scholarly journals Super-Strict Implications

2021 ◽  
Author(s):  
Eugenio Orlandelli ◽  
Guido Gherardi
Keyword(s):  
Modal Logics ◽  
Relational Semantics ◽  
Strict Implication ◽  
Material Implication ◽  
Structural Rules ◽  
Relational Frames

This paper introduces the logics of super-strict implications, where  a super-strict implication is  a strengthening of  C.I. Lewis' strict implication that avoids not only the paradoxes of material implication but also those of strict implication. The semantics of super-strict implications is obtained by strengthening the (normal) relational semantics for strict implication. We consider all logics of super-strict implications that are based on relational frames for modal logics in the  modal cube. it is shown that all  logics of super-strict implications are connexive logics in that they validate Aristotle's Theses and (weak) Boethius's Theses. A proof-theoretic characterisation of logics of super-strict implications is given by means of G3-style labelled calculi, and it is proved that the structural rules of inference are admissible in these calculi. It  is also shown that validity in the $$\mathsf{S5}$$-based logic of super-strict implications is equivalent to validity in  G. Priest's negation-as-cancellation-based  logic. Hence, we also   give a cut-free calculus for Priest's logic.

Extensions of the Lewis system S5

1951 ◽  
Vol 16 (2) ◽  
pp. 112-120 ◽  
Author(s):  
Schiller Joe Scroggs
Keyword(s):  
Broad Class ◽  
Characteristic Matrix ◽  
Normal Extension ◽  
Strict Implication ◽  
Finite Characteristic ◽  
Material Implication ◽  
Sentential Calculus ◽  
The Third ◽  
Simple Class ◽  
Definition Of

Dugundji has proved that none of the Lewis systems of modal logic, S1 through S5, has a finite characteristic matrix. The question arises whether there exist proper extensions of S5 which have no finite characteristic matrix. By an extension of a sentential calculus S, we usually refer to any system S′ such that every formula provable in S is provable in S′. An extension S′ of S is called proper if it is not identical with S. The answer to the question is trivially affirmative in case we make no additional restrictions on the class of extensions. Thus the extension of S5 obtained by adding to the provable formulas the additional formula p has no finite characteristic matrix (indeed, it has no characteristic matrix at all), but this extension is not closed under substitution—the formula q is not provable in it. McKinsey and Tarski have defined normal extensions of S4* by imposing three conditions. Normal extensions must be closed under substitution, must preserve the rule of detachment under material implication, and must also preserve the rule that if α is provable then ~◊~α is provable. McKinsey and Tarski also gave an example of an extension of S4 which satisfies the first two of these conditions but not the third. One of the results of this paper is that every extension of S5 which satisfies the first two of these conditions also satisfies the third, and hence the above definition of normal extension is redundant for S5. We shall therefore limit the extensions discussed in this paper to those which are closed under substitution and which preserve the rule of detachment under material implication. These extensions we shall call quasi-normal. The class of quasi-normal extensions of S5 is a very broad class and actually includes all extensions which are likely to prove interesting. It is easily shown that quasi-normal extensions of S5 preserve the rules of replacement, adjunction, and detachment under strict implication. It is the purpose of this paper to prove that every proper quasi-normal extension of S5 has a finite characteristic matrix and that every quasi-normal extension of S5 is a normal extension of S5 and to describe a simple class of characteristic matrices for S5.

scholarly journals Dynamic sequent calculus for the logic of Epistemic Actions and Knowledge

10.29007/mwpp ◽  
2018 ◽  
Author(s):  
Giuseppe Greco ◽  
Alexander Kurz ◽  
Alessandra Palmigiano
Keyword(s):  
Sequent Calculus ◽  
Relational Semantics ◽  
Sequent Calculi ◽  
Structural Rules ◽  
Epistemic Actions ◽  
Dynamic Logics ◽  
Display Calculi

We develop a family of display-style, cut-free sequent calculi for dynamic epistemic logics on both an intuitionistic and a classical base. Like the standard display calculi, these calculi are modular: just by modifying the structural rules according to Dosen’s principle, these calculi are generalizable both to different Dynamic Logics (Epistemic, Deontic, etc.) and to different propositional bases (Linear, Relevant, etc.). Moreover, the rules they feature agree with the standard relational semantics for dynamic epistemic logics.

Modal Logics of Strict Implication

2000 ◽  
pp. 117-169
Author(s):  
Dov M. Gabbay ◽  
Nicola Olivetti
Keyword(s):  
Modal Logics ◽  
Strict Implication

scholarly journals Pooling Modalities and Pointwise Intersection: Semantics, Expressivity, and Dynamics

2022 ◽  
Author(s):  
Frederik Van De Putte ◽  
Dominik Klein
Keyword(s):  
Expressive Power ◽  
Modal Logics ◽  
Relational Semantics ◽  
Modal Operators

AbstractWe study classical modal logics with pooling modalities, i.e. unary modal operators that allow one to express properties of sets obtained by the pointwise intersection of neighbourhoods. We discuss salient properties of these modalities, situate the logics in the broader area of modal logics (with a particular focus on relational semantics), establish key properties concerning their expressive power, discuss dynamic extensions of these logics and provide reduction axioms for the latter.

On the interpretation of the sign ‘⊃’

1953 ◽  
Vol 18 (1) ◽  
pp. 60-62 ◽  
Author(s):  
John Myhill
Keyword(s):  
The Other ◽  
Strict Implication ◽  
Material Implication ◽  
Strict Interpretation ◽  
False Proposition ◽  
Positive Logic ◽  
Law Of Excluded Middle ◽  
Image Position ◽  
Formal Fallacies ◽  
Excluded Middle

The sign ‘⊃’ (or ‘→’ or ‘C’) functions in many logical systems in a way which precludes its interpretation as either strict or material implication. For example, in the systems of Heyting, Johansson, Fitch and Bernays (positive logic), the following are theorems:Now if ‘⊃’ were interpreted as strict implication, ⊃2 would mean ‘if p is true, then p is strictly implied by every proposition’, i.e. ‘if p is true, it is necessarily true’, which is false for contingently true p. If on the other hand ‘⊃’ were interpreted as material implication, ⊃1 would reduce to ‘~p ∨ p’, i.e. to the law of excluded middle, which is conspicuously lacking in the systems mentioned. The reader is likely in practice to veer between these two interpretations. Thus in Fitch or Heyting on realizing that ‘~p⊃▪ p⊃q’ is a theorem, one thinks of it as meaning ‘a false proposition implies everything’ and regards the implication as material; but the presence of ‘p⊃p’ as a theorem, even for choices of p which do not satisfy excluded middle, inclines one again to the strict interpretation. This vacillation, while it need not lead to the commission of any formal fallacies, tends to hamstring one's intuition and thus waste time. The purpose of this paper is to suggest an interpretation of ‘⊃’ which will prevent such havering.Let two formulae A and B be called interdeducible if A ⊢ B and B ⊢ A.

An incomplete nonnormal extension of S3

1978 ◽  
Vol 43 (2) ◽  
pp. 211-212
Author(s):  
George F. Schumm
Keyword(s):  
Modus Ponens ◽  
Modal Logics ◽  
Relational Semantics ◽  
General Semantics ◽  
Normal Logic ◽  
Enormous Number

Fine [1] and Thomason [4] have recently shown that the familiar relational semantics of Kripke [2] is inadequate for certain normal extensions of T and S4. It is here shown that the more general semantics developed by Kripke in [3] to handle nonnormal modal logics is likewise inadequate for certain of those logics.The interest of incompleteness results, such as those of Fine and Thomason, is of course a function of one's expectations. Define a “normal” logic too broadly and it is not surprising that a given semantics is not adequate for all normal logics. In the case of relational semantics, for example, one would want to require at least that a normal logic contain T, the logic determined by the class of all normal frames, and that it be closed under certain (though perhaps not all) rules of inference which are validity preserving in such frames. The adequacy of that semantics will otherwise be ruled out at the outset.For Kripke a logic is normal if it contains all tautologies, □p→p and □ (p → q)→(□p → □q), and is closed under the rules of substitution, modus ponens and necessitation (from A infer □A). T is the smallest normal logic, and this fact, together with the “naturalness” of the definition and the enormous number of normal logics which have been shown to be complete, made it plausible to suppose that Kripke's original semantics was adequate for all normal logics. That it is not is indeed surprising and would seem to reveal a genuine shortcoming.

Independent axiom schemata for S5

1956 ◽  
Vol 21 (3) ◽  
pp. 255-256
Author(s):  
Alan Ross Anderson
Keyword(s):  
Strict Implication ◽  
Material Implication ◽  
Independent Axiom ◽  
Image Position

Leo Simons has shown that H1—H6 below constitute a set of independent axiom schemata for S3, with detachment for material implication “→” as the only primitive rule. He also showed that addition of the scheme (◇ ◇ α ⥽ ◇ α) yields S4, and that these schemata for S4 are independent. The question for S5 was left open. We shall show (presupposing familiarity with Simons' paper) that H1—H6 and S, below, constitute a set of independent axiom schemata for S5, with detachment for material implication as the only primitive rule.Let S5′ be the system generated from H1—H6 and S with the help of the primitive rule. It is easy to see that Simons' derivations of the rules (a) adjunction, (b) detachment for strict implication, and (c) intersubstitutability of strict equivalents, may be carried out for S5′. We know that (1) (∼ ◇ ∼ α ⥽ ◇ α) is provable in S2, hence also in S3 and S5′; and (1) and S yield (2) (α ⥽ ∼ ◇ ∼ ◇ α). Perry has shown that addition of (2) to S3 yields S5, so S5 is a subsystem of S5′. And it is easy to prove S in S5; hence the systems are equivalent.

An extended joint consistency theorem for a family of free modal logics with equality

1984 ◽  
Vol 49 (1) ◽  
pp. 174-183 ◽  
Author(s):  
Raymond D. Gumb
Keyword(s):  
Infinite Family ◽  
Modal Logics ◽  
Relational Semantics ◽  
Accessibility Relation ◽  
Barcan Formula ◽  
Distinct Individual ◽  
Individual Variables

In this paper, we establish an extended joint consistency theorem for an infinite family of free modal logics with equality. The extended joint consistency theorem incorporates the Craig and Lyndon interpolation lemmas and the Robinson joint consistency theorem. In part, the theorem states that two theories which are jointly unsatisfiable are separated by a sentence in the vocabulary common to both theories.Our family of free modal logics includes the free versions of I, M, and S4 studied by Leblanc [5, Chapters 8 and 9], supplemented with equality as in [3]. In the relational semantics for these logics, there is no restriction on the accessibility relation in I, while in M(S4) the restriction is reflexivity (refiexivity and transitivity). We say that a restriction on the accessibility relation countenances backward-looping if it implies a sentence of the form ∀x1 …xn(x1Rx2 &…&xn ⊃ xkRxj) (1 ≤ j < k ≤ n ≥ 2), where the xi (1 ≤ i ≤ n) are distinct individual variables. Just as reflexivity and transitivity do not countenance backward-looping, neither do any of the restrictions in our family of free modal logics. (The above terminology is derived from the effect of such restrictions on Kripke tableaux constructions.) The Barcan formula, its converse, the Fitch formula, and the formula T ≠ T′ ⊃ □T ≠ T′ do not hold in our logics.

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