Intuitionistic logic

Constructive knowledge

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• published: 11 Oct 2015
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Sergei Artemov, “Constructive knowledge” Logic Colloquium 2015, Helsinki http://www.helsinki.fi/lc2015/materials/slides_artemov.pdf The constructive approach considers a proposition true only when there is a conclusive proof of it. This leads to the Brouwer-Heyting-Kolmogorov semantics and a variety of intuitionistic logic systems. Likewise, constructive knowledge is viewed as a result of verification, not necessarily producing an explicit proof of what has been verified. Since conclusive proofs can serve as verifications, constructive truth, under the natural assumptions that the corresponding conclusive proof is available and recognized as such, yields constructive knowledge. Within the framework of intuitionistic logic augmented by the “knowledge modality” K this can be formulated as the fundamental co-reflection principle: ‘constructive truth yields constructive knowledge’ p → Kp. This helps to resolve the knowability paradox by Church and Fitch which appears due to an unwarranted classical reading of constructive epistemic principles [3, 4]. Furthermore, the well-known Russell [1] and Gettier [2] “counterexamples” to the principle ‘justified true belief yields knowledge,’ (1) sometimes attributed to Plato, do not survive constructive analysis. Moreover, we argue that the following constructive version of (1) does hold: ‘constructive truth and belief on the basis of the corresponding conclusive justification/proof yields knowledge.’ These and other observations led to formal systems of intuitionistic epistemic logic IEL, developed jointly with Tudor Protopopescu [4]. [1] B. Russell, “The problems of philosophy”, London: Williams and Norgate; New York: Henry Holt and Company, 1912. [2] E. Gettier, “Is justified true belief knowledge?” Analysis, vol. 23 (1963), no. 6, pp. 121–123. [3] S. Artemov and T. Protopopescu, “Discovering knowability: a semantic analysis”, Synthese, vol. 190 (2013), no. 16, pp. 3349–3376. [4] S. Artemov and T. Protopopescu, “Intuitionistic epistemic logic”, arXiv preprint arXiv:1406.1582, 2014.