Iranian Association of Logic
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Volume 1, Issue 1 (2026)
3 2026, 1(1): 15-21 |
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Request Type: Original Research |
Received: 2026/10/21 | Accepted: 2027/01/2 | Published: 2027/01/2
Received: 2026/10/21 | Accepted: 2027/01/2 | Published: 2027/01/2
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Valizadeh A. Intuitionistic Logic and the Meaning of Proof. 3 2026; 1 (1) :15-21
URL: http://daneshafarand.org/article-9-87269-en.html
URL: http://daneshafarand.org/article-9-87269-en.html
Isfahan University
Abstract: (1 Views)
Aims This article introduces intuitionistic logic as a philosophical position about the nature of mathematics and as a precise logical system, and explains why it remains central to the philosophy of mathematics and to computer science.
Approach The discussion follows the development of the subject from Brouwer's critique of classical reasoning, through Heyting's formalization and the proof interpretation of Brouwer, Heyting and Kolmogorov, to Kripke's semantics and Dummett's meaning-theoretic arguments.
Main Points Intuitionistic logic arises when the meaning of a mathematical statement is explained by what counts as a proof of it rather than by truth conditions that may transcend our knowledge. On this view the law of excluded middle is not a logical law, while most of ordinary reasoning remains valid. Classical and intuitionistic logic are closely related by translations, and intuitionistic logic has natural semantics in terms of growing states of knowledge.
Conclusion Intuitionism is not only a historical episode in the foundations of mathematics. Its central idea, that meaning is given by proof, remains a live option in the philosophy of language and underlies modern constructive mathematics and type theory.
Approach The discussion follows the development of the subject from Brouwer's critique of classical reasoning, through Heyting's formalization and the proof interpretation of Brouwer, Heyting and Kolmogorov, to Kripke's semantics and Dummett's meaning-theoretic arguments.
Main Points Intuitionistic logic arises when the meaning of a mathematical statement is explained by what counts as a proof of it rather than by truth conditions that may transcend our knowledge. On this view the law of excluded middle is not a logical law, while most of ordinary reasoning remains valid. Classical and intuitionistic logic are closely related by translations, and intuitionistic logic has natural semantics in terms of growing states of knowledge.
Conclusion Intuitionism is not only a historical episode in the foundations of mathematics. Its central idea, that meaning is given by proof, remains a live option in the philosophy of language and underlies modern constructive mathematics and type theory.
Keywords: Intuitionism, Constructive Mathematics, BHK Interpretation, Kripke Semantics, Philosophy of Mathematics
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