Interpolation for first order S5

Inspired by an interesting quotation from the literature, we propose four modalities, called ‘sane belief’, ‘insane belief’, ‘reliable belief’ and ‘unreliable belief’, and introduce logics with each operator as the modal primitive. We show that the four modalities constitute a square of opposition, which indicates some interesting relationships among them. We compare the relative expressivity of these logics and other related logics, including a logic of false beliefs from the literature. The four main logics are all less expressive than the standard modal logic over various model classes, and the logics of sane and insane beliefs are, respectively, equally expressive as the logics of unreliable and reliable beliefs on any class of models. The logics of reliable and unreliable beliefs are then combined into a bimodal logic, which turns out to be equally expressive as the standard modal logic. Despite this, we cannot obtain a complete axiomatization of the minimal bimodal logic, by simply translating the axioms and rules of the minimal modal logic $\textbf{K}$ into the bimodal language. We then introduce a schematic modality which unifies reliable and unreliable beliefs and axiomatize it over the class of all frames and also the class of serial frames. This line of research is finally extended to unify sane and insane beliefs and some axiomatizations are given.

There has been a significant interest in modal logics with intersection, prominent examples including epistemic and doxastic logics with distributed knowledge, propositional dynamic logic with intersection, and description logics with concept intersection. Completeness proofs for such logics tend to be complicated, in particular on model classes such as S5 used, e.g., in standard epistemic logic, mainly due to the undefinability of intersection of modalities in standard modal logic. A standard proof method for the S5 case uses an “unraveling-folding” technique to achieve a treelike model to deal with the problem of undefinability. This method, however, is not easily adapted to other logics, due to its reliance on S5 in a number of steps. In this paper we demonstrate a simpler and more general proof technique by building a treelike canonical model directly, which avoids the complications in the processes of unraveling and folding. We illustrate the technique by showing completeness of the normal modal logics K, D, T, B, S4 and S5 extended with intersection modalities. Furthermore, these treelike canonical models are compatible with Fischer-Ladner-style closures, and we combine the methods to show the completeness of the mentioned logics further extended with transitive closure of union modalities known from PDL or epistemic logic. Some of these completeness results are new.

Reasoning about preferences is a fundamental component of rationality, and therefore central in AI and computational social choice. Most logic-based frameworks for representing and reasoning about preferences assume that preferences are arbitrarily given, typically as a ranking of a set of alternatives or using utilities, with little concern about how preferences are formed or where they come from. Recent work in rational choice theory, however, has devoted attention to giving more internal structure to the notion of rationality, focusing more on the faculties of individual agents, such as their mood, mindset, and motivating reasons. In this paper we develop a modal logic for reasoning about preferences that depend on a set of motivationally salient properties, based on recent work on reasons behind preferences by Dietrich and List. The main result is that we show how the problem of reasoning in this logic can be translated to reasoning in a standard modal logic (KT with universal modality), and consequently that reasoning systems and algorithms developed for modal logic (with universal modality) can be employed for reasoning about reason-based preferences.

In this paper I give conditions under which a matrix characterisation of validity is correct for first order logics where quantifications are restricted by statements from a theory. Unfortunately the usual definition of path closure in a matrix is unsuitable and a less pleasant definition must be used. I derive the matrix theorem from syntactic analysis of a suitable tableau system, but by choosing a tableau system for restricted quantification I generalise Wallen's earlier work on modal logics. The tableau system is only correct if a new condition I call “alphabetical monotonicity” holds. I sketch how the result can be applied to a wide range of logics such as first order variants of many standard modal logics, including non-serial modal logics.

14 Hybrid logics

Quantified modal inferences interest logicians, linguists, and computer scientists, but no previous psychological study of them appears to be in the literature. Here is an example of one: All those artists are businessmen. Paulo is possibly one of the artists. What follows? People tend to conclude: Paulo is possibly a businessman (Experiment 1). It seems plausible, and it follows from an intuitive mental model in which Paulo is one of a set of artists who are businessmen. Further deliberation can yield a model of an alternative possibility in which Paulo is not one of the artists, which confirms that the conclusion is only a possibility. The snag is that standard modal logics, which deal with possibilities, cannot yield a particular conclusion to any premises: Infinitely many follow validly (from any premises) but they do not include the present conclusion. Yet, further experiments corroborated a new mental model theory's predictions for various inferences (Experiment 2), for the occurrence of factual conclusions drawn from premises about possibilities (Experiment 3) and for inferences from premises of modal syllogisms (Experiment 4). The theory is therefore plausible, but we explore the feasibility of a cognitive theory based on modifications to modal logic.

In standard modal logic, □ ≡ ∼◊ ∼ and ◊ ≡ ∼□∼. I will, first, examine why in tense-logic, Arthur Prior thinks that ∼ ◊ ∼ is weaker than □ and ∼ □ ∼ is weaker than ◊. I will, then, examine whether there are similar motivations in modal logic to take ∼ ◊ ∼ to be weaker than □ and ∼ □ ∼ to be weaker than ◊. The upshot will be that, just as certain metaphysical views within the philosophy of time (e.g., Presentism) motivate one to deny the standard tense equivalences, certain metaphysical views within the metaphysics of modality (e.g., Contingentism, nonmodalism) motivate one to deny the standard modal equivalences.

In “Must ...stay ...strong!” (von Fintel and Gillies in Nat Lang Semant 18:351–383, 2010. https://doi.org/10.1007/s11050-010-9058-2), we set out to slay a dragon, or rather what we called The Mantra: that epistemic must has a modal force weaker than expected from standard modal logic, that it doesn’t entail its prejacent, and that the best explanation for the evidential feel of must is a pragmatic explanation. We argued that all three sub-mantras are wrong and offered an explanation according to which must is strong, entailing, and the felt indirectness is the product of an evidential presupposition carried by epistemic modals. Mantras being what they are, it is no surprise that each of the sub-mantras have been given new defenses. Here we offer them new problems and update our picture, concluding that must is (still) strong.

A key issue in the philosophy of biology is evolutionary contingency, the degree to which evolutionary outcomes could have been different. Contingency is typically contrasted with evolutionary convergence, where different evolutionary pathways result in the same or similar outcomes. Convergences are given as evidence against the hypothesis that evolutionary outcomes are highly contingent. But the best available treatments of contingency do not, when read closely, produce the desired contrast with convergence. Rather, they produce a picture in which any degree of contingency is compatible with any degree of convergence. This is because convergence is the repeated production of a given outcome from different starting points, and contingency has been defined without reference to the size of the space of possible outcomes. In small spaces of possibilities, the production of repeated outcomes is almost assured. This paper presents a definition of contingency which includes this modal dimension in a way that does not reduce it to the binary notion of contingency found in standard modal logic. The result is a conception of contingency which properly contrasts with convergence, given some domain of possibilities and a measure defined over it. We should therefore not ask whether evolution is contingent or convergent simpliciter, but rather about the degree to which it is contingent or convergent in various domains, as measured in various ways.

There are two typical ways to formalize reasoning over time. The first is in a modal logic where time is characterized implicitly through tense operators. The second addresses time explicitly in a first-order theory, where the reasoning is itself essentially first order. In this paper we shall introduce an amalgamated temporal logic which combines the tense operators of the modal approach with the referential aspects of the first-order approach.The modal operators of the new logic, called IQ-C, include the familiar tense operators plus a few new ones; together they allow complex temporal propositions to be represented in a compact form. Unlike standard modal temporal logics, IQ-C contains terms which refer to times; this allows the logic to characterize certain temporal properties and operators which cannot be axiomatized or defined by conventional modal operators. IQ-C is expressively complete over any connected flow of time, and has several other interesting theoretical properties, some of which will be developed in this paper.Most interval logics take one of two approaches to temporal assertions; either they analyze assertions for intervals in terms of assertions at points, or they treat intervals as primitive, allowing assertions for intervals but not at points. IQ-C allows assertions both for intervals and at points, and treats them as independent; this yields a special degree of semantic discrimination and at the same time simplifies the representation of several temporal properties. In this paper we shall outline the IQ-C framework, specifying its special features and defining its semantics. We shall also extend a method of skolemization, which is designed to support a resolution based approach to the proof theory. This will be sketched in some detail, with a view to providing a preliminary account of the basic proof theory. A complete specification of the deductive theory, together with relevant proofs, will be developed elsewhere.KeywordsIntervalsPointsTemporal Representation and ExpressivenessHomogeneitySkolemizationand Modal Theorem Proving

We propose a notion of focusing for nested sequent calculi for modal logics which brings down the complexity of proof search to that of the corresponding sequent calculi. The resulting systems are amenable to specifications in linear logic. Examples include modal logic \(\mathsf {K}\), a simply dependent bimodal logic and the standard non-normal modal logics. As byproduct we obtain the first nested sequent calculi for the considered non-normal modal logics.KeywordsModal LogicInference RuleModal RuleLinear LogicSequent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In classical quantificational logic, every individual constant is assigned a value from the domain of discourse, thus ensuring that every instance of \(\exists x(x=\alpha )\) is valid and so a theorem of a complete logic. Standard tense and modal logics validate a rule of necessitation, according to which, crudely, every theorem is always and necessarily true. Combining these two generates the result that everything always and necessarily exists. In a number of works from the late 1950s through to his death in 1969, Prior worked to develop tense and modal logics that avoided these results. Prior’s key idea was to reject the rule of necessitation. In this essay, I present and criticize Prior’s account of the contingency of existence and offer a more satisfying account that shares many of the virtues of Prior’s account.

Many authors have noted that a number of English modal sentences cannot be formalized into standard first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who’s actually rich could have been poor.” In response, many authors have introduced an “actually” operator @ into the language of first-order modal logic. It is occasionally noted that some of the example sentences still cannot be formalized with @ if one allows only actualist quantifiers, and embedded versions of these example sentences cannot be formalized even with possibilist quantifiers and @. The typical justification for these claims is to observe that none of the most plausible candidate formalizations succeed. In this paper, we prove these inexpressibility results by using a modular notion of bisimulation for first-order modal logic with “actually” and other operators. In doing so, we will explain in what ways these results do or do not generalize to more expressive modal languages.

The Semantics of Entailment

Beth's Theorem and Self-Referential Sentences