If V is a set of arithmetical sentences satisfying the following conditions:

Proof. By (b) there is a formula V(v) of L such that (c) V(n) is true if and only if n is the code number of a sentence in V if and only if AnV, for every n. Consider the formula ¬T(v). Substitute the expression s(v, v) for all free occurrences of the variable v in ¬T(v). The resulting formula is ¬T(s(v, v). Denote it by B(v). Suppose that m is its code number. Thus, B becomes the sentence Am. Let p be the natural number such that the sentence p = s(m, m) is true; thus, p is the code number of the sentence Am(m). Denote it by G. Therefore, p is the code number of G, that is, G is Ap. Observe now that we have the sequence of equivalences:

(E) forces the conclusion that G asserts of itself that it does not belong to V. We also conclude that G is true, because, in the opposite case, it would be true as a member of V. Thus, G is true and GV. It implies that ¬G is false and it does not belong to V, since this set consists of true sentences only. This completes the proof of (UT’). If V is taken as the set of all arithmetical truths, we obtain (AUT), The property being a true arithmetical sentence is not arithmetically definable. A terminological comment is in order. Bell labels (UT’) as the arithmetical truth-theorem and he calls (AUT) the Tarski undefinability theorem. This seems not quite proper, because (UT) was proved by Tarski and his collaborators for all arithmetizable theories. Thus, (UT’) is just a counterpart of (UT) in Bell’s language and (AUT’) is the truth undefinability theorem for arithmetic.

The second account (see Smullyan 1992, p. 104, Murawski 1999, p. 204) is the simplest one. First, we observe that the fixed-point lemma holds for PA (the symbol PA refers to Peano arithmetic), that is, we have:

Assume that there is a formula A(v) such that for any sentence S of L(PA) PAimage SA(Sdn). By (FPLPA) there is a sentence S’ such that PAimage S’ ⇔ ¬A(S’dn). This immediately leads to a contradiction, because we conclude: PAimage A(S’dn) ⇔ ¬A(S’dn). Thus, there is no formula of L(PA) such that it is the truth-definition for PA. It is, of course, the theorem of the undefinability of truth for PA. It can be generalized for any theory, which contains PA as its subtheory. Suppose that the notion of proof is formalized and we have (*) if every member of a set X of sentences is true, then all sentences provable from X are true; (**) if the property of being a member of the set X is arithmetically definable, then the property of a sentence provable from X is also arithmetically definable. Let X* be the set of all sentences provable from X, provided that the latter consists of truths. Thus, by (*) and (**), X* consists of true sentences and the property of being its ← 19 | 20 → member is arithmetically definable. This allows us to apply (UT’) and conclude that there is a true sentence G such that neither it nor its negation belongs to X*. In other words, we have the first Gödel incompleteness theorem:

in the semantic setting (Bell calls it “Gödel’s weak incompleteness theorem”). Its syntactic (or strong) formulation requires much more advanced methods. In particular, the assumption of the truth of X is replaced by the supposition that this set is ω-consistent (or even consistent after Rosser). The advantage of the strong version is that its proof provides an example of an unprovable sentence, but the weak version only states that such sentences exist. However, the sentence G asserts its own unprovability in both versions, although G used in the proof provides a of (UT’) expresses its own falsehood. The limitations asserted by (UT) and (GT1) are not accidental. Although they can be eventually repaired by adding the ω-rule as a new inference rule or admitting that the considered theories have non-recursive axiomatizations, these moves demolish the finitary character of deductive means.

I shall now return to the fixed-point lemma (theorem). In mathematics, for instance, in topology, we frequently encounter the situation that fx = x, where f is a function. If it is the case, we say that x constitutes a fixed point of fx. The equality sign in the formula fx = x admits a variety of interpretations, including its reading as ⇔. In fact, (FPL) and (FPLPA) employ just this interpretations. Roughly speaking (see Smullyan 1992, pp. 102–103), (FPL), as a general metamathematical theorem, says that, for any formula FA asserting that A satisfies the clause F, FA has a fixed point A, that is, the equivalence FAA holds, provided that the diagonal function is definable in a given theory. If d is definable in TH, which is a formal theory (for a survey of fixed-point constructions in logic, see Sommaruga-Rosolemos 1991), this means that THimage d(A) ⇔ A. In particular, THimage d(λ) ⇔ λ, which leads to LA. It is interesting that we do not need to invoke self-reference directly. The fixed-point lemma and diagnonalization in formalized theories act similarly to self-reference in natural language, at least as far as the matter concerns general effects. Instead of eliminating self-referential context, we must decide whether to produce all possible T–equivalences (define the function d as total) or to define the set V of true sentences. Combining both these tasks inevitably leads to antinomies. Otherwise speaking, the choice is either a partial truth-definition for a given L with the truth-predicate belonging to this language or the full truth-definition but formulated in ML. If (CT) is a condition for a correct truth-definition, it blocks maximalism consisting in defining V and ← 20 | 21 → d together; compare the content of (UT). Clearly, the introduced formal machinery refines Tarski’s analysis of natural languages.

Stephen Yablo (see Yablo 1985) argues that Tarski’s original approach does not disprove all versions of LA. Before presenting Yablo’s reasoning, I shall recall another version of LA produced by the pair of sentences A1: A2 is true, A2: A1 is false. Assume that A1 is true. Thus, A2 is true as well, but A1 appears as false, contrary to the assumption. Assume that A1 is false. This entails that A2 is false and, then, that A1 is false, once again contrary to the adopted assumption. The circle of liars (or the loop of liars) shows that self-reference can be indirect. In fact, none of {A1, A2} is self-referential in itself, but it is only their combination that has this property. Yablo proposes to consider an infinite sequence S of sentences: A0, A1, A2, …, Ai, …, where A0 = ‘Ak is false for any k > 0’, A1 = ‘Ak is false for any k > 1’, A2 = ‘Ak is false for any k > 2’, …, Ai = ‘Ai is false for any k > i’, …. Fix an arbitrary i. By assumptions, Ai is false for all k > i. When we then apply the T-scheme, we obtain that (α) Ai is true if and only if Ai is false for all k > i. Assume that Ai is true. This entails that Ai +1 is false, Ai +2 is false, etc. Now, if Ai +1, then Ai +2 is true, contrary to the right part of (α). The last assertion entails that Ai is false for all k > i. Consequently, the left side of (α) is false as well, contrary to our assumption. The supposition that the formula ‘Ai is true’ is false immediately leads to contradiction, because it entails that at least one Ak is true.

Yablo labels the paradox constituted by S as the ω-Liar and claims that it has nothing in common with self-reference and language-hierachies, even indirectly. More precisely (see Yablo 2006), he argues that the typical strategy of anti-circularity recommended by the Tarskian-style approach does not apply to the ω-Liar. Regardless of Yablo’s own recipes, his critical remarks against Tarski are not quite correct. Consider the sequence S once again. He has the stock of sentences A0, A1, A2, …, Ai and the condition that for any k > i (k = 0, 1, 2, …, i, …), Ai asserts that Ak is false. A consistent description of S requires that A0 is true, because in the opposite case there is at least one Ak which is true. Thus, we can conclude that Ao indirectly asserts its own truth. Similarly, we argue that since no sentence Ai (i > 0) can be true, each of these sentences indirectly refers to its falsehood. A more important observation points out that since the ω-Liar heavily uses the T-scheme, it depends on the (FPL). Consequently, the ω-Liar shows that the set of true sentences is not definable in the language in which this antinomy is formulated. Thus, diagonalization and (FPL) appear as much more fundamental phenomena than self-reference.

What about self-reference to truth? The sentence

is called the Truth-Teller. It is self-referential and informs about its truth; (τ) expresses. Is (τ) paradoxical? Common wisdom says: No. The justification is as follows. (FPL) produces the formula (β) Tr (τ) ⇔ τ as provable in arithmetic. Moreover (see Löb 1955), if τ is provable, it is true. This fact does not lead to an antinomy and is consistent with (UT). The equivalence (β) expresses one of the so-called reflection principles; it reflects about itself, its own truth in this case. However (see Woleński 1995 for details), we can formulate the paradox of Truth-Teller by assuming the rejection logic or dual logic as our formal frame. Roughly speaking, classical logic as understood in (C) distinguished truth as a logical value. This means that the consequence operation associated with logic preserves truth and formalizes the process of assertion of sentences on the basis of other sentences, already asserted. However, nothing, at least from a purely formal point of view, prevents using logic which preserves falsehood and operates on rejected sentences. For example, if A or B is rejected, then AB is rejected; if AB is rejected, A is rejected and B is rejected. Both these rules are formulated in the metalanguage of the assertion logic. Yet, we can build the rejection logic in L and ML without referring to the language of assertion. In this new formal framework, which has the duals of all standard metamathematical results (λ) is harmless, but (τ) leads to a paradox. This means that we have no general criteria to decide a priori, which cases of self-reference are innocent and which are not. An accessible empirical evidence suggests that dangerous self-referential cases are semantic in their essence (they involve categories like truth, reference, etc.) and occur together with reduction schemes as the T-scheme allowing a compression of self-referential sentences. Once again, the functions d and (FPL) play a crucial role. In fact, they liberate us from appealing to self-reference, at least in well-(formally)behaved systems. Self-reference becomes essential, when we pass to natural languages.

My previous considerations suggest that the assertion “semantic loops are created by self-reference” should be taken cum grano salis and only as the first approximation. A closer analysis via metamathematical devices shows that diagonalizations and (FPL) produce such loops (also syntactic) in well-defined formal theories. Self-reference appears when we work inside natural languages in which such concepts as truth, provability or definability function as not quite determined categories. At the same time, however, there is a fairly close correlation between both sources of loops, that is, precise mathematical construction and self-referential contexts, because both produce inconsistencies, unless some restrictions are imposed on the performed constructions. Returning to the beginning of this paper, we should observe that some loops are open, but other closed. ← 22 | 23 → By definition, open loops are consistent, whereas closed ones result in antinomies. Hence, we need a way to open closed loops in order to save our systems from inconsistencies. Tarski’s idea was to introduce language hierarchies and eliminate self-referential sentences. The claim that concepts like satisfaction, truth or denotation should be defined in metalanguages became a by-product of this strategy. Some people consider this device as artificial and propose various alternative methods (see above; see also, for instance, Kripke 1976, Barwise, Etchementy 1987, Barwise, Moss 1996), including strategies employing fixedpoints constructions, although combined with reforms of logic or set theory. I would like to propose another look at opening closed loops, which, at least in my opinion, is a straightforward application of Tarski’s way of thinking.

Let us examine (UT) once again. It says that the diagonalization function and the set of truth are not together definable in a consistent, rich formal system (containing arithmetic). We can say that diagonalization produces a semantic loop, which is closed by any truth-definition satisfying (CT). The way out is simple: resign from defining truth for L in L and pass to ML. In other words, (UT) opens the closed loop in question. Similarly, the incompleteness theorem opens the closed loop produced by the claim that arithmetic is complete (or finitely axiomatizable). Both limitative results save consistency. The difference between the “Tarskian loop” and the “Gödelian” loop consists in the fact that although the latter can be opened in L, the former cannot. Simply speaking, this fact reflects a very deep difference between syntax and semantics. The very amazing result, which formally exhibits this situation (see Murawski 1999, pp. 284–285), states that the concept of truth does not belong to the arithmetical hierarchy, but it belongs to the analytical one, although the notion of provability can be incorporated into the former. Due to the well- known features of natural languages, we cannot apply DG or (FPL) to them, even indirectly. However, we can say that (CT) determines our use of the predicate ‘is true’ in the ordinary speech and every correct truth-definition should imply all instances of the T-scheme. Since it is demonstrable that some well-formed ordinary sentences lead to antinomies, something must be done in order to avoid inconsistency. We claim that truth is undefinable and that the hierarchy of language should be introduced. The advantages of the proposed picture are as follows. Firstly, the strategy is very simple, simpler than its alternatives. Secondly, self-reference becomes de-mystified (see also Smorynski 1985, Chapter 0). Thirdly, self-referential assertions do not need to be treated as ill-formed or ungrammatical.

It remains to touch on some general philosophical points suggested by the foregoing arguments. I deliberately abstain from saying that formal logical con ← 23 | 24 → structions entail (in the strict logical sense) definite philosophical consequences, because, according to my view, the relation between sciences (including logic and mathematics) is very complex and requires much reflection. However, let me speculate on this issue. Assume that we have a language L suitable to express the entire knowledge in it. This means that ML is a part of L. Due to (UT) the set of true sentences of L has no definition in this language. Since no other language is available, the extension of ‘is true in L’ cannot be capture by any L-formula. It appears natural to say that the concept of being corresponds to the set of all truths of L. However, the undefinability of truth implies the undefinability of being. These assertions agree with the theory of transcendentals developed by medieval philosophers (see Woleński 1997). The next philosophical observation refers to the nature of deductive devices used in metamathematics. Although our actual deductive capabilities are finitary, arithmetic can be made complete by adding the ω-rule, which is infinite as far as the matter concerns the cardinality of the set of its premises. Of course, due to our deductive nature, this does not mean that we are able to use the ω-rule effectively, although we can speak about its effects. At the same time, however, (UT) cannot be overcome by supplementing our deductive machinery by infinitary devices. Otherwise speaking, the concept of truth (and being a fortiori) remains undefinable after adding infinite rules of inference. This remark suggests a modesty in speculations about epistemic potentialities of “infinite minds”, very frequently offered by theologians. The same warning concerns theories of everything.

Heller (see Heller 1999) makes some more general suggestions concerning loops (not necessarily semantic) as occurring in physics. This direction seems interesting and deserves a continuation. I would like to offer a small contribution to this line of thinking. The semantic truth-definition (see above) defines ‘is true in L and M.’ These relativizations are frequently considered as artificial and at odds with our intuitions. However, think about L and M as semantic frames of reference, similar to those used in physics, for example, in the special theory of relativity. The claim that truth should be defined without any reference to languages and models is similar to the pretension that time flows absolutely and the same way for arbitrary frames. More importantly, the relativizations used in the semantic analysis of truth do not deprive this concept of the mark of objectivity, just as time is an objective category of physics. In fact, physical loops and semantic loops provided some theoretical constraints. They remain open, although the observers and observables interact in both cases. Metaphorically speaking, defining truth for L in L without additional postulates generates a “semantic speed” that exceeds “the speed of semantic light”. ← 24 | 25 →

II
Logic as Calculus vs. Logic as Universal Medium, and Syntax vs. Semantics

The distinction between logic as calculus (LoC) and logic as language (LoL) was introduced by van Heijenoort (see Van Heijenoort 1967). He regarded this distinction as a very useful device for the history of logic, closely resembling Leibniz’s famous pair of calculus ratiocinator and lingua characteristica. According to van Heijenoort, logic under Frege’s understanding was rather a kind of lingua characteristic, contrary to the view of Boole or Schröder seeing logic as a calculus. Russell and Whitehead followed Frege’s view, making some changes forced by the theory of logical types. A new tradition, more related to Schröder than to Frege, was initiated by Löwenheim in 1915 (see also Badesa 2004), who proved the first model-theoretic theorem (if a formula of first-order logic is valid in a denumerable domain, it is valid in every domain). Generally speaking, the opposition between LoC and LoL was replaced by the axiomatic approach to logic and its set-theoretic account. Skolem, Gödel and Herbrand made essential contributions to this new world of logic. In particular, van Heijenoort finds Herbrand’s work as a possible third way. Van Heijenoort seems to consider LoC and LoL as ideal types. For instance, he stresses that Frege characterized logic not only as a lingua characteristica but also as a calculus ratiocinator. Although van Heijenoort does not give any general characteristics of both types of logic, he points out some of their important features, particularly of Frege’s conception. For Frege, logic is universal and concerned the stable (not changeable) universe. Since logic functions as the universal language, there is no stance outside of it. Hence, no metalogical problems can arise, because it would result in transcending the language limits, which is, by definition, impossible.

Van Heijeenort’s conceptual dichotomy of LoC and LoL was generalized by Hintikka (see Hintikka 1989, Hintikka 1997) and Kusch (see Kusch 1989), who introduced and elaborated the distinction between language as a calculus (LaC) and language as a universal medium (LaM). Kusch (Kusch 1967, pp. 6–7) proposes the following characterization: ← 25 | 26 →

LaC LaM
1. Semantics is accessible.

1.’ Semantics is inaccessible.

2. Different systems of semantic relations are conceivable.

2’. Different systems of semantic are inconceivable.

3. Model theory is accepted.

3’. Model theory is rejected.

4. Semantic Kantianism is rejected.

4’. Semantic Kantianism is adopted.

5. Metalanguage is not legitimate.

5.’ Metalanguage is legitimate.

6. Truth as correspondence is intelligible.

6’. Truth as correspondence is not intelligible.

7. Formalism is linked with the thesis 1.

7’. Formalism is linked with the thesis 1’.

Clearly, since language cannot be reduced to logic, LaC generalizes LoC and LaM provides a generalization of LoL.

The pair <LaC, LaM> enlightens some aspects of the history of logic and philosophy in the 20th century (see Hintikka 1988, Woleński 1997a). Apart from Frege, Wittgenstein was perhaps the purest representative of LaM. The statements 5.6 (5. 6: Die Grenzen meiner Sprache beduten dir Grenzen meiner Welt – The limits of my language mean the limit of my world) and a part of 5.61 (Die Logik erfüllt die Welt; die Grenzen der Wlt sind auch ihre Welt – Logic fills the world: the limits of the world are also its limits) in Wittgenstein 1922 are probably the most extreme expressions of semantic Kantianism. Works by Husserl, Carnap and Tarski, in turn, can be interpreted as representative examples of the LaC conception. In general, Hintikka identifies LaC with the modeltheoretic tradition in logic. Although I agree that both distinctions in question considerably help us to gain a better understanding of various crucial episodes of contemporary philosophy – both analytic and continental – I claim that they are less important than their proponents think. I shall restrict my remarks to the philosophy of logic, because any attempt to investigate the role of language in the entire philosophy (or even in its part) exceeds the scope of a single paper. The main device I apply is the distinction between syntax and semantics which I believe to be the most important tool for any correct account of the nature of logic. More specifically, I will argue that logic, including metalogic (metamathematics), is both a calculus and a language. Rephrasing Kant’s famous dictum, syntax without semantics is empty, but semantics without syntax is blind. Or, using van Heijenoort’s distinction between logic as a calculus and logic as a language, we can say that the former is empty without the latter and LoL is blind without LoC. Yet, this colorful framing of the problem does not entail semantic Kantianism. ← 26 | 27 →

It is intriguing that van Heijenoort mentions Hilbert only occasionally (Van Heijenoort, p. 15, footnote 6):

The word ‘again’ suggests that Hilbert followed someone else in bringing the concept of formal system into the forefront. In fact, van Heijenoort to some extent considers Hilbert as a follower of the axiomatic approach to logic initiated by Frege and Russell. Hintikka ascribes much more importance to Hilbert than van Heijenoort does (see Hintikka 1988, pp. 108–110) and says that the founding father of formalism in the foundational studies mastered the axiomatic approach to logic and mathematics. According to Hintikka, Hilbert’s axiomatics was an essential contribution to the model-theoretic tradition in logic. However, although, pace Hintikka, Hilbert was very far from representing a “blind” formalist, i.e. a philosopher of mathematics neglecting the content of formal systems, his main merit for the foundations of mathematics consists in initiating the systematic metamathematics, that is, mathematical investigations of formal systems. Thus, the problems of consistency and completeness became mathematical problems due to Hilbert’s foundational program (see Sieg 2000 for a survey of Hilbert’s way toward finitary proof theory, which is the essence of metamathematics).1 It is legitimate to consider the finitistic approach to formal systems from the perspective of the finitist proof theory as syntactics. Thus, for Hilbert a formal system is a purely syntactic object entirely defined by its vocabulary (a stock of symbols) and the recursive rules which prescribe how to construct complex strings of symbols from the initial atoms. Such a picture of formal systems perfectly fits the idea of LoC and LaC.

Even if we agree that Hilbert’s main contribution belongs to the formalist tradition in the foundations of mathematics, we should remember that he used semantic concepts, like domain or satisfaction, in an informal way. Since I identify the model-theoretic tradition with the semantic tradition, the rise of formal semantics is a crucial fact for this perspective. There are currently two uses of the term ‘formal semantics’ (I follow Woleński 2004a). The first, becoming more and more popular among linguists, refers to a formal (mathematical) approach to natural languages (see Cann 1983, pp. 1–2) and their relation to the world. ← 27 | 28 → Formal semantics in this meaning accepts the following theses: (a) Chomsky’s thesis that natural language is a formal system; (b) Montague’s thesis that there is no principal difference between formal and natural languages. However, the real scope of the validity of (a) and (b) is still disputed. Hence, it is not clear to what extent speaking about the formal semantics of natural languages is justified. Since I am not going to discuss this question, let me now pass to the second meaning of formal semantics, according to which it consists in a mathematical approach to formal languages and their semantic features, such as truth, reference, etc. Formal semantics in this sense is the same as model theory. Roughly speaking, to give a semantic characteristics of a language L means to point out the class C(M) of models of L, i.e. the class of algebraic structures of L such that the sentences of L are true in every M belonging to C(M). Thus, we have L as a formal language and C(M) as a formal set-theoretical object. Since both are mathematically characterized, everything is formal in formal semantics of formal languages.

However, the outlined picture is oversimplified. The following opposites applied to languages are relevant to our problem: (A) natural – artificial; (B) informal – formal; (C) unformalized – formalized; (D) interpreted – uninterpreted. At first glance it seems that members of the sequence <natural, informal, unformalized, interpreted> express the same property, which can also be attributed to ‘ordinary’, ‘colloquial’, etc. Consequently, the words ‘artificial’, ‘formal’, ‘formalized’ and ‘uninterptreted’ appear to indicate the same feature, contrasted with the property expressed by such terms as ‘natural’, ‘informal’, etc. A closer inspection shows that these extensional identifications are dubious. If we say that a language is artificial, we mean that it was created to perform some specific tasks. Although an artificial language contain special symbols, it does not need to be formalized. According to a basic intuition, a language is formal if it can be described independently of the content of its expressions and appeals only to their syntactic form. Finally, a formalized language is a result of a special process or operation, called formalization. Nothing precludes that a formalized language arose as a formalization of an informal one. For instance, formal mathematics arises a a result of formalization of more or less informal mathematics. Furthermore, nothing precludes a formalized language from being interpreted. This was strongly stressed by Tarski (see Tarski 1956, pp. 166–167; the work was published in Polish already in 1933):

For Tarski, doing formal semantics for L requires that the language in question is interpreted, although formalized. Tarski himself offers a very general description of formalized languages as those in which the sense (meaning) of every expression is “unambiguously determined by its form” (Tarski 1956, pp. 165–166). On the one hand, this account certainly leaves important questions open. In particular, one can ask: what is the sense? On the other hand, he rather relied on carefully listed features of formalized languages; for instance, the fact that all constructions in such languages are performed according to structural properties and rules.

Perhaps, it is the most important outcome of the foregoing discussion is the conclusion that such features as formalized and interpreted can coexist. This raises the question of how the interpretation of a formalized language L (for simplicity, I assume that L is a first-order language without functions, symbols and the identity predicate) proceeds. The first step to describe L is to specify its alphabet. Without going into details, the alphabet of L contains logical constants (propositional connectives, quantifiers), individual variables and constants, and predicate letters. The set of formulas of L includes atomic items of the type P(t1, …, tn), where P is a predicate letter and t1, …, tn are terms (individual variables or individual constants) and compound formulas built from connectives and quantificational closures, for instance ¬Px or ∃xPx. In order to speak about semantic relations, we have to define an interpretation of L, i.e. to specify an interpretation function v. The first step consists in choosing the universe U fixing the set of values for individual variables and the scope of quantifiers. The next step is to define the function v (the valuation function) which ascribes determined elements of U to individual constants and denotations (subsets of U or relations defined within U; I will call them attributes expressible in L) of predicate-letters. Models of L are those interpretations in which sentences of the language in question are true.

It is not true that a purely formal language (at least, as defined here) does not yield any information about its possible models. Every language (more precisely, its alphabet) has its signature generated by the arity of its individual constants and predicates. Generally speaking, the arity indicates the number of ← 29 | 30 → arguments. By definition, the arity of constants is equal to 0, the arity of monadic predicates is equal to 1 and the arity of dyadic predicates is equal to 2. Since the signature of the alphabet is unique, it also determines some structural properties of interpretation. If the signature of L is <0; 1, 2>, we know in advance that any interpretation of L must contain at least one distinguished object (since v is not a bijection, more than one name can refer to the same object) and two attributes, one monadic and one dyadic. However, connections between languages and their possible interpretative structures generated by the signatures do not yield interpretations. Although the function v interprets the alphabet, it suffices to interpret the whole L, provided that L is semantically compositional. The compositionality means here that the semantic value of a complex expression is the value of a function, denoted as v, of the semantic values of their explicit components. This is the case if composite expressions are formed by applying syntactic rules to already constructed items. One could maintain that the requirement of compositionality is too strong, as such a claim creates difficulties in mathematical analysis of intensional contexts, which in many cases are not compositional. This is true and also explains notorious difficulties encountered in formal analyses of modal propositions or epistemic reports. In fact, compositionality displays an essential feature of formal semantics. Languages and interpretations should be handled by mathematical devices operating according to recursive procedures. It requires that both correlate entities, i.e. languages and their interpretations, should have parallel properties. Additionally, compositionality allows the use of recursive and inductive procedures in syntax and semantics.

The introduced apparatus is not sufficient for interpreting sentences. It is an important point, because languages are set of sentences. We can interpret sentences in at least two different ways. Firstly, we can supplement a model M of L with two (I adopt the principle of bivalence) new distinct objects, namely 1 (truth) and 0 (falsehood) and assume that if a sentence A belongs to L, then v(A) = 1 or v(A) = 0. Truth is not defined by this construction, but taken as a new primitive concept. This is Frege’s way out, consistent with his view that logic is a language. Secondly, following Tarski, we can define the concept of truth in a well-known way, that is, as satisfaction of a sentence by all infinite sequences of objects; 1 and 0 can be thus dropped. An important difference between these two approaches, Fregean and Tarskian, consists in the fact that the former does not use a metalanguage while for the latter it is essential. Moreover, as Tarski explicitly showed on the occasion of forming the semantic truth-definition, the metalanguage cannot be reduced to the object language as far as the issue con ← 30 | 31 → cerns semantic considerations. Consequently, if L is a language, and ML plays the role of its metalanguage, we have LML, but not (L = ML).

What is formal in performing interpretations of formalized languages? Of course, the answer depends on how the term ‘formal’ is understood. Suppose that something is formal if it is independent from content and solely dependent on form. This old meaning is fairly useful, but not very precise. As far as the matter concerns L, it is all simple, because the structure and signature of L are given in a purely syntactical way. The meaning (sense) of expressions of L is irrelevant here. Since we assume the signature parallelism between L and its models, the formal structure of the latter is displayed by the signature. Let us say that, for a given language L, its syntax and its signature display the form of interpretation given by the function v. On the other hand, v itself is not exclusively formal, because it depends on how expressions of L are understood. Let us take a language L with the alphabet {a1, a2, P1, P2.} Assume that: U is a set of philosophers, a1 = ‘Plato’ a2 = ‘Aristotle’, P1 = ‘a rationalist’, P2 = ‘is older than’. The interpretation which is faithful to the history of philosophy correlates Plato with ‘Plato’, ‘Aristotle’ with Aristotle. Thus, the sentence ‘Plato is a rationalist’ is true, the sentence ‘Aristotle is a rationalist’ is false, the sentence ‘Plato is older than Aristotle’ is true and the sentence ‘Aristotle is older than Plato’ is false. Clearly, we can consider other interpretations in which things look differently, for example, possible worlds in which Plato was younger than Aristotle and Aristotle was a rationalist. However, regardless of whether or not our interpretations are faithful to empirical data and ordinary language, the function v behaves accordingly to some contents. Thus, we can say that the content of an interpretation is given through correlating L with some objects by the function v. Even if we say that this work is formal to some extent, because it is described by mathematical devices, like functions, sets, relations, etc., the meaning of the term ‘formal’ is different here. Anyway, the content of interpretation is obtained informally to some extent, contrary to the form of interpretation, which is defined inside syntax.

A general framework for the relation between the form and content of an interpretation can be outlined by appealing to one point in Tarski’s construction of the truth-definition. It concerns the T-scheme, i.e. the equivalence S is true if and only if S*, where the letter S denotes a name of the sentence in question and the symbol S* denotes a metalinguistic translation of the sentence denoted by ‘S’. If the sentence denoted by ‘S’ belong to L, the appropriate specification of (T) belongs to ML. The simple rule is as follows: if L is formal (formalized), ML is not. There is no other way to establish interpretation than to use more or ← 31 | 32 → less informal metalanguage. Thus, the content of interpretation goes from the top to the bottom, that is from ML to L. Beth (Beth 1962, p. XIV) once made the following remark:

This means that (a) hermeneutics consists in any decision how to interpret L, and (b) it is impossible to do formal semantics without a certain amount of hermeneutics or, equivalently, formal semantics is always embedded into some hermeneutics. Using the distinctions mentioned at the beginning of this paper, LoC and LaC are not sufficient for doing semantics of formalized. Even if we formalize ML in order to give a pure account of semantics of L, we must use an informal MML. This resembles the relation between formal and informal mathematics. We need the latter in order to develop the former.

Hintikka (see Hintikka 1988) is right to observe that the model-theoretic tradition considers calculi as re-interpretable systems, but it must be noted that every interpretation or re-interpretation requires a sufficiently powerful and partially informal metasystem. Needless to say, although logic as a calculus must be supplemented with logic as a language, the latter is the universal medium in Frege’s or Wittgenstein’s sense, at least if we assume the Tarskian way of doing semantics. Regardless of whether we work with the hierarchy of languages L1, L2 (= ML1), L3 (= ML2), … (where Ln = MLn-1) or claim that since everything can be captured by first-order languages, we need such languages and their extensions to higher-order ones, both hierarchies are open, provided that the last achieved level is partly informal. In light of metalogic, the Frege-Wittgenstein view that there are ultimate or universal borders of logic is mistaken. Although it is true, pace Wittgenstein, that logic itself does not exclude any possibility, this statement trivially follows from the metalogical theorem that pure logic does not distinguish any extralogical content. On the other hand, we can favour or exclude any possibility by performing a suitable interpretation in ML. Van Heijenoort says (Van Heijenoort 1967, p. 13) that Frege excludes any change of the universe. This is true, because the logical universe for Frege consists of the True and the False. It cannot be reduced to a single element due to a danger of a contradiction and cannot be expanded. On the other hand, the logical universe (the set of logical values) and the model-theoretic universe (the set of objects admissible as values of variables) are completely distinct objects and should be not confused. The present-day philosophy of logic can accommodate Frege’s picture of the logical world without any difficulty. However, it has to reject his ← 32 | 33 → view concerning the universe of all possible objects. Finally, the universality of logic means that logical laws are universally valid, but it does not mean that logic sets the limits of extralogical discourse.

The distinction between LoC and LoL, which turned out to be useful in the history of logic, if it is taken as absolutely sharp, obscures the present logical scene. In particular, the limitative results of Gödel, Tarski and Church demonstrate limitations of syntax (calculus) and firmly support the view that semantics is prior to syntax or, in other words, that language is just prior to calculus. The very relation between syntax and semantics has some general consequences, which are very significant for the philosophy of logic and which cannot be expressed via the distinctions LoC/LoL and LaC/LaL. Let us suppose that one considers logic as the sum of syntax and semantics. It is clear that this sum is not homogeneous. Syntax or calculus are given effectively, which means that there is an effective procedure (algorithm) deciding in a finite number of steps whether a given syntactic construction is correct or not. Due to the parallelism of syntax and semantics effectivity also holds a property of the latter, but only partially. Some semantic properties, at least in classical logic, require infinitary proof methods. For example, the completeness theorem for first-order logic is not provable by finitary methods. Hence, the model-theoretic proofs of consistency of rich formal theories go beyond devices admissible for constructivists. Even if one formalizes semantics completely, this problem will automatically reoccur on the level of metasemantics. Suppose that one wants to justify a system of logic conceived as calculus, i.e. a deductive machinery by appealing to semantic properties. It is usually done by proving that some given logic is sound (provided that the premises are true, its rules lead to true conclusions) and complete (every tautology is provable). However, this justification must use methods going beyond the part of logic which is justified. Thus, logic as a syntactic and thereby effective construction, that is, formed according to recursive procedures, is justified by less effective constructions. At the same time, it is difficult to imagine a purely syntactic way of proving that some logic is sound and complete. It would be a paradox if logic was understood only as a calculus and such a proof would be impossible for logic conceived as the universal medium. This is the main reason why the distinction between syntax and semantics is more important than other distinctions indicated in the title of this paper. Finally, the Kantian metaphor that syntax without semantics is empty, but semantics without syntax is blind, means that the preciseness of calculus sharpens the semantic eye, although semantics brings the content into correct formulas. ← 33 | 34 → ← 34 | 35 →


1       Emil Post’s proof of the completeness of propositional calculus in 1920 (see Post 1921) is a phenomenon deserving a separate historical explanation. All references in Post’s paper indicate that he was exclusively inspired by Principia Mathematica by Russell and Whitehead; Hilbert is not mentioned by Post.

III
Do We Need to Reform the Old T-Scheme?

Read 2010 almost verbatim reproduces Read 2008. I was invited to make comments about the later paper (see Woleński 2008). In what follows I shall partly repeat my earlier critical comments about Read’s interpretation of the T-scheme, while also adding new remarks and improving the old ones; some of the additions are inspired by other contributors participating in Rahman, Tulenheimo, Genot 2008, including Stephen Read’s replies in this volume (see Read 2008a) as well as the previous discussion in Discusiones Filosóficas (see Miller 2010, Read 2011, Sandu 2010).

At first, let me remind Leśniewski-Tarski’s diagnosis of the Liar paradox. They pointed out (see Tarski 1944) that the derivation of the paradox uses: (I) self-referential sentences asserting semantic properties; (II) the T-scheme, and (III) classical, that is, bivalent logic. Hence, we can conceive three strategies in order to solve the paradox: (i) to exclude self-referentiality; (ii) to reject or modify the T-scheme; (iii) to change logic. It would be mistaken to maintain that there is a solution free of costs or some artificialities. It concerns Tarski who choose (i), Kripke (and many other logicians) who opted for (ii) (Sandu makes several remarks about this way out), and Read who tries to reform the T-scheme. More precisely, Read argues that the old (Tarskian) T-scheme is inaccurate and proposes its modification. I would like to show that Read’s reading of Tarski is incorrect and that his (Read’s) leads to some difficulties.

Read says that, according to Tarski, every instance of the T-scheme, that is the formula

   (T) x is true if and only if p,

is true; Read even says that his understanding of (T) is “an unquestioned orthodox.” However, Tarski focused on the provability of T-sentences from his truth-definition, but not on their truth. In Read’s reply to my criticism (see Read 2008a, p. 218), he agrees with my standpoint, but he adds “But he [Tarski] clearly imposed that requirement because he thought those instances were true”. I should note that Read uses the proper formulation in another place of his main paper (see Read 2010, p. 122). He probably thinks that both formulations are equivalent. However, they are not, unless we assume that the metatheory of truth-theory is ω-complete. Anyway, Read’s statement about “an unquestioned orthodoxy” is certainly incorrect. For example, Miller and Sandu in their con ← 35 | 36 → tributions published in Discusiones Filosóficas formulate (T) via provability, not truth.

I see no place in Tarski’s writings which could justify the view that he “clearly imposed”, etc. The observation that provable instances of (T) are true is trivial and has no relevance for Tarski’s proposal of how to solve the Liar paradox. Nonetheless, the provability of T-sentences matters very much and the role of this fact has the best illustration in the problem of the definability of the T-predicate. Combining the fixed-point theorem and the Tarski undefinability theorem, leads to the unprovability of some instances of (T). This fact concerns the Liar sentence, independently of its formulation, as ‘this sentence is false’ or ‘this sentence is not true’, both recorded via arithmetization. The situation is clear in formal arithmetic of natural numbers, but the results that hold for arithmetic cannot be directly applied to natural language. However, pace Tarski, excluding the Liar sentence from the stock of permitted formulas of natural language plays a quite similar role to showing that some formalized instances of (T) are not provable. In fact, this is an analogical move as the exclusion of division by 0 in arithmetic. Although the formula m/n = 0 is grammatically correct and perfectly understandable, it must be rejected as producing inconsistency. As far as I know, Tarski never said that the Liar sentence is nonsensical, meaningless, etc. He only recommended that so-called closed languages, that is, languages containing a semantic concept used self-referentially, should be avoided. This restriction is by no means that counter-examples to (T) are excluded by fiat, as Read suggests (see Read 2010, p. 127). On the contrary, they are avoided by a subtle and elegant reasoning.

That T-sentences are assumed to be provable is important for their status. Read says (see Read 2010, p. 125) that according to Tarski (T) is “a merely a material equivalence.” Consequently, the instances of (T) have the same status. Although Tarski himself was not quite explicit about this issue, it is rather obvious that provable theorems of the form AB are something more than material equivalences. In particular, one cannot replace A or B by their material equivalences. For instance, the equivalence: the sentence ‘Stephen Read wrote a paper on T-scheme’ is true if and only if Stephen Read wrote a paper about T-scheme’ cannot be replaced by the equivalence: ‘Stephen Read wrote a paper about T-scheme if and only if Jan Woleński commented on Read’s paper’, although both equivalences are true, and thus materially equivalent, and, moreover, both consist of true constituents. I do not suggest that Read considers such a replacement as possible or justified. Yet, I claim that the difference between merely material equivalences and provable material equivalences is important, particularly for a proper interpretation of Tarski’s truth-theory. ← 36 | 37 →

Read entirely neglects some properties of languages for which the semantic concept of truth is defined. Firstly, such a language L is formalized or at least has a specified structure; this latter concept was introduced in Tarski 1944. Without entering into details, L must be well-described as a syntactic object. Firstly, we should know what belongs to the vocabulary of L and how the class of its sentences is defined. Secondly, and more importantly, L is an interpreted language. Let me quote the following words of Tarski (Tarski 1933, p. 166/167):

Tarski did not explain what he understood by ‘meaning’. However, it is clear that meanings of words, ordinary or artificial, for instance, the convention that ‘black’ expresses the property of being white, dictate semantic interpretations. Speaking more precisely, and using the model-theoretic terminology, meanings generate so-called interpretation functions which ascribe denotations of individual terms and predicates in given models. Although Tarski was skeptical as far as the matter concerns consistent semantic constructions for natural language, the difference between it and the interpreted, formalized or structurally specified L is less radical than it is frequently assumed by interpreters and critics of Tarski.

Taking into account certain facts about L, one can easily demonstrate that all of the counterexamples given by Read fail. This concerns the sentences:

The argument is that the words ‘I’, ‘that’ and ‘any’ can have different meanings on the left and right sides of the respective equivalences. However, this cannot happen by definition in the case of properly interpreted languages. In particular, such languages do not require a special principle of uniformity (see Dutilh Novaes 2008), blocking ambiguities and other defects of expressions, because they are automatically excluded by technology of the semantic interpretation. ← 37 | 38 → The valuation is a function and ‘I’, ‘that’ and ‘any’ have necessarily the same interpretations in all their occurrences in (1)–(3). This is particularly clear for indexicals, like ‘I’ and ‘that’. An ostensive specification of references (for example, supplementing the act of using ‘I’ or ‘that’ by a gesture) fixes the ascribed objects. On the other hand, if the references of such words are not made precise, they function as variables and the problem of truth (falsehood) of (1)–(3) does not arise at all. I do not argue that ambiguities do not occur in ordinary language, but only point out that the semantic definition of truth assumes that L for which it works is formalized (or has a specified structure) and its interpretation is fixed, even when L is selected as a suitable (not closed!) part of colloquial parlance. Read proposes to replace (T) by

This scheme codes an intuition expressed by

Since what a sentence says is covered by all its implications, (A) formalizes (S), provided that the symbol ⇔ denotes strict equivalence. Read says (Read 2010, p. 125) that “(A) is a logical equivalence” (expressed by the symbol ⇔) contrary to (T) as a material equivalence. However, Read’s qualification of (A) as a the logical equivalence is vague. If we say that A is a logical sentence, we can mean various things. Firstly, (a) A is logical if and only if it is coded by signs belonging to the language of logic, but, secondly, (b) A is logical if and only if it is a logical theorem. Now, (T) and (A) are logical equivalences in the same sense, if the first understanding is assumed, but they both are not logical theorems. Thus, neither (a) nor (b) qualifies (A) as a logical equivalence, contrary to (T). Perhaps, Read intends to say that (A) is a logical equivalence, because strict connectives are logically stronger than material ones. However, due to my previous remarks about the status of the instances of (T), provable equivalences are “something more” than material equivalences. In fact, the difference between the formulas X image AB and AB seems secondary, although, at least in my view, the former is much clearer than the latter.

Read points out that (A) is intensional for ‘says that’, but he does not see any problem with it. In particular, he seems to think that everything is solved by the closure of x:p by “allowing substitution only of logical equivalents” (Read 2010, p. 124). Unfortunately, the matter is not so simple, because ‘says that’ is strongly intensional. Consider two equivalent formulations of a mathematical axiom, let say, \the parallel postulate. Denote them by F and F’, respectively. They are ← 38 | 39 → logically equivalent. Assume that we have a person O who does not know that F and F’ are provably equivalent. Thus, the formulas x:F and x:F’ do not say the same for O, although Read claims that they do. In fact, ‘says that’ forms contexts are not, contrary to Read, closed in being replaceable by their logical equivalents. Defining ‘says that’ as satisfying such a kind of closure is, in my opinion, at odds with the ordinary meaning of this operator.

There are other implausible consequences of (A). Consider (I do not assume that Peano arithmetic is first-order):

Peano axioms are true in both standard and non-standard models of arithmetic. Let N be the standard model and N’ a nonstandard one. Take the sentence (*) ‘all natural numbers have finitely many predecessors’, which is true in N, but false in N’. Intuitively speaking, Peano axioms say (*) in N, but its negation in N’. Thus, ‘says that’ requires a relativisation to a model, but (T) without it has false instances. This also shows that defining the context x: p by the consequences of p may be insufficient in some cases, because a reference to models is required.

There is also a problem with (A) as applied to falsehoods (this is also pointed out by Miller and Sandu). The definition of F (‘is false’) corresponding to (A’) can be recorded as

As an example, we have

However, (5) has its instantiation in

Now assume that someone knows that a person O knows that the sentence ‘the greatest Polish city is the capital of France’ is false and that the right side of (6) is a correct (true) instantiation of the right side of (5). Hence, he or she knows that the sentence ‘Warsaw is the capital of France’ is also false; we use here the principle ‘if an instantiation of A is false, A is false too’. However, to justify that, one must assume that ‘Warsaw’ and ‘the greatest Polish city’ are co-denotative (I neglect that the latter is a description). This consideration shows that we do not need to worry whether the sentences ‘Warsaw is the capital of France’ and ‘the greatest Polish city is the capital of France’ say the same or, eventually, in which circumstances they cover the same content, because it is sufficient to known the ← 39 | 40 → values of nominal expressions. Thus, even if we agree that that logically equivalent sentences say the same thing, this observation does not close the issue, because it can happen, as in the case of (5) and (6), that sentences are equivalent in theories or some language systems modulo denotative conventions, although they are not equivalent on purely logical grounds. Thus, the interpretation of a language has a crucial importance for establishing what sentences say and when they are true or false.

Further, the negation of (x: p) can be interpreted either as ¬(x: p) or as (x: ¬p) (the latter is stronger than the former). It matters in the case of negative sentences because

looks more plausible than

Perhaps the most important critical observation concerns metalogical properties of T. The law of the excluded middle can be stated as

which is an instance of

However, the relation of ¬TA and T¬A is not clear under Read’s definition. In general, we have

and in the classical bivalent (two-valued) case the equivalences

hold. Now T¬A has its interpretation (according to Read’s definition) in

However, (13) does not imply

although we have (or should have)

I do not claim that (12) is indispensable, but only note that important relations are unclear in light of Read’s proposals. In particular, the functor of negation commutes with truth (in its semantic understanding), but not with ‘says that’.

Returning to ‘x says that p’, a full analysis of this phrase seems to be much more complicated than Reads maintains. The general problem is that sentences can say quite different things for different persons or even for the same person depending on various pragmatic factors. For instance, we should distinguish direct (explicit) and indirect (implicit) ‘saying that’. Every sentence entails infinitely many logical consequences. Even if we assume that the direct linguistic content of a sentence A is definable (it is a very optimistic presumption), the implicit content is much more vague and the amount of its grasping, always partial, cannot be accounted in advance. Reads seems to consider the intensionality of (A) as its advantage, but it is a very dubious view. I am inclined to think that the intensionality of (A) prevents a satisfactory definition of saying wholly as things are. This is very important, because the real virtue of the scheme introduced by Read essentially depends on such a definition.

(T) is essentially different from (A), because the former is purely extensional. In spite of this difference, I will argue that both schemes record almost the same intuitions. It is interesting that Tarski’s starting point was similar to that proposed by Read. The initial intuition of the semantic definition of truth was presented by the following formulation (see Tarski 1933/1956, p. 155; Tarski followed Tadeusz Kotarbiński):

The original Polish formulation is much closer to (A), because it runs (in English translation): “a true sentence is a sentence which expresses that things are so and so, and things indeed are so and so”. Tarski explicitly formalized (*) by T, presumably also for eliminating ‘expresses’ as an intensional factor. That L is interpreted and that its expressions have “intelligible meanings” illuminate the issue at stake. (T) can be supplemented by assertions of the type ‘x says that p’, ‘x means that p’, ‘x asserts that p’, etc. All are external with respect to the instance of (T) and serve to fix, define, explain, etc. the valuation function connecting L-expressions with their denotations. However, ‘x:p’, although internally embedded into (A), plays exactly the same role, provided that semantics a la Tarski is associated with this scheme. However, Read explicitly says (see Read 2010, p. 125) that this is just the case. In my opinion, (T) has similar intuitive advantages as (A), including the correspondence platitude, that is, the ability to express the basic content of adequatio rei et intellectus. ← 41 | 42 →

At the same time, the Tarski scheme avoids all problems caused by the internal intensionality of (A), in particular, the treatment of ‘is false’ (remember Russell’s requirement that any satisfactory theory of truth should be also a theory of falsehood). In order to explain this point, let me refer to Miller 2010, p. 223). He attributes to Tarski the following scheme (I am using the notation of the present paper):

The first conjunct in the right side of (T’) indicates that the sentence named by x belongs to L. Miller define ‘is false’ by

Since Tx and Fx are mutual negations in classical logic, the same relation must hold between the right sides of (T’) and (F’). However, (xL) ∧ p and (xL) ∧ ¬p do not negate each other. The negation of the former formula yields the sentence ¬(xL) ∨ ¬p. Assume that this disjunction is true in favour of ¬(xL). Now, since x does not belong to L, it is not a sentence. Consequently, it is neither true nor false, because only sentences of L can be true or false. Clearly, Fx and ¬Tx are not equivalent under (T’) and (F’). This example additionally shows difficulties of defining truth by conjunctions of conditions. It is much better to treat clauses, like xL, as external with respect to (T).

Read claims that (A) is better, because it correctly solves the Liar paradox. According to Read, (T) is false, because it has a false instance (the letter l refers to the Liar sentence):

Still, (A) produces a correct truth-condition of l in the form:

However. this success is entirely apparent. The formula (#) does not give a complete analysis of the Liar sentence. First of all, (#), as inconsistent, is not provable in the semantic theory of truth. Since (##) is true for Read, the formula Tl is either false or inconsistent. Yet, both these cases must be excluded. A simple transformation of (##) gives:

which is not satisfactory, because it says that a tautology is not true, that is, inconsistent. In fact, adding ¬Tl (this is an unprovable formula!) to the stock of theorems of propositional logic immediately abolishes the Post (absolute) consistency of this logic. The actual situation of l is displayed by the formula ← 42 | 43 →

Roughly speaking, every semantic assumption about l, that is, every assumption concerning its truth-value modulo the bivalence, entails a contradiction. It clearly shows, firstly, why some instances of (T) are excluded as unprovable and, secondly, that (##) does not formulate any truth-condition. This concurs with Sandu’s remark that (see Sandu 2010, p. 289) it is not clear what the Liar sentence says, although I am inclined to a more radical conclusion, namely that l says nothing. To sum up, if (A) is reduced to (T), the former solves the Liar paradox in the same way as the latter does, but if is not reduced, the issue is still open, because the proponent of (A) must decide whether this schema has true, but unprovable instances.

Finally, I would like to note that the problem of a philosophical significance of the T-scheme is still open. Does it code the correspondence intuition or not? Should we modify (T) in order to express the correspondence platitude? I do not address these and similar questions in this paper. I hope to give an account of the philosophical significance, if any, of Tarski’s theory of truth in my Woleński (in preparation. ← 43 | 44 → ← 44 | 45 →

IV
Truth is Eternal if and only if It is Sempiternal

The problem addressed in this paper goes back to Aristotle and his considerations about tomorrow’s sea battle. In a famous passage in De Interpretatione (19a 25–30; after The Works of Aristotle, vol. 1: Categoriae and De Intepretatione, tr. by E. M. Edghill, Oxford University Press, Oxford 1928), the Stagirite says:

   Everything must be either be or not be, whether in the present or in the future, but it is not always possible to distinguish and state determinately which of these alternatives must necessarily come about.

   Let me illustrate. A sea-fight must take place to-morrow or not, but it is not necessary that it either should not take place to-morrow, neither it is necessary that it should not take place, yet it is necessary that it either should or should not take place to-morrow. Since propositions correspond with facts, it is evident that when in future events there is a real alternative, and a potentiality in contrary directions, the corresponding affirmation and denial have the same character.

These words initiated a considerable discussion about the relation between truth and time. Is truth relative and dependent on temporal coordinates, or is it absolute and timeless? The debate concerns several problems, in particular, the validity of some logical principles, fatalism, God’s omniscience, free-will and determinism (see Bernstein 1992, Cahn 1967, Gaskin 1995, Hintikka 1977, Lucas 1989, Prior 1953, Vuellimin 1996). This paper concentrates almost entirely on logical issues.

Interpretations of Aristotle’s quoted passage essentially vary. Did he revise logic or not? Literally understood, his text defends the principle of excluded middle (EM for brevity), but it also tells us something about the logical value of future contingents, that is, sentences about accidental events which will happen or not. Should we distinguish the predicates ‘true’ and ‘true at a time-moment t’? This problem was also noted by the Stoics. Their celebrated Master Argument tried to prove that everything that is possible will happen in the closer or further future. Thus, EM is valid and truth has no temporal coordinates. Jan Łukasiewicz (see Łukasiewicz 1930) interpreted the Stoics as defending the principle of bivalence (BI). According to him, we should distinguish the logical and metalogical versions of EM (MEM). Whereas the former is a logical theorem coded by the formula A ∨ ¬A (or its another notational variant), the latter says that every sentence is either true or false. BI conjoins MEM with ← 45 | 46 → the metalogical law of non-contradiction (MNC), that is, the assertion that no sentence is both true and false. Łukasiewicz interpreted Aristotle as suggesting that future contingents are neither true nor false. Thus, in light of Łukasiewicz’s interpretation, Aristotle limited BI.

Łukasiewicz’s own solution followed his reading of the Stagirite. He rejected BI as universally valid and envisaged three-valued logic (later, many-valued, countable or even uncountable, systems). According to him (I restrict the discussion to three-valued logic, that is, Ł3), if A is a sentence about a future contingent event, A possesses the third logical value; let us call it neutrum (Łukasiewicz’s first idea consisted in considering neutrum with possibility). EM is not a logical theorem. Numerically speaking, if A possesses neutrum as its logical value, v(A) = ½. Furthermore, v(A) = ½ ⇔ vA) = ½. The valuation for disjunction in Ł3 establishes that if v(A) = v(B) = ½, v(AB) = ½ as well. Hence, EM is not a theorem of Ł3, because A is a theorem of this logic if and only if v(A) = 1, for any valuations of components of A. The logical law of non-contradiction, that is, the formula ¬(A ∧ ¬A) (NC) shares the fate of EM in Ł3, because if v(A) = v(B) = ½, v(AB) = ½. Yet Ł3, as any other logical system, has to satisfy the condition of consistency. Thus, MNC holds in the metatheory of Ł3. This is a very strong reason to distinguish NC and NMC. However, Łukasiewicz still considered truth as absolute. According to him (I restrict the discussion to three-valued logic, that is, Ł3), if A is a sentence about a future contingent event, A possesses the third logical value; let call it neutrum (Łukasiewicz’s first idea consisted in considering neutrum with possibility). It will true or false in the future. Truth and falsity are eternal, but not sempiternal. This means that truth and falsity cannot change in their opposite or other logical values. On the other hand, sentences valued by the neutrum can become true or false; Łukasiewicz did not consider such changes, deeming logical values as necessary. This means that, at least for Łukasiewicz, future contingents can become true or false, but it is not necessary. Łukasiewicz’s view has important and deep consequences for understanding the absoluteness of truth. His position was moderate, because eternality suffices for absoluteness.

The equivalence ‘truth is absolute if and only if truth is eternal’ can be called the weak truth-absoluteness thesis (WTAT). The strong truth-absoluteness thesis (STAT) says that truth is absolute if and only if truth is eternal and sempiternal. STAT can be stated as

This statement has two parts:

Consider a concrete example, namely the sentence (*) I will go to Warsaw on April 26, 2012; the date of writing this sentence is April 26, 2011. Assume that I will stay in Warsaw on April 2012. Thus, (*) is true on t = April 26, 2012. Consequently, we have:

Clearly, if A is valued by the neutrum, being an indefinite sentence, v(A) = ½), (1a) holds, but (1b) loses its valour.

STAT was defended in Poland by Kazimierz Twardowski (see Twardowski 1902) and Stanisław Leśniewski (see Leśniewski 1912) even before Łukasiewicz envisaged many-valued logic. Twardowski used philosophical (epistemological) arguments. He criticized all forms of truth-relativism, not only the view that truth is essentially determined by temporal coordinates (see Kokoszyńska 1948, Kokoszyńska 1951 for a modern shaping of Twardowski’s arguments). As far as the matter concerns logic, Twardowski observed that the relativity of truth is at odds with the principle non-contradiction (he did not distinguish its logical and metalogical formulations; the same concerns Leśniewski in his early writings). Leśniewski offered a proof that, assuming the law of non-contradiction, ET and ST are equivalent, that is, STAT holds. His argument is as follows. Assume that a sentence (**) ‘S is P’ is true, but not sempiternally true. Thus, there is a time t in which (**) is not true, although it is true at some t1t. This assertion implies that (***) ‘S is not P’ is true at t. Leśniewski uses the principle of non-contradiction at this point and concludes that since (***) is always false, it is also not true at t. Thereby, (**) must be true at t, contrary to our initial assumption. However, this produces inconsistency. Leśniewski’s proof that eternality entails sempiternality proceeds analogously.

Leśniewski’s reasoning is informal. One of its steps can be questioned, namely the one in which Leśniewski assumes that falsehood is sempiternal. One might observe that if we assume that the property of ‘being false’ obtains sempiternally, the same concerns truth automatically. Fortunately, the outlined proof has its contemporary setting, which does not require any appeal to the sempiternality of falsehood. Assume that (a) X is the true set of sentences about the past; (b) A is a future contingent; (c) A is independent of X. The supposition (c) is necessary, because the dependency of A would imply STAT. The assumption (a) implies that X is consistent and thereby possesses a model. Due to (c), the sets X1 = X ∪ {A} and X2 ∪ {¬A} are consistent as well and have models, let us call ← 47 | 48 → them M1 and M2. We can think about M1 and M2 as models of maximally consistent sets of sentences; this is perhaps the best intuition related to the concept of possible worlds. In virtue of Lindembaum’s theorem on maximalization, every consistent set of sentences has its maximal extension. Thus, the set X has at least two such extensions which contain the sets X1 and X2. Similarly, both models M1 and M2 extend the model M. More technically, M1 and M2 are submodels of M. Clearly, sentences true in M remain true in M1 and M2.

Another construction which justifies (c) is as follows. We define (see Asser 1972, pp. 168–169) the concept of branchability by:

Since we have no reason to assume that X is a complete set of sentences, we say that X branches at A. This is displayed by the diagram (Δ).

image

An ontological interpretation is also possible. Let A be a sentence uttered at the moment fixed as present (denoted by t) and referring to a future event E, which will happen at t1 (E(t1) or will not happen at t1 (-E(t1)); the segment indicated by the PAST covers everything what happened before t (including this moment). We transform the diagram (Δ) into (Δ’)

image

W1 and W2 are possible worlds, that is, possible continuations of PAST as their initial segment. Otherwise speaking, W1 and W2 enlarge PAST, but in radically different (or ontologically inconsistent) ways, because the former validates A and the latter verifies ¬A.

The status of M as a submodel of M1 and M2 justifies ET. In fact, if A will be true at the point t1 later than t, its truth becomes stable for ever. Thus, ET is automatically valid. For the sake of the further argument, we can consider the value of A at any temporal point t earlier that at t1, not necessarily at the moment in which this sentence is (was) issued. Assume that A is eternally true, that ← 48 | 49 → is, if A is true at t, it remains true at any t1t. This means that A is true in M1. However, the sentence in question cannot be false in any submodel of M1, in particular in M. Hence, we immediately conclude that if A is eternally true, it is sempiternally true as well. The same line of reasoning applies to the assumption that ¬A as true in M2, implies that if A is false in M2, it is also false in M. Thus, both logical values are stable over time, ET and ST are equivalent, and STAT finds its support. Now, we have exact tools for comparing many-valueness, BI and truth-absoluteness. The argument for STAT essentially uses BI. The converse dependence holds as well. Hence, BI and STAT are equivalent, at least in the model-theoretic semantics. Nonetheless, if the truth-absoluteness is reduced to eternality, BI has no explicit connection with ET. One can assume that any indefinite sentence becomes true or false sooner or later, but the supposition that at least some indefinite sentences remain such for eternity is also possible. In both cases, however, ET can be defended. Thus, Łukasieiwicz could consider truth as absolute, but provided that this property would be understood in a weaker sense, that is, according to WTAT.

Many philosophers, and Łukasiewicz is a good example in this respect, accept the eternality of truth as uncontroversial, but complain about sempiternality. The argument for the logical equivalence of (ET) and (ST) shows that something is wrong in this view, because if sempiternality is felt as non-intuitive, the same should concern eternality. I only indicate this problem briefly. Its full analysis requires taking in account ontological questions related to determinism and its various forms. For example, radical determinism can be displayed by the diagram (Δ’’)

image

This line symbolizes the uniform “flow” of reality from the past though the present moment t to the future. This flow can be organized by the strict (one-to-one) causal connection. In fact, Łukasiewicz argued that sempiternality is implied by MEM plus causality. However, the outlined argument shows that STAT can be defended on purely logical grounds and without an appeal to ontology. Thus, objections against sempiternality as an intuitive property of truth seem to come as a result of a conflation of logic with ontology.

Let me add that that my reasoning concerns truth and falsehood in their model-theoretic sense. Perhaps, Leśniewski’s demonstration is semantic in principle. Anyway, his argument implicitly uses bivalence, namely by employing the reductio ad absurdum mode. Thus, the conjunction of the reduction principle and the principle of non-contradiction has the same metalogical consequences as the product of MEM and MNC. Anyway, classical logic is closely related to the ← 49 | 50 → absolute treatment of both logical values. Moreover, the semantic definition of truth is an example, which suggests an absolutists interpretation. In fact, some authors (see Kokoszyńska 1936, Woleński 1994) argue that Tarski defined the absolute concept of truth.2 Leśniewski (see Leśniewski 1929) suggested – and this remark applies to the semantic concept of truth – that we should not use the predicate ‘is true at t’, but rather the construction ‘is P at t’ is true (false)’. In other words, truth has no temporal coordinates, but predication has or might have them. But different solutions are also possible. Borkowski (see Borkowski 1991) claims that Łukasiewicz simultaneously worked with two different semantic valuations represented by the pairs {true, false} and {true at t, false at t). Suszko (see Suszko 1975) distinguished the logical valuation (true/false) and algebraic valuations consisting in introducing various objects (in particular, situations) denoted by sentences. These constructions provide other tools for defending STAT as related to the basic valuation of sentences by the unqualified truth and the unqualified falsehood. All mentioned strategies are explicitly model-theoretic or convertible to such. Let me, however, remark that there is a question of how the relativity/absoluteness debate over the concept of truth could be applied to correspondence theories, for example, Hochberg’s explication (see Hochberg 1984a) of the relation between fact and truth, particularly if the concept of truth-makers is introduced. I see some problems here associated with the absence and presence of facts, because truth-makers, considered as facts, states of affairs, situations or other objective (ontological) correlates of sentences, seem to require additional valuations. ← 50 | 51 →


2       What is more, Kokoszyńska identified the classical definition of truth with the absolutist account of this notion (see Kokoszyńska 1936).

V
Is Identity a Logical Constant and are there Accidental Identities?
3

Propositional connectives and quantifiers are logical constants without any doubt. However, we speak about first-order logic with or without identity. Even this way of speaking suggests that identity has a special status to some extent. In fact, the status of identity is controversial. Wittgenstein says (Wittgenstein 1922, 5.5303):

   Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all.

According to Wittgenstein, identity is not a relation. This view rises an important point: does identity hold between objects, which are numerically different, for example, two occurrences of ‘e’ in the word ‘different’? Tarski’s view of identity was radically opposite to that of Wittgenstein. The difference is well illustrated by the following quotation (Tarski 1941, p. 49):

   Among the logical concepts not belonging to sentential calculus, the concept of IDENTITY, or, of EQUALITY, is perhaps the one which has the greatest importance.

Wittgenstein’s and Tarski’s statements about identity can be rephrased without essential changes by replacing ‘identity as a relation’ by ‘identity as a predicate’ (I prefer the second way of speaking).

Formally speaking, identity is introduced into first-order logic by the axioms (I omit quantifiers in the front of formulas):

   (A1) x = x;

   (A2) (x = y) ⇔ (y = x);

   (A3) (x = y) ∧ (y = z) ⇔ (x = z),

together with the rule of replacement (for simplicity, I restrict it to monadic predicates, but its is obvious how to generalize this rule from arbitrary formulas), which can be formulated as:

   (RR) (x = y) ⇔ (PxPx/y). ← 51 | 52 →

Thus, first-order logic with identity (FOLI) is determined by the propositional calculus (PC; more precisely, by its codification via axioms or rules of inference). Pure (the meaning of ‘pure’ in this context will be explained below) first-order logic without identity (FOL) is codified by axiomatization or rules of inference and the set {(A1)–(A3), (RR)}.

As it is well known, the identity predicate is not definable in first-order logic. The situation changes in second-order logic via the Leibniz rule:

which says that identical objects have the same properties. In fact, the implication

suffices for defining identity. The reverse implication

expresses the famous principium identitatis indiscernibilium (the principle of identity of indiscernibles). Hence, (LR) is the conjunction of (1) (the principle of indiscernibility of identicals) and principium identitatis indiscernibilium.

Although formal properties of identity are (or seem to be) clear, the concept of identity provides several problems for logicians and philosophers. One of them is captured by the already mentioned question: ‘Is identity a logical constant?’ The arguments for the affirmative answer point out that fundamental metalogical results (semantic completeness, compactness, undecidability, the Löwenheim-Skolem theorem, the Lindström theorem) are valid for FOLI. In particular, the last result seems important because it provided a characterization of first-order logic as contrasted with higher-order logic. Consequently, the the Lindström theorem determines the borderline between ‘being the logical’ and ‘being the extralogical’, that is, the first-order thesis (see Woleński 2004a for details and a discussion). Speaking metalogically, all theorems with the identity-predicate derivable in FOLI are universally valid. Speaking more philosophically, these theorems are necessary in the strongest sense, because logic represents an uncontroversial kind of necessity. Tarski (see 1986a) argued that identity is a logical notion because it is invariant under all transformations of a domain into itself.

However, there are some problems with considering identity as a purely logical item. Having identity, we can define numerical quantifiers of the type ‘there are n objects’, where n is an arbitrary natural number. Consequently, we can characterize finite domains, although first-order logic is too weak to define the ← 52 | 53 → concept of finiteness. Now, if we add the sentence ‘there are n objects’ to firstorder logic, its theorems are valid not universally, but in domains that have exactly n elements. Hence, it seems that identity brings some extralogical content to pure logic, contrary to the view (it can be expressed by a suitable metalogical theorem) that logic does not distinguish any extralogical content. Perhaps, this is the very reason why the label ‘the identity-predicate’ is used, although logicians simultaneously remark that this is a very special predicate. Anyway, a qualification of identity as logical or extra-logical is conventional to some extent.

Another reason to see identity as an extralogical problem stems from the so-called inflation and deflation theorems (see Massey 1970), both closely related to the definability of finite domains in FOLI. The former says that if a formula, say A, is satisfied in a non-empty domain D with n elements, it is also satisfied in any domain D’ with at least n elements. The deflation theorem asserts that if A is satisfied in D, it is also satisfied in any D’ with at most n elements. Although these theorems hold for FOL, they fail for FOLI. The formula xy(x = y) provides a counterexample to the inflation theorem, because it is true in the one-element domain and no other, but the formula xy(x ¹ y) is false in the domain with one element. On the one hand, both theorems hold in FOL. This is a reason for applying the adjective ‘pure’ to first-order logic without identity. On the other hand, if we look at PC and FOL, we can note some metalogical difference between both systems. In particular, PC is decidable, but FOL has no decision procedure. Furthermore, PC is Post-complete, but FOL (with numerical quantifiers) lacks this property. This shows that the concept denoted by the phrase ‘being the logical’ has a different strength in particular subsystems of FOL.

One additional problem requires a clarification. According to early Frege (see Frege 1879) and Wittgenstein (see Wittgenstein 1922), identity operates on signs. The view that the formula x = y concerns objects became standard in contemporary logic. However, the notation used in (A1) – (A3) (as well as in other quoted formulas) is ambiguous to some extent. In fact, under the objectual treatment of identity, we should formulate (RR) as

This formula means: if the object denoted by the letter x (the denotation of x) is identical with the object denoted by the letter y (denotation of y), then if the denotation of x has a property P, the letter x can be replaced by the letter y. Note that the antecedent of (RR’) concerns objects, but its consequent deals with objects and signs. A suitable rephrasing of other listed formulas is straightforward. The proposed reading of the replacement rule underlines its semantic character. We should think about identity as determined by an interpretation of terms in ← 53 | 54 → models; denotations depend on valuation functions ascribing objects to terms (individual constants and individual variables; I omit valuations of predicated letters). Note that (A1) is the only axiom, which is an unconditional formula contrary to (A2), (A3) and (RR) [or (RR’)]. I assume the objectual framing of identity in what follows without using the symbolism employed in (RR’).

Kripke (see Kripke 1971) presents an argument intended to show that there are no accidental identities or that every identity is necessary. This view is supported by the following reasoning. Assume (RR) and (A1) in the following form (I insert quantifiers, because their position plays a relevant role in the argument; the box expresses necessity):

Now, interpret P as the property ‘necessarily identical with’. (RR) gives

Since image(x = x) is universally valid, it can be omitted. Thus, we obtain:

that is, the conclusion that if two objects are identical, they are necessarily identical. However, this result seems non-intuitive, because the identity of London and the capital of UK looks as accidental.

How convincing is this reasoning? First of all, let us change it by using the provability operator image. Since (A1) – (A3) are logical axioms (I assume here that identity is a logical constant), we can add image. As far as the matter concerns (RR) (translated into a formula), we obtain (quantifiers added):

Since the provability operator is monotonic, (RR’’) entails

Two things are to observed. Firstly, the antecedent inside (5) has the sign image Is it possible to skip this element in (5). It would be at odds with the practice of using identity in inferences. For instance, mathematicians derive conclusions about properties of identical objects, assuming that their identity is provable in mathematical theories. Secondly, we cannot interpret P as expressing provability. Now, if provability is understood as a kind of necessity, Kripke’s argument cannot by repeated. We can only obtain:

Inserting or omitting the formula image(x = x) is completely pointless in this reasoning. Let us strengthen (5) to the formula:

We can think about (7) as a scheme capturing the first-order version of the Leibniz rule. Disregarding whether the provability of the right part of (7) is realistic or not, this equivalence leads to:

which is not very exciting, because it asserts that if an identity is necessary, it is necessary.

I will not discuss Kripke’s solution of the puzzle produced by (5) in details (he accepts that some identities are accidental and a posteriori). I only note that his view assumes few things, in particular, the distinction between rigid and non-rigid designators and essentialism as well as admissibility of switching from de dicto necessities to de re ones. There is no essential difference between (A1) and (A1’), although the latter has the de re form (the quantifiers precede the box). However, replacing P by ‘necessarily identical with’ relevantly uses the de re formulation. My proposal consciously ignores all exrtalogical circumstances except the claim that necessity should be used de dicto and as related to the provability. This blocks the passing from image(x = y) to x = y. Without assuming x = y as independent of image or image, the conclusion that all identities are necessary does not follow. And yet we can still keep the difference between the unconditional, like (A1), identity validities and the conditional ones, namely (A2), (A3). Facts of interpretation are, of course, accidental and a posteriori; for instance, the term ‘the capital of UK’ could be valued not by London, but by another British city, say, Manchester. However, if an interpretation is fixed, its consequences are conditionally necessary. Since unconditional identity is a special case of conditional, we obtain a uniform treatment of =, independent of the view whether it is a logical constant or not. These conclusions are obvious, if we adopt the objectual understanding of identity. ← 55 | 56 → ← 56 | 57 →


3       This paper uses and extends some considerations in Woleński 2005 and Woleński 2008.

VI
Naturalism and Genesis of Logic

It is traditionally accepted that we differentiate between logica docens and logica utens, that is, between theoretical logic (logic as theory) and applied or practical logic. Both can be defined with the use of the concept of logical consequence. The first is a set of consequences of an empty set, symbolically LOGT = Cn∅, provided that the operation Cn satisfies the well-known general Tarski’s axioms, i.e. denumerability of the language (a set of sentences) L, XCnX (the inclusion axiom; X, Y are sets of sentences of L), if XY, then Cn X ⊆ CnY (monotonicity of Cn), CnCnX = Cn (idempotence of Cn) and, if ACnX, then there is a finite set YX such that ACnY (Cn is finitary). Cn is a mapping of the type 2L 2L, that is, transforming subsets of L into its subsets. In order to make things simpler, I assume that Cn is based on classical logic. LOGT can be also defined as the only common part of the consequences of all sets of sentences. Otherwise speaking, logic is the only non-empty intersection of the family of all subsets of L. What follows from it is that logic is included in the consequences of each set of sentences, which underlines its universal character. If CnXX, then X = CnX due to the inclusion axiom. Moreover, if CnXX, we say that that X is closed by the consequence operation of logical consequence. This is the definition of a deductive system (a deductive theory). Thus, in the case of deductive systems, Cn does not extend X beyond itself. The concept of logical consequence belongs to the syntax of language. The notion of logical following (entailment) is a semantic counterpart of Cn. The properties of both of these notions are such that if ACnX and X consists of true sentences, the sentence A must be also true. If X is a theory and the set CnX coincides with a set of true sentences (specifically: true in a determined model M or relevant class of models) of X, this theory is semantically complete.

The statement that Cn closes sets of sentences as long as Cn XX suggests some analogies with topology, since certain properties of this operation satisfy Kuratowski’s axioms for topological spaces. Let Cl denote the closure operation of a topological space, and X, Y – any subspaces (subsets); I intentionally use the same letters for denoting the set of sentences and sets investigated by topology. Then (see Duda 1986, p. 115): (1) Cl∅ = ∅; (2) (2) XClX, (3) ClClX = ClX; (4) Cl(XY) = ClXClY. Operations Cn and Cl differ from each other as far as the matter concerns axioms (1) and (3), because, in the case of logic, the set Cn∅ is non-empty and CnXCnYCnY but the reverse inclusion ← 57 | 58 → does not hold. The first difference is founded on the specific definition of logic, which does not possess a clear topological sense (I will return to this question below), while the other one indicates a partial analogy between closed sets in the topological sense and deductive systems in the logical sense, because (3) does not hold for arbitrary sets of sentences. Thus, “logical” closure is weaker than the topological one. The set of theses of logic is for sure non-empty and it is a system. It can be treated as a specifically closed topological space, with individual theorems as its points.

Topology{∅ = Cl∅, X} is minimal (see Wereński 2007, p. 124) in the sense that the smallest one cannot be examined. Next, Cl∅ ⊆ ClX, since for each X, ∅ ⊆ X. Let us agree (this is a convention) that Cl∅ is a topological equivalent of logic. The motivation for this convention consists in taking into consideration that a proof of logical theorems does not require any assumption. The evident artificiality of this convention can be essentially weakened by the acknowledgment that closing an empty set produces any theorem of logic. It can be shown that if A and B are theses of logic, then Cn{A} = Cn{B}, which means that any two logical truths are deductively equivalent. Let us assume that X (this time as a set of sentences) is consistent and consists of the set X’ of logical tautologies and a set X’’ of theorems outside logic. Thus, X’ = Cn∅ and X’’J – X’. Sets X’ and X’’ are disjoint and constitute mutual complements in the set (space) X. Since the set X’ is closed, its complement, i.e. X’’, is an open set. The introduced convention about Cl∅ allows to “topologize” the properties of sets of theses; in particular, it makes it possible to treat the set X (of theses) as a clopen set. From the intuitive point of view, the operation of logical consequence encodes inference rules of deriving some sentences from other sentences, that is, the procedure of deduction of conclusions from defined sets of premises. Deduction, at the same time, is infallible, that is, it never leads from truth to falsity.

What is applied logic or logica utens? When X is any non-empty set of sentences, then applied logic LOGA(X) of this set can be associated with the operation Cn applied to X. This is applied logic in a potential sense. This understanding of logica utens is, however, decidedly unrealistic, since its user applies only these rules that he needs, regardless of whether or not he does so in a conscious way. In other words, real applied logic of a given set is a stock of these logical laws (or rules) that are used in a concrete inferential work. This circumstance makes it impossible to give an abstract definition of real applied logic. It is worth to observe that Cn can be based on a non-classic logic, e.g. intuitionistic, many-valued or modal logic. Furthermore, we can neglect the monotonicity condition in order to obtain a non-monotonic logic. These remarks point ← 58 | 59 → to the fact that non-classical logics is similarly definable as the classical system. Since applied logic operates on closed-open sets, they, by this assumption, contain extra-logical sentences beside theorems of logic. The inclusion condition decides that logic can be deduced from any set of sentences. This fact has a serious methodological importance. If deduction within closed sets ‘leads’ to accumulation points in the topological sense, adding new extralogical sentences can be executed in an extra-deductive manner. This corresponds to the definition of an open set as such that includes all of its neighbourhoods. To put it in a different way, the transition to neighbourhoods of sentences as points in spaces in the set X”, that is, the extension of this set, can be non-deductive. The above considerations suggest that there was logica docens ‘at the beginning’ and it became logica utens through application. According to this image, logic is thus applied in the same way as mathematical schemes in a concrete physical theory, e.g. Euclid’s geometry in classical mechanics or non-Euclidean geometry in a specific theory of relativity. This circumstance makes a naturalistic interpretation of logic difficult, and even impossible, because, generally speaking, laws of logic are considered abstract to the highest degree and as such are thought to belong to Plato’s world of forms.

A contemporary follower of Plato says that naturalism is helpless with respect to the domain of abstracts for two reasons. Firstly, because the naturalistic view acknowledges the existence of temporal-spatial objects as the only ones (there exclusively exist temporal-spatial and changeable objects), while the logical realm exists out of time and space. According to Platonism, this is the main ontological difficulty of naturalism. Secondly, the naturalist also faces an epistemological problem, since as a genetic empiricist with respect to sources of cognition he or she cannot elucidate the genesis of the genuine universal and infallible knowledge represented by logic and mathematics. In particular, the follower of Plato adds that no empirical procedure is able to generate logical theorems as true, independent of empirical circumstances. Platonism is – as a matter of fact – a historical and metaphorical label in above remarks. From the systematic point of view, it is much better to use transcendentalism (or antinaturalism) as the opposition against naturalism, since every criticism of naturalism (sooner or later) makes references to transcendental arguments in the sense of Kant. It is in this way that, for example, criticism of psychologism (as a version of naturalism) was executed by Frege and Husserl; one could say the same about Moore’s arguments against reducibility of axiological predicates to non-axiological ones. In general, transcendentalists reproach naturalists with what Moore defined as a naturalistic fallacy on the occasion of his criticism of ← 59 | 60 → reduction of moral values to utility. Dualisms of facts and values or logical and extra-logical theorems are not the only ones which naturalism has difficulties with. Other oppositions, from which – in the transcendentalists’ opinion – naturalists are cut off in the sense of impossibility of their satisfactory explaining are, for example, the following: physical information – semantic information or quantity – quality.

This, of course, is fairly true that naturalism must meet various difficulties. The criticism of this view, however, overlooks problems of anti-naturalism, which Moore had already drawn attention to. The arguments he used were that the super-naturalistic (Moore used this qualification) grounding of morality as rooted in the supernatural world is a similar error to that of reducing axiological predications to ones definable in purely natural categories. Another problem of transcendentalism arises with respect to the so-called Benacerraf argument indicating the enigmatic character of cognition of mathematical objects, provided that each cognition consists in causal interaction on the part of the object of epistemic acts, while numbers – on the power of their nature according to Platonism – do not interact in a causative way with people. How, then, can an anti-naturalist explain the genesis of logic? He or she can either assume – as Plato did – that the world of abstract forms is eternal, or argue – like Descartes – that certain ideas are inborn, or still – like some theists – that man obtained logic as a gift from God when he was created as the imago Dei. The Platonic and Cartesian paths are ad hoc, whereas that of theists is based on extra-scientific premises. In any case, the situation of an anti-naturalist is not to be envied as it must resort to secret beings (souls, spirits, ideas) and secret kinds of cognition (intellectual intuition, etc.). The naturalist can paraphrase the title of Hoimar von Ditfurth’s book Der Geist fiel nicht vom Himmel (The Ghost Has Not Fallen From Heaven) by saying that logic has not fallen from the other world, Platonic or others (as regards the defence of naturalism in other contexts compare Woleński 2016).

For a positive naturalist account of the genesis of logic, it is indispensable to combine the dichotomy logica docens – logica utens with the notion of logical competence, modelled on grammatical competence in Chomsky’s sense. Both abilities play a similar role. The grammatical competence generates the right usage of linguistic devices, while the logical competence determines the application of logical rules in inferential processes. Nevertheless, the analogy is not complete, at least according to my own concept of the question. Whole Chomsky defines grammatical competence simply as grammar, the distinction which I am going to use differentiates logic, both theoretical and applied, from logical competence. The last category refers to a determined disposition of the ← 60 | 61 → biological organisms which are capable to perform mental functions (compare further comments below). Speaking more precisely, logical competence is the ability to use operation Cn. Thus, a logical theory is not logical competence, but its articulation. The dispositional character of logical competence does not settle whether each element of logic as a theory finds its coverage in its natural generator. By the way, a negative answer is rather obvious as the development of logical theories was and is heavily dependent on nature and needs of communicative interactions within human society. Further considerations in this paper will be devoted to the genesis of logical competence. They refer to the genesis of logic inasmuch as without the possibility to create and apply rules of logic there would not appear logic in either of the two distinguished senses. In other words, logica docens and logica utens are derivatives (precisely speaking – one derivative) of logical competence. This circumstance justifies the title “Naturalism and the genesis of logic”. Anyway, one of the main theses of this paper says that logical competence is not eternal, it appeared in the Cosmos at some time and is rooted in the biological structure of organisms. However, I have to make it clear at once that I do not treat my comments as relating to the biological question as empirical. My cognitive interest is of the philosophical nature and remains within evolutionary epistemology. However, in compliance with my metaphilosophical convictions, I have to take into consideration the output of empirical sciences, biology, in particular, in the analysis of philosophical problems. Otherwise speaking, philosophical analysis, though somehow speculative in its character, is superstructured on empirical knowledge.

In accordance with the above explanations, logical competence precedes logic, both theoretical and applied; nevertheless, there is also a feedback because theoretical reflection on logic and its applications to concrete questions can enhance the logical competence. Everything points to the fact that the logical theory required a prior application of rules of logic and the development of language. In the case of the Mediterranean culture, the applied logic appeared, for sure, with Greek mathematicians and philosophers. When Anaximander said that there does not exist the principle of closeness, since it would create the boundary of apeiron which is boundless, he made use of a rule similar to regressum ad absurdum. Pythagoras proved the existence of irrational numbers and his reasoning was a proof by reduction in the modern sense. Various paradoxes formulated by the Eleats were of a similar character. The first logical theory, that is, Aristotle’s syllogistics, originated much later, although on the basis of extensive practical material accumulated earlier. It was also, in a vital way, linked to the structure of sentences of the Greek language. One cannot, however, say ← 61 | 62 → that carrying out logical operations requires knowledge of the language because they are typical of infants (see Langer 1980), and the latter do not have linguistic material at their disposal yet.

The question of the sense of understanding in animals, other than humans, is controversial, yet the following example (which can be treated as an anecdote) is only too suitable in this place (see Aberdein 2008). In 1615, in Cambridge, there was a debate devoted to the dog logic, which was attended by King James I. The problem concerned the question whether hunting dogs which were used for locating game during hunting, applied logic, in particular the so-called law of disjunctive syllogism in the form “A or B, thus if non-A, then B” (this question had already been considered by Chrysippus). Let us suppose that a hound reaches a fork in the roads. The trailed game runs away to the right or to the left. The hound establishes that it is not to the left, and therefore runs to the right. The debate had a very serious character and a truly academic form. John Preston (a lecturer of Queens’ College) defended the thesis that dogs apply logic, whereas his opponent, namely Matthew Wren (of Pembroke College), argued that hounds are directed solely by the scent and it is the only reason why they choose the right direction. The role of the moderator was played by Simon Reade (of Christ’s College). When the latter acknowledged M. Wren to be right, the King, himself a great enthusiast of hunting and relying on his own hunter’s experience, observed that the opponent, however, should have a better opinion about dogs and lower about himself. Wren, very skilfully managed to get out of the tight situation by saying that the King’s hounds – in contrast to others – were exceptional, since they hunted upon the ruler’s order. This compromising solution is said to have satisfied everybody. After all, even if hunting dogs do apply disjunctive syllogism occasionally, they certainly do not do this making use of a language.

The debate held in the presence of the King of England is a good illustration of a certain difficulty regarding an analysis of the genesis of logic. There appears the question of what evidence could help in this case. The debaters in Cambridge considered the dogs’ behaviour and drew conclusions from that. In the case of humans, we can observe signs of inferential processes in people or base ourselves on the written evidence of the past. Anyway, the empirical base is greatly limited. Not much can be inferred from the inscriptions found on the walls of caves inhabited by our distant predecessors. All the information through which human logical competence manifested itself has been recorded in a language developed to a such degree that it made it possible to encode the deductions carried out factually, even if it did not suffice to formulate a logical theory. ← 62 | 63 → In this respect, the genesis of logic appears to be more mysterious than the appearance of mathematics (see Dehaene 1997) or language (see Botha 2003, Johansson 2005, Larson, Déprez, Yamakido 2010, Talerman, Gibson 2012). In both mentioned domains, especially in the latter one, there have appeared a host of works and theories. In particular, the question of whether animals can count and make use of a language, or at least a protolanguage, has been discussed (see Hauser 1997, Bradbury, Vehrencamp 1998).

Studies in the origins of logic are limited to various inquiries into the logical competence of children going through their pre-language period or that of people living in primitive societies. This provides solely epigenetic and ontogenetic material, whereas phylogenetic is taken into account only to the extent in which the traditional and strongly speculative Haeckel’s assumption that ontogenesis reproduces phylogeny is accepted. Nevertheless, considerations concerning the origins of the calculating and languafe competence are important also to the discussion on the origins of logic. This concerns, in particular, the concept of origin and the development of grammar and sign systems (see Heine, Kuteva 2007). According to a fairly common conviction, signs were the earliest to appear, especially expressive ones, then there were iconic signs, followed by symbols. This corresponded to the evolution of grammatical structures from nominal through sentential-extensional to sentential-intensional. Thus, the development of language progressed on the basis of s transition from a-semantic or little-semantic objects to fully-semantic ones (intensionality symbolism). The origin of language has always been the object of an animated interest on the part of philosophers (compare Stam 1976 for a review of earlier theories). In 1866, the French Linguistic Society decided that considerations on this subject should be excluded from sciences. As a matter of fact, a renaissance of studies of the appearance and evolution of language was observed in the mid-20th century. One can speculate that works dealing with the genesis of logic, if they had been written on a mass scale in the first half of the 19th century, would have shared the fate of linguistic dissertations on the origins of language, which were deemed as too speculative.

It is not without significance to model microbiological and neurological processes, for instance, through cell automata (see Ilachinski 2001), or even with the help of advanced mathematical techniques (see Bates, Maxwell 2005) and computational ones (see Lamm, Unger 2011). These enterprises indicate that the very organisms themselves, and whatever is happening inside them, possess properties which can be formulated mathematically. However, far-fetched methodological carefulness is indispensable. The title of one of the quoted books ← 63 | 64 → runs as follows: DNA Topology. It can be understood in a dual way: firstly, it suggests that, for instance, DNA has a looped structure in certain circumstances; secondly, this can be understood in a weaker manner, i.e. in such a way that a topological notion of a loop models certain properties of DNA. Reading the book by Bates and Maxwell inspires to conclude that the authors make use of both meanings. My opinion on this problem consists in recommending the other sense of modelling. It is only assumed here that the world is mathematizable (that is, describable mathematically) due to its certain properties, yet it is not mathematical. Works in the field of the evolution of language and those devoted to modelling biological phenomena, as a rule, accept naturalism, silently or explicitly. It is well-expressed by the appearance of biosemiotics (see Barbieri 2003, Barbieri 2008, Hoffmeyer 2008, Favareau 2009), cognitive biology (see Auletta 2011; this author declares the theistic Weltanschauung, yet suspends it in his book) or the more and more popular physicalization of biology (see Luisi 2006, Nelson 2008). Since syntheses of biology and semiotics or biology and cognitive science can be conducted, there is no reason not link logic to biology.

The only advanced attempt at the naturalistic grounding of logic that I am familiar with derives from William Cooper (see Cooper 2001), who considers the following sequence: (*) mathematics, deductive logic, inductive logic, theory of decision, history of life strategies, evolution theory. The relations between elements (*) are such that from the evolution theory to mathematics we deal with implication, whereas reduction proceeds in the opposite direction. As far as deductive logic is concerned, it is directly implied by inductive logic and reduces itself to the latter. The evolution theory is the ultimate basis, both for implication and reduction. Briefly speaking, deductive logic arose at a certain stage of evolution (Cooper does not make it precise in detail) through a natural selection and adaptive processes. Cooper’s schema leaves a lot to be desired. Omitting the lack of a more detailed definition of the ‘production’ of logic through the process of evolution, which was indicated earlier, the notions of implication and reduction are not clear in Cooper’s model. Since deductive logic (that is, Cn∅) is implied by any set of sentences, the role of inductive logic (I neglect here disputes relating to its existence; however, see below) is not specific. In consequence, the reduction of deductive logic to inductive logic appears to be highly unclear. Moreover, the phrase ‘logic as part of biology’ (the subtitle of Cooper’s monograph) is ambiguous. It may mean that logica docent is part of the biological theory (more precisely: the evolution theory) or, also, that deductive competence (Cooper does not use this name) is an element of the biological equipment of human being. Indeed, in compliance with the well-known maxim of Theodo ← 64 | 65 → sius Dobzhansky, nothing makes sense in biology if it is not considered in the context of evolution, but this does not mean that everything can be explained on the basis of the evolution theory. Cooper, in his analysis, ignores genetics completely and this is, perhaps, the most serious deficiency of the model.

A purely evolutionistic classical approach towards the establishment and development of human mental competences, such as the ability to use a language or reasoning, is – in an outline – as follows (it can be found in countless publications dealing with the theory of evolution and its application to different specific problem areas; compare, for instance, Lieberman 2005, Tomasello 2010). The Universe appeared about 15 billion years ago (all the dates here are given in approximation). The age of our Earth is 4.5 billion years. The first cell appeared a billion years later, and multicellular organisms after the next 2.5 billion years. Plants have been around for 500 million years, reptiles – for 340 million years, birds – 150 million years, and apes – for 7 million years. The species of homo appeared two million years ago, homo erectus – from 1 million to 700 thousand, and homo sapiens – 200 thousand. The cultural-civilizational evolution marked with the language (in the understanding of our modern times), the alphabet and writing began 8.500 years ago. Three and a half billion years from the moment of the appearance of the first cell to that of the appearance of civilization and culture were completely sufficient to form the mind capable of performing typical intellectual activities, in particular, to carry out a logical operation. Homo sapiens must have been able to do so much earlier, maybe it happened at the beginning of this species. It cannot be ruled out that rudiments of logical competence had already been available to homo erectus. Establishing the date of the appearance of logical competence in the course of evolution, and attributing it to organisms other than human, does not seem particularly important here. A safe evolutionist hypothesis in this respect can claim, for instance, that the inferential ability appeared by way of a randomly acting mutation, and because it proved to be an effective adaptive tool, it was developed by homo sapiens, also by virtue to available and more and more perfect linguistic instruments. The logical theory appeared as the final product of the long evolution process. This is an adaptation of the classical concept of the evolution of language (compare, however, the conclusions at the end of the paper).

Neo-Darwinian evolutionism connects the appearance of life and its further evolution with entropic phenomena (see Brooks, Wiley 1986, Küppers 1990). This perspective leads to the need for indicating anti-entropic phenomena, i.e. mechanisms which maintain stability of organisms and their internal order, and thereby determine the continuance of their existence (see Kauffman 1993). The ← 65 | 66 → decisive event to enhance a serious revision of the evolution theory was the discovery of DNA structure by Crick and Watson in 1953 (the model of double helix), as well as further research into genetic encoding. Those results demonstrated the necessity of a more profound linking of evolution with genetics. The notion of genetic information and manners of its transferring became the key instruments of a new biological synthesis, obviously, while keeping suitably modified classical categories of the evolution theory. Notice that formal analogies between information and entropy caused biologists and philosophers of biology to become interested more closely in relations between the first notion and the course of biological processes since as early as in the 1920s, (see Yockey 2005).

Several facts established by molecular biology are significant from the point of view of this paper (for a while I mention them without a ‘metalogical’ commentary; I entirely omit the physical-chemical questions, likewise the mechanism of hereditariness). Firstly, passing genetic information is directed from DNA through RNA (more precisely: mRNA – the letter ‘m’ denotes that RNA is in this case a messenger, that is, an agent passing information) to proteins. This observation makes the so-called main dogma of molecular biology. There are, as a matter of fact, certain exceptions in this respect (e.g. in the case of viruses), but at least in the so-called eukaryotic organisms (humans belong to this biological group) transmission of information is in compliance with this dogma. Secondly, genetic information is passed in ordered, linear, discreet, and sequential a manner. Thirdly, DNA particles are subject to replication (copying) and recombination (regrouping). Fourthly, the intracellular information system encodes and processes information, which causes the encoding in question to be interpreted as a computational system and to be modelled accordingly. Fifthly, passing of genetic information is not deterministic but random in its very nature, which is why there may appear genetic novelties. This last fact is vital from the point of view of the evolution theory because it explains the way in which mutation appears on the microbiological level.

The view that genetic information is of the linguistic character is only too tempting. Indeed, it is very often that we can see it treated as a language. And thus, we can speak about alphabets, words, syntax, codes and encoding, or about translations (in the sense of transfer from the genetic language into another one); this is done especially by representatives of bio-semantics, who – in the genetic information – detect a semantic dimension or, at least, its germs. Such an approach is, however, very debatable (compare the discussion in Kay 2000; Sarkar 1996a rejects the notion of genetic code, but this solution seems too radical). In true fact, technical elaborations of genetics avoid comparing the genetic code ← 66 | 67 → with the language (see, for example, Klug, Cummings, Spencer 2006). Regardless of the applied language, for instance, some write about ‘words’ as components of the genetic code, surely using quotation marks to indicate a certain metaphorical investing of genetic information with the linguistic dimension, while others do so about words, we can easily find here the problem of relation of physical information as something quantitative to the semantic information as qualitative. The mathematical theory of information concerns the former, and only indirectly relates to the latter. The well-known Shannon’s statement on the capacity of channels of transmitting information and limiting the so-called information noise has a sense only with reference to its quantitative understanding. The genetic information is a kind of physical, not semantic information. On the other hand, processing the former, i.e. quantitative, into the other, i.e. qualitative, is a notorious fact, for example, reading a book – as long as we understand the language in which it has been written, we rapidly process the physical stimulus into semantic units, i.e. such that we understand according to their linguistic sense. For the time being, we do not know the mechanism of this transformation and it makes the biggest anthropological puzzle (see Hurford 2007). Perhaps, the properties of the genetic information lie at the foundations of what may be called semiotization of mental processes, yet this is a fairly speculative assumption from the biological assumptions, though one could consider it as philosophically justified to some degree.

At first sight, if the genetic information were a language in the full or merely approximated sense or, we could look for the genesis of logical competence directly on the microbiological level. After all, the properties of the genetic code, whatever it is, stand far from those that can serve to define operation Cn. Nevertheless, these properties can be tied to logic in their understanding of today. Before I pass on to show this relation, I will draw attention to certain theoretical questions. Kazimierz Ajdukiewicz (see Ajdukiewicz 1955) divided inferences into deductive, increasing the probability (inductive in a broad sense) and logically worthless. The first are based on operation Cn which holds between the premises and the conclusion (the conclusion results logically from the accepted assumptions), the second increase the probability of the conclusion on the basis of the premises, and the third ones are devoid of any logical relation between the links, e.g. ‘if Kraków lies on the Vistula, then Paris is situated in France’. Logic, in this context, is understood in a broader way than at the beginning of the paper, since it includes also induction rules. We can, too, extend respectively the notion of logical competence, but I do not wish to consider such a generalization. Treating the thing from the information point of view (see above), while ← 67 | 68 → deduction does not broaden the information included in the premises, although it does not allow it to be lost, the conclusion is false, which disperses (in the sense of entropy) the information acquired earlier, whereas logical inference that is worthless is redundant from the information point of view.

The infallibility of the rules generated by operation Cn derives from the fact that they correspond to theorems of logic, i.e. to sentences (formulas) that are true in all circumstances. One of the axioms of probability calculus is the assumption that there exists an event whose probability obtains the value 1 (the whole space on which the probability measure is defined constitutes this event). An interesting interpretation of this axiom consists in acknowledging that it prevents levelling (dispersing) of probabilities ascribed to particular occurrences, that is subsets of the whole space. In other words, this axiom saves the differences in the amount of information, which conditions its flow. Thus, it performs the anti-entropic function, i.e. blocks the dispersion of information, it protects it in this way. Operation Cn can be understood also as an instrument of protection of information from its dispersion, since it prevents the formation of false information on the basis of true information. As I have already mentioned, logic of induction is debatable, yet – on the other hand – nobody contradicts the fact that at least certain induction rules, e.g. those of statistical induction, are rational. It is true that they do not exclude the dispersion of information, but they are still able to somehow normalize its flow and in this way save or control it. Inferences that are logically worthless do not play any role in the processes of information protection.

Saving information (obviously it is not problems of the legal or moral nature that I mean here), both physical and semantic, appears as a vital function of all organisms, which operate with a given type of code. Since we treat operation Cn as an information-protective instrument, saving the possessed content (in the sense of information content, not necessarily meaningful in the sense of intensional semantics), then – at least from the naturalistic point of view – the logical consequence has the biological rooting. In relation to this issue, I will return to the properties of the genetic codes and genetic information mentioned earlier, this time in the metalogical context. Here are the features of the genetic code (see Klug, Cummings, Spencer 2006, p. 307; I keep abstracting from the nature of elements of the code with one exception of amino acids due to the comprehensiveness of certain formulations): (i) genetic code is written in the linear form; (ii) if we assume that mRNA consists of ‘words’, then each of such words has three ‘letters’; (iii) each three-letter group, that is, the codon, determines another element in the form of an amino acid; (iv) if the code is unambiguous, it delineates one and only one amino acid; (v) if the code is degenerated, ← 68 | 69 → the given amino acid can be determined by more codons; (vi) the genetic code includes the initial signal and the terminal one in the form of codons initiating and finalizing the processes of passing the genetic information; (vii) the genetic code does not contain punctuation characters (commas); (viii) elements of the genetic code do not overlap, that is, a concrete ‘letter’ can be part of only one codon; (ix) the genetic code is nearly universal, i.e. apart from few exceptions, the same ‘dictionary’ of encoding serves all viruses, procariotic and eucariotic organisms. Completing the remarks offered earlier, I would like to add (see Klug, Cummings, Spencer 2006, pp. 264 – 265) that replication of DNA (which forms two new strings of the primary helix) can be semi-conservative (each replicated particle of DNA has one old string and one new one), conservative (the parent string is conserved as a result of synthesis in two new strings), or dispersed (old strings are dispersed in new ones). The most frequent is the case of semi-conservation. Nevertheless, the genetic information that exists earlier is inherited by helixes formed by way of replication.

It follows from properties (i) – (ix) that the ‘syntax’ of the genetic code is rigorous. It is based on a detailed specification of simple elements (‘letters’) and their combinations (codons). The lack of commas points to the fact that it is a series of concatenations. A code is unambiguous inasmuch as it is not degenerated. This property can be likened to syntactic correctness, while degeneration to a lack of it. ‘Letters’ are atoms in the same sense as a simple expression, which is non-decomposable any further. Transformation of a codon into an amino acid is a function, unless the code is degenerate. The beginning and the end of the procedure realized by the code is clearly marked with separate ‘words’. I marked some expressions with letters so as not to suggest treating the code as a language. The linguistics-oriented terminology could easily be avoided through speaking about configurations and their elements. Genetic codes treated in this manner can be, and are, similar to electric nets or cellular automata, which – as a matter of fact – is underlined by above-mentioned modelling of genetic phenomena. The essence of things relies on that the outline ‘syntax’ is of an effective character and is trivially recursive, since operations realized by the codes are of the terminal character.

Biographical notes

Jan Woleński (Author)

Jan Woleński is professor at the University of Information, Technology and Management in Rzeszów. He is a member of the Polish Academy of Sciences, the Polish Academy of Arts and Sciences, and the International Institute of Philosophy and Academia Europea. He publishes on law and philosophy.

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Title: Logic and Its Philosophy