This collection of essays examines logic and its philosophy. The author investigates the nature of logic not only by describing its properties but also by showing philosophical applications of logical concepts and structures. He evaluates what logic is and analyzes among other aspects the relations of logic and language, the status of identity, bivalence, proof, truth, constructivism, and metamathematics. With examples concerning the application of logic to philosophy, he also covers semantic loops, the epistemic discourse, the normative discourse, paradoxes, properties of truth, truth-making as well as theology, being and logical determinism. The author concludes with a philosophical reflection on nothingness and its modelling.
XIX. Some Liar-Like Paradoxes
XIXSome Liar-Like Paradoxes
The classical Liar Antinomy or Paradox (LA, for brevity) runs as follows (it is a slightly modified version proposed in Poland by Jan Łukasiewicz and employed by Alfred Tarski):
(1) The sentence denoted by (1) is false;
(2) (1) ⇔ (1) is true;
(3) The sentence (1) is false ⇔ The sentence (1) is true;Contradiction!
Remark: How to understand the premise (2)? Some commentators say that it is a nonsense, because it assumes that the equality (1) = (1) is true, but this equation is plainly false, if identity is understood in its official logical meaning. Imagine, however, that we correlate some objects with numbers (numerals), for example, saying that Lionel Messi has the number 10 as a soccer player in FC Barcelona or the Argentinian national team. Consequently, we are entitled to say 10 = Lionel Messi in such and such context. The same can be said in the case of LA. What is important, is that the equivalence (b) is justifies by this convention.
Stanisław Leśniewski, followed by Tarski, offered a diagnosis of Leśniewski–Tarski. According to them, we can identify three sources of LA:
(i) Self-referentiality of the predicate ‘is false’;
(ii) T-scheme, that is, the formula A is true ⇔ A;
(iii) Classical logic.
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