Intuitionistic Epistemic Logic, Kripke Models and Fitch’s Paradox

Proietti, Carlo

doi:10.1007/s10992-011-9207-1

Cited by 16 publications

(8 citation statements)

References 7 publications

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“…25 Intuitionistic responses to the paradox so far have accepted this informal interpretation of the Church-Fitch construction, and hence been committed to showing that an intuitionistic conception of truth and knowledge does not yield co-reflection, and hence is not committed to omniscience, e.g. [35,36,79,88,89,93,94,95,96]. We argue that the proper intuitionistic response is simply that there is no paradox; 26 intuitionistically the 'knowability paradox' is a pseudo-problem which holds only from a classical standpoint.…”

Section: Iel and Intuitionistic Responses To The Knowability Paradoxmentioning

confidence: 99%

“…A more recent development of the basic approach taken by DeVidi and Solomon [23] is found in Proietti [79], who develops an intuitionistic epistemic logic based on a Kripkean semantics.…”

Section: Proiettimentioning

confidence: 99%

“…For instance, Williamson (1992), in outlining his system of intuitionistic epistemic logic affirms that KA → A holds. Similarly Proietti (2012) argues that knowledge is factive in his system of intuitionistic epistemic logic. Wright states that an operator could not be a knowledge operator if it were not factive Wright (1994).…”

Section: Intuitionistic Reflection and The Truth Condition Onmentioning

confidence: 99%

“…The Church-Fitch 'knowability paradox' is an informal interpretation of a classical derivation in bimodal logic with the modalities K and 3 discovered by Church (2009) and reported by Fitch (1963) The fact that all truths are knowable is taken to be definitive of intuitionistic truth, hence the 'knowability paradox' appears to be a reductio ad absurdum of the very idea of intuitionism. 25 Intuitionistic responses to the paradox so far have accepted this informal interpretation of the Church-Fitch construction, and hence been committed to showing that an intuitionistic conception of truth and knowledge does not yield coreflection, and hence is not committed to omniscience, e.g., Proietti (2012); Dummett (2001Dummett ( , 2009; Tennant (1997Tennant ( , 2009; Williamson (1982Williamson ( , 1988Williamson ( , 1992Williamson ( , 1994. We argue that the proper intuitionistic response is simply that there is no paradox; 26 intuitionistically the 'knowability paradox' is a pseudoproblem which holds only from a classical standpoint.…”

Section: Modeling Knowledge Vs Beliefmentioning

confidence: 99%

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Intuitionistic Epistemic Logic

Artëmov

Protopopescu

2016

The Review of Symbolic Logic

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We outline an intuitionistic view of knowledge which maintains the original Brouwer-Heyting-Kolmogorov semantics for intuitionism and is consistent with the wellknown approach that intuitionistic knowledge be regarded as the result of verification. We argue that on this view co-reflection A → KA is valid and the factivity of knowledge holds in the form KA → ¬¬A 'known propositions cannot be false'.We show that the traditional form of factivity KA → A is a distinctly classical principle which, like tertium non datur A ∨ ¬A, does not hold intuitionistically, but, along with the whole of classical epistemic logic, is intuitionistically valid in its double negation form ¬¬(KA → A).Within the intuitionistic epistemic framework the knowability paradox is resolved in a constructive manner. We argue that this paradox is the result of an unwarranted classical reading of constructive principles and as such does not have the consequences for constructive foundations traditionally attributed it.

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Section: Iel and Intuitionistic Responses To The Knowability Paradoxmentioning

confidence: 99%

“…A more recent development of the basic approach taken by DeVidi and Solomon [23] is found in Proietti [79], who develops an intuitionistic epistemic logic based on a Kripkean semantics.…”

Section: Proiettimentioning

confidence: 99%

Section: Intuitionistic Reflection and The Truth Condition Onmentioning

confidence: 99%

Section: Modeling Knowledge Vs Beliefmentioning

confidence: 99%

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Intuitionistic Epistemic Logic

Artëmov

Protopopescu

2016

The Review of Symbolic Logic

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“…For other approaches to intuitionistic epistemic logic cf. alsoWilliamson [Wil92] or Proietti[Pro12] and for a more dynamic take, see Kurz and Palmigiano[KP13].31 Litak[Lit14] denotes ¬P⊥ as (nv)-non-verum.…”

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confidence: 99%

Lewis meets Brouwer: Constructive strict implication

Litak¹,

Visser

2018

Indagationes Mathematicae

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C. I. Lewis invented modern modal logic as a theory of "strict implication" . Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than P and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction has been discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of intuitionistic in terms of preservativity in extensions of HA, i.e., Heyting's Arithmetic. 65The basic option is to read § § 2-4 to get the basics of motivational background, the Kripke semantics and an impression of possible reasoning systems. ? The reader who wants more solid treatment of Kripke semantics can extend the basic option with § 6. | The computer science package consists of the basic option and § 7.= The reader who wants to go somewhat more deeply into the history of the subject can extend the basic option with Appendix D. « The reader who wants to understand the basics of arithmetical interpretations can extend the basic option with § 5. » An extended package for arithmetical interpretations combines « with § 8.-The full arithmetical package extends » with Appendices A, B and C. The rise and fall of the house of Lewis "The error of philosophers"We are reflecting on L.E.J. Brouwer's heritage half a century after his passing. Given his negative views on the rôle of logic and formalisms in mathematics, it seems somewhat paradoxical that these days the name of intuitionism survives mostly in the context of intuitionistic logic. 3 One is reminded in this context of what Nietzsche called the error of philosophers:The philosopher believes that the value of his philosophy lies in the whole, in the structure. Posterity finds it in the stone with which he built and with which, from that time forth, men will build oftener and better-in other words, in the fact that the structure may be destroyed and yet have value as material. 4

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