Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when (...) dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness. (shrink)

The aim of the paper is to present two natural deduction systems for Intuitionistic Sentential Calculus with Identity ( ISCI ); a syntactically motivated \(\mathsf {ND}^1_{\mathsf {ISCI}}\) and a semantically motivated \(\mathsf {ND}^2_{\mathsf {ISCI}}\). The formulation of \(\mathsf {ND}^1_{\mathsf {ISCI}}\) is based on the axiomatic formulation of ISCI. Its rules cannot be straightforwardly classified as introduction or elimination rules; ISCI -specific rules are based on axioms characterizing the identity connective. The system does not enjoy the standard subformula (...) property, but due to the normalization procedure non-subformulas can label only leaves of proofs. In \(\mathsf {ND}^2_{\mathsf {ISCI}}\), we propose only two general identity-related rules, in reference to the treatment of the identity connective in First-Order Logic. (shrink)

We prove that the intuitionistic sentential calculus is Ł-decidable, i.e. the sets of these of Int and of rejected formulas are disjoint and their union is equal to all formulas. A formula is rejected iff it is a sentential variable or is obtained from other formulas by means of three rejection rules. One of the rules is original, the remaining two are Łukasiewicz's rejection rules: by detachement and by substitution. We extensively use the method of Beth's (...) semantic tableaux. (shrink)

Intuitionistic sentential logic is shown to be sound and complete with respect to a semantics centered around extensions of atomic bases (i.e. sets of inference rules for atomic sentences). The result is made possible through a non-standard interpretation of disjunction, whereby, roughly speaking, a disjunction is taken to hold just in case every atomic sentence that follows from each of the disjuncts separately holds; it is argued that this interpretation makes good sense provided that rules in atomic bases (...) are conceived of as being accepted hypothetically rather than categorically. (shrink)

In this paper we obtain a finite Hilbert-style axiomatization of the implicationless fragment of the intuitionistic propositional calculus. As a consequence we obtain finite axiomatizations of all structural closure operators on the algebra of {–}-formulas containing this fragment.

An assignment is a function f that assigns subsets of N to some atoms. Then f is extended to f* which sends every formula A of HPC to a subset of S(A).

R. Suszko’s Sentential Calculus with Identity \( SCI \) results from classical propositional calculus \( CPC \) by adding a new connective \(\equiv \) and axioms for identity \(\varphi \equiv \psi \) (which we interpret here as ‘propositional identity’). We reformulate the original semantics of \( SCI \) using Boolean prealgebras which, introduced in different ways, are known in the literature as structures for the modeling of (hyper-) intensional semantics. We regard intensionality here as a measure for (...) the discernibility of propositions (and hyperintensionality as a high degree of intensionality). As concrete examples of \( SCI \) -based intensional modeling, we review and study algebraic semantics of some Lewis-style modal logics in the vicinity of \( S3 \) and present conditions under which those modal systems can be restored, in a precise sense, as certain axiomatic extensions of \( SCI \). This generalizes work of Suszko which is focused on the modal systems \( S4 \) and \( S5 \). Our approach is particularly intended as a proposal to consider and to further study \( SCI \) (and its extensions) as a general framework for the modeling of (hyper-) intensional semantics. (shrink)

With each superintuitionistic propositional logic L with a disjunction property we associate a set of modal logics the assertoric fragment of which is L . Each formula of these modal logics is interdeducible with a formula representing a set of rules admissible in L . The smallest of these logics contains only formulas representing derivable in L rules while the greatest one contains formulas corresponding to all admissible in L rules. The algebraic semantic for these logics is described.

Questions connected with the admissibility of rules of inference and the solvability of the substitution problem for modal and intuitionistic logic are considered in an algebraic framework. The main result is the decidability of the universal theory of the free modal algebra imageω extended in signature by adding constants for free generators. As corollaries we obtain: there exists an algorithm for the recognition of admissibility of rules with parameters in the modal system Grz, the substitution problem for Grz and (...) for the intuitionistic calculus H is decidable, intuitionistic propositional calculus H is decidable with respect to admissibility . A semantical criterion for the admissibility of rules of inference in Grz is given. (shrink)

Explicit description of maps definable by formulæ of the second order intuitionistic propositional calculus is given on two classes of linear Heyting algebras—the dense ones and the ones which possess successors. As a consequence, it is shown that over these classes every formula is equivalent to a quantifier free formula in the dense case, and to a formula with quantifiers confined to the applications of the successor in the second case.

The intention of the presented paper is to establish within a certain modal semantic based on the situational ontology a description of the phenomenon of substantial change, which originally had been formulated within Aristotelian metaphysics – a theory based in reistic ontology. We understand substantial changesto be such changes whose subjects are primary substances (πρωται ουσι αι ) conceived as actually existing individual essences. The analysed changeability is of an existential character - it pertains to the existence of those substances. (...) The formalised description of the chosen changes, constructed in part (1) of the study, takes into consideration the necessary condition for their occurrence: the disappearing and coming into being of successively becoming substances. Part (2) presents the manner in which it is possible to reformulate the obtained characteristic of substantial change in order to create a description of a particular variety of situational changes, which belong, in turn, to the intended interpretation of modal sentential logic of change LCG - a calculus in which change is a primitive concept precised with the help of two one-argument operators C and G, read respectively as: it changes that…, a change is enabled by this, that ….. (shrink)