Sequent Calculi for Orthologic with Strict Implication

In this study, new sequent calculi for a minimal quantum logic (\(\bf MQL\)) are discussed that involve an implication. The sequent calculus \(\bf GO\) for \(\bf MQL\) was established by Nishimura, and it is complete with respect to ortho-models (O-models). As \(\bf GO\) does not contain implications, this study adopts the strict implication and constructs two new sequent calculi \(\mathbf{GOI}_1\) and \(\mathbf{GOI}_2\) as the expansions of \(\bf GO\). Both \(\mathbf{GOI}_1\) and \(\mathbf{GOI}_2\) are complete with respect to the O-models. In this study, the completeness and decidability theorems for these new systems are proven. Furthermore, some details pertaining to new rules and the strict implication are discussed.

Another way to look at counterfactuals

Counterfactuals such as If the world did not exist, we would not notice it have been a challenge for philosophers and linguists since antiquity. There is no generally accepted semantic analysis. The prevalent view, developed in varying forms by Robert Stalnaker, David Lewis, and others, enriches the idea of strict implication by the idea of a “minimal revision” of the actual world. Objections mainly address problems of maximal similarity between worlds. In this paper, I will raise several problems of a different nature and draw attention to several phenomena that are relevant for counterfactuality but rarely discussed in that context. An alternative analysis that is very close to the linguistic facts is proposed. A core notion is the “situation talked about”: it makes little sense to discuss whether an assertion is true or false unless it is clear which situation is talked about. In counterfactuals, this situation is marked as not belonging to the actual world. Typically, this is done in the form of the finite verb in the main clause. The if-clause is optional and has only a supportive role: it provides information about the world to which the situation talked about belongs. Counterfactuals only speak about some nonactual world, of which we only know what results from the protasis. In order to judge them as true or false, an additional assumption is required: they are warranted according to the same criteria that warrant the corresponding indicative assertion. Overall similarity between worlds is irrelevant.

Super-Strict Implications

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.

From intuitionism to Brouwer’s modal logic

We try to translate the intuitionistic propositional logic INT into Brouwer’s modal logic KTB. Our translation is motivated by intuitions behind Brouwer’s axiom p →☐◊p as discussed in [4]. The main idea is to interpret intuitionistic implication as modal strict implication, whereas variables and other positive sen-tences remain as they are. The proposed translation preserves fragments of the Rieger-Nishimura lattice which is the Lindenbaum algebra of monadic formulas in INT. Unfortunately, INT is not embedded by this mapping into KTB.

Lyndon’s interpolation property for the logic of strict implication

The main result proves Lyndon’s and Craig’s interpolation properties for the logic of strict implication ${\textsf{F}}$, with a purely syntactical method. A cut-free G3-style sequent calculus $ {\textsf{GF}} $ and its single-succedent variant $ \textsf{GF}_{\textsf{s}} $ are introduced. $ {\textsf{GF}} $ can be extended to a G3-variant of the sequent calculus GBPC3 for Visser’s basic logic. Also a simple syntactic proof of known embedding result of $ {\textsf{F}} $ into $ {\textsf{K}} $ is provided. An extension of $ {\textsf{F}} $, namely $ \textsf{FD}, $ is considered as well.