$${\in_K}$$ : a Non-Fregean Logic of Explicit Knowledge

Lewitzka, Steffen

doi:10.1007/s11225-011-9304-8

Cited by 13 publications

(49 citation statements)

References 10 publications

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“…We finish this section with a discussion on self-referential propositions. Having in mind the intended meaning of the identity connective (see Lemma 3.15), we are able to express self-referential statements by means of equations (this kind of modeling self-reference was proposed in [17] and subsequently used in, e.g., [19,10,11]). For instance, the equation (12) x ≡ (x → ⊥) defines a version of the liar proposition.…”

Section: Axiomatization and Algebraic Semanticsmentioning

confidence: 99%

“…All biconditionals of the form(10) ϕ ↔ (ϕ ≡ ⊤) are theorems of our modal logics.Proof. Of course, ϕ → ⊤ is a theorem of IPC.…”

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confidence: 99%

“…This along with (Id2) yields (ϕ ≡ ψ) → ( (ϕ ↔ ϕ) → (ϕ ↔ ψ)). Since (ϕ ↔ ϕ) is a theorem, also (ϕ ≡ ψ) → (ϕ ↔ ψ)is a theorem 10. Note, however, that the expression ϕ ≡ ⊤ is defined in terms of , i.e.…”

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confidence: 99%

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Reasoning about proof and knowledge

Lewitzka

2019

Annals of Pure and Applied Logic

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In previous work [Lewitzka, Log. J. IGPL 2017], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in [Lewitzka, J. Log. Comp. 2015]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL [Artemov and Protopopescu, Rev. Symb. Log. 2016]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer-Heyting-Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verificationbased intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems.1 These natural axioms ensure that the identity connective ≡ is a congruence relation modulo any given theory. We read ϕ ≡ ψ as "ϕ and ψ have the same meaning (denotation, Bedeutung)" or "ϕ and ψ denote the same proposition". We strictly distinguish between formulas (syntactical objects) and propositions (semantic entities): a formula denotes a proposition. This is in accordance with our non-Fregean, intensional, view on logics: the semantics of a formula is, in general, more than its truth value. The Fregan Axiom (ϕ ↔ ψ) → (ϕ ≡ ψ) is not valid (see [18]).

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Section: Axiomatization and Algebraic Semanticsmentioning

confidence: 99%

“…All biconditionals of the form(10) ϕ ↔ (ϕ ≡ ⊤) are theorems of our modal logics.Proof. Of course, ϕ → ⊤ is a theorem of IPC.…”

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confidence: 99%

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Reasoning about proof and knowledge

Lewitzka

2019

Annals of Pure and Applied Logic

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“…Many further properties of knowledge and belief, such as the distribution law K(ϕ → ψ) → (Kϕ → Kψ), can be modeled semantically by imposing suitable closure conditions on the set BEL of each model. 4 Special attention deserves the bridge theorem ϕ → Kϕ which establishes the connection between the modal and the epistemic part of the logic. Actually, we will need a slightly stronger bridge axiom, namely ϕ → Kϕ, in order to warrant that the axioms of propositional identity, in particular (Id3), also hold in the extended epistemic language (see Lemma 2.1).…”

Section: Introductionmentioning

confidence: 99%

“…In[4] are modeled also more complex epistemic concepts such as common knowledge in a group of agents.…”

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confidence: 99%

Epistemic extensions of combined classical and intuitionistic propositional logic

Lewitzka

2017

Logic Journal of the IGPL

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Logic L was introduced by Lewitzka [7] as a modal system that combines intuitionistic and classical logic: L is a conservative extension of CPC and it contains a copy of IPC via the embedding ϕ → ϕ. In this article, we consider L3, i.e. L augmented with S3 modal axioms, define basic epistemic extensions and prove completeness w.r.t. algebraic semantics. The resulting logics combine classical knowledge and belief with intuitionistic truth. Some epistemic laws of Intuitionistic Epistemic Logic studied by Artemov and Protopopescu [1] are reflected by classical modal principles. In particular, the implications "intuitionistic truth ⇒ knowledge ⇒ classical truth" are represented by the theorems ϕ → Kϕ and Kϕ → ϕ of our logic EL3, where we are dealing with classical instead of intuitionistic knowledge. Finally, we show that a modification of our semantics yields algebraic models for the systems of Intuitionistic Epistemic Logic introduced in [1]. 1 Instead of (Id3), Suszko presents a set of three axioms which together are equivalent to (Id3). By a proposition we mean the denotation of a formula. Instead of proposition, Suszko uses the term situation.2 Certain connections between non-Fregean logic and the modal systems S4 and S5 were already investigated by Suszko and Bloom (see [2,9]). 3 Recall that there is no known intuitive semantics for S1.

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Tableau-based Decision Procedure for Non-Fregean Logic of Sentential Identity

Golińska-Pilarek

Huuskonen

Zawidzki

2021

Automated Deduction – CADE 28

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Sentential Calculus with Identity ($$\mathsf {SCI}$$ SCI ) is an extension of classical propositional logic, featuring a new connective of identity between formulas. In $$\mathsf {SCI}$$ SCI two formulas are said to be identical if they share the same denotation. In the semantics of the logic, truth values are distinguished from denotations, hence the identity connective is strictly stronger than classical equivalence. In this paper we present a sound, complete, and terminating algorithm deciding the satisfiability of $$\mathsf {SCI}$$ SCI -formulas, based on labelled tableaux. To the best of our knowledge, it is the first implemented decision procedure for $$\mathsf {SCI}$$ SCI which runs in NP, i.e., is complexity-optimal. The obtained complexity bound is a result of dividing derivation rules in the algorithm into two sets: decomposition and equality rules, whose interplay yields derivation trees with branches of polynomial length with respect to the size of the investigated formula. We describe an implementation of the procedure and compare its performance with implementations of other calculi for $$\mathsf {SCI}$$ SCI (for which, however, the termination results were not established). We show possible refinements of our algorithm and discuss the possibility of extending it to other non-Fregean logics.

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$${\in_K}$$ : a Non-Fregean Logic of Explicit Knowledge

Cited by 13 publications

References 10 publications

Reasoning about proof and knowledge

Reasoning about proof and knowledge

Epistemic extensions of combined classical and intuitionistic propositional logic

Tableau-based Decision Procedure for Non-Fregean Logic of Sentential Identity

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