On Formalizing Logical Modalities

This paper is in the scope of the philosophy of modal logic; more precisely, it concerns the semantics of modal logic, when the modal elements are interpreted as logical modalities. Most authors have thought that the logic for logical modality—that is, the one to be used to formalize the notion of logical truth (and other related notions)—is to be found among logical systems in which modalities are allowed to be iterated. This has raised the problem of the adequacy, to that formalization purpose, of some modal schemes, such as S4 and S5 . It has been argued that the acceptance of S5 leads to non-normal modal systems, in which the uniform substitution rule fails. The thesis supported in this paper is that such a failure is rather to be attributed to what will be called “Condition of internalization.” If this is correct, there seems to be no normal modal logic system capable of formalizing logical modality, even when S5 is rejected in favor of a weaker system such as S4, as recently proposed by McKeon.

Meta-inferences and Supervaluationism

Many classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. Thus, supervaluationists can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the consequence relation that preserves truth in a model of the non-normal modal logic NT. Finally, we extend our approach to a first-order setting and show that supervaluationism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented.

Outline of a theory of justification

Core theses of the novel account of justification to be developed are first stated: one has propositional justification for p just in case one is in no position to know that one is in no position to know p; and one has doxastic justification for p just in case one is in no position to know that one does not know p. Unlike other theories that conceive of justification in terms of the metaphysical possibility of knowing, the present account thus construes it as a distinctive kind of epistemic possibility. It treats propositional justification as non-factive, both its presence and its absence as luminous conditions, and by assuming a weak non-normal modal logic for knowledge and being in a position to know, validates principles of positive and negative introspection for it. The account thereby attributes features to justification that internalists care about. But it does so without construing justification as an internal condition. The account allows one to systematically distinguish between the condition of being justified and the metaphysical grounds for its obtaining, thereby heeding externalist insights into the difference between the good cases and the bad cases envisaged by radical scepticism. Lines of argument that show the account’s potential, e.g. in dealing with the preface and lottery paradoxes, are previewed, and so are lines of defence against challenges and objections, including prominent anti-luminosity arguments.

A CORRECT POLYNOMIAL TRANSLATION OF S4 INTO INTUITIONISTIC LOGIC

We show that the polynomial translation of the classical propositional normal modal logic S4 into the intuitionistic propositional logic Int from Fernández is incorrect. We give a modified translation and prove its correctness, and provide implementations of both translations to allow others to test our results.

A Defence of Information Closure

In this chapter, the principle of information closure (PIC) is defined and defended against a sceptical objection similar to the one discussed by Dretske in relation to the principle of epistemic closure. If successful, given that PIC is equivalent to the axiom of distribution and that the latter is one of the conditions that discriminate between normal and non-normal modal logics, one potentially good reason to look for a formalization of the logic of ‘S is informed that p’ among the non-normal modal logics, which reject the axiom, is also removed. This is not to argue that the logic of ‘S is informed that p’ should be a normal modal logic, but that it could still be, insofar as the objection that it could not be, based on the sceptical objection against PIC, has been removed. In other words, this chapter argues that the sceptical objection against PIC fails, so such an objection provides no ground to abandon the normal modal logic B (also known as KTB) as a formalization of ‘S is informed that p’, which remains plausible insofar as this specific obstacle is concerned.

BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.

Remarks about the unification type of several non-symmetric non-transitive modal logics

The problem of unification in a normal modal logic $L$ can be defined as follows: given a formula $\varphi$, determine whether there exists a substitution $\sigma$ such that $\sigma (\varphi )$ is in $L$. In this paper, we prove that for several non-symmetric non-transitive modal logics, there exists unifiable formulas that possess no minimal complete set of unifiers.

STABLE MODAL LOGICS

Stable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas.