Ordinary Modal Logic Research Articles

Several significant equalities on intuitionistic fuzzy operators

There are various operators and operations in intuitionistic fuzzy set theory. The roles of these operators and operations are very significant as they show a deeper interconnection between the two ordinary modal logic operators. It can be well noted that modal operators can change intuitionistic fuzzy sets into fuzzy sets easily. Considering all of these, we establish several equalities on intuitionistic fuzzy sets.

Free choice disjunction as a rational speech act

The so-called free choice inference (from You may take an apple or a pear to You may take an apple) is mysterious because it does not follow from ordinary modal logic. We show that this inference arises in the Rational Speech Act framework (Frank & Goodman 2012). Our basic idea is inspired by exhaustification-based models of free choice (Fox 2007) and by game-theoretic accounts based on iterated best response (Franke 2011). We assume that when the speaker utters You may take an apple or a pear, the hearer reasons about why the speaker did not choose alternative utterances such as You may take an apple. A crucial ingredient in our explanation is the idea of semantic uncertainty (Bergen, Levy & Goodman 2016). Specifically, we assume that the speaker is uncertain whether or not the hearer will interpret You may take an apple as forbidding them from taking a pear. This uncertainty can be thought of as resulting from Fox’s (2007) optional covert exhaustification. Uttering the disjunction is a way for the speaker to prevent the hearer from concluding that any fruit is forbidden to take. Knowing this, the hearer concludes that they may choose either fruit.

A Critical Examination of the Historical Origins of Connexive Logic

It is often assumed that Aristotle, Boethius, Chrysippus, and other ancient logicians advocated a connexive conception of implication according to which no proposition entails, or is entailed by, its own negation. Thus Aristotle claimed that the proposition ‘if B is not great, B itself is great […] is impossible’. Similarly, Boethius maintained that two implications of the type ‘If p then r’ and ‘If p then not-r’ are incompatible. Furthermore, Chrysippus proclaimed a conditional to be ‘sound when the contradictory of its consequent is incompatible with its antecedent’, a view which, in the opinion of S. McCall, entails the aforementioned theses of Aristotle and Boethius. Now a critical examination of the historical sources shows that the ancient logicians most likely meant their theses as applicable only to ‘normal’ conditionals with antecedents which are not self-contradictory. The corresponding restrictions of Aristotle’s and Boethius’ theses to such self-consistent antecedents, however, turn out to be theorems of ordinary modal logic and thus don’t give rise to any non-classical system of genuinely connexive logic.

Why does the proof-theory of hybrid logic work so well?

This is primarily a conceptual paper. The goal of the paper is to put into perspective the proof-theory of hybrid logic and in particular, try to give an answer to the following question: Why does the proof-theory of hybrid logic work so well compared to the proof-theory of ordinary modal logic? Roughly, there are two different kinds of proof systems for modal logic: Systems where the formulas involved in the rules are formulas of the object language, that is, ordinary modal-logical formulas, and systems where the formulas involved in the rules are metalingustic formulas obtained by attaching labels representing possible worlds to ordinary modal-logical formulas. Systems of the second kind often also involve an explicit representation of the accessibility relation. From a proof-theoretic point of view, modal-logical systems of the first kind are less well-behaved than systems of the second kind. It turns out that this can be remedied by hybridization, that is, hybridization of modal logics enables the formulation of well-behaved proof systems without involving metalinguistic machinery. What has happened is that the metalinguistic machinery has been internalized in the object language. This gives an answer to the initial question, which is that the proof-theory of hybrid logic works so well because the metalinguistic semantic machinery has been internalized in the object language.

Proof-Theoretic Functional Completeness for the Hybrid Logics of Everywhere and Elsewhere

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.

Natural Deduction for Hybrid Logic

In this paper we give a natural deduction formulation of hybrid logic. Our natural deduction system can be extended with additional inference rules corresponding to conditions on the accessibility relations expressed by so-called geometric theories. Thus, we give natural deduction systems in a uniform way for a wide class of hybrid logics which appears to be impossible in the context of ordinary modal logic. We prove soundness and completeness and we prove a normalization theorem. We finally prove a result which says that normal derivations in the natural deduction system correspond to derivations in a cut-free Gentzen system.

Compatible complement in Piron’s system and ordinary modal logic

Previous critiques of the relevant of Piron’s questions-propositions system as a generated of quantum mechanics by interpretation are reinforced by confrontation with the de Morgan laws and with the ordinary modal logic.

Talk about Knowing

In this article I wish to suggest a new approach to an old question, the question whether knowing something entails the impossibility of being mistaken about its truth. It will lead to a negative answer to the question, but by a line of argument which will also contain a possible (and I think neglected) explanation why an affirmative answer is so tempting. I shall assume that knowledge is based on evidence, or (to speak more neutrally) restrict myself to such knowledge. The nature and cogency of the evidence is only indirectly my concern; so the term 'conclusive' will be used to indicate neither more nor less than the fact that someone's evidence for p is sufficient evidence for him to know p. What is being asked is whether, if someone has such evidence (and, let us say, infers p) he can falsely believe the truth of p. This is not always the same as asking whether p can, still less asking whether it could, be false, for that might be asking merely whether it was contingent. But of course 'it may be false that p' commonly is equivalent to may be wrong', namely when neither the first is meant as the trivial consequence of 'p is contingent' nor the second as that of 'you are right is contingent'. Certainly it is a necessary truth that if 'p' is known 'p' is true; and no doubt some people have confused this with a claim that if 'p' is known 'p' is a necessary truth.' But I hope to shew that there is more behind the demand for conclusiveness to exclude the possibility of error, and hence for a distinction between knowing and any kind of believing, than that confusion. The reason is that the rules for 'must be' and 'may be' in ordinary discourse do not coincide with those for the necessity and possibility of ordinary modal logic.2 That on the contrary they sometimes represent 'epistemic' modalities has of course been claimed by various philosophers (for instance, Frege), and everyone is indebted to Jaakko Hintikka for a systematisation of them.3 In this article the purpose is different. It is to shew how the usage of ordinary discourse does differ from that of modal logic without having to represent epistemic modalities. It is that difference, independent of any particular concepts of knowledge, which bears on our question of the relation of conclusive evidence to infallibility. So as to make the material clear it is presented not as reports of what can be thought about some situation involving or thought to involve knowledge and evidence but as explicit expressions of such thoughts by participants in the situation. To give the simplest of examples, we are going to be presented with conversations in which somebody says that

Application of modal logic to programming

The modal logician's notion of possible world and the computer scientist's notion of state of a machine provide a point of commonality which can form the foundation of a logic of action. Extending ordinary modal logic with the calculus of binary relations leads to a very natural logic for describing the behavior of computer programs.

In so many possible worlds.

Ordinary modal logic deals with the notion of a proposition being true in at least one possible world. This makes it natural to consider the notion of a proposition being true in n possible worlds for any nonnegative integer n. Such a notion would stand to Tarski’s numerical quantifiers as ordinary possibility stands to the existential quantifier. In this paper I present several logics for numerical possibility. First I give the syntax and semantics for a minimal such logic (sections 1 and 2); then I prove its completeness (sections 3 and 4); and finally I show how to extend this result to other logics (section 5).