Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

Many languages have past-and-counterfactuality markers such as English simple past. There have been various attempts to find a common definition for both uses, but I will argue in this paper that they all have problems with ruling out unacceptable interpretations, or accounting for the contrary-to-fact implicature of counterfactual conditionals, or predicting the observed cross-linguistic variation, or a combination thereof. By combining insights from two basic lines of reasoning, I will propose a simple and transparent approach that solves all the observed (...) problems and offers a new understanding of the concept of counterfactuality. (shrink)

The epistemology of modality has focused on metaphysical modality and, more recently, counterfactual conditionals. Knowledge of kinds of modality that are not metaphysical has so far gone largely unexplored. Yet other theoretically interesting kinds of modality, such as nomic, practical, and ‘easy’ possibility, are no less puzzling epistemologically. Could Clinton easily have won the 2016 presidential election—was it an easy possibility? Given that she didn’t in fact win the election, how, if at all, can we know whether she easily could (...) have? This paper investigates the epistemology of the broad category of ‘objective’ modality, of which metaphysical modality is a special, limiting case. It argues that the same cognitive mechanisms that are capable of producing knowledge of metaphysical modality are also capable of producing knowledge of all other objective modalities. This conclusion can be used to explain the roles of counterfactual reasoning and the imagination in the epistemology of objective modality. (shrink)

The purpose of the paper is to rethink the role of actuality in the branching model of possibilities. We investigate the idea that the model should be enriched with an additional factor—the so-called Thin Red Line—which is supposed to represent the single possible course of events that gets actualized in time. We believe that this idea was often misconceived which prompted some unfortunate reactions. On the one hand, it suggested problematic semantic models of future tense and and on the other, (...) it provoked questionable lines of criticism. We reassess the debate and point to potential pitfalls, focusing on the semantic dimension of the Thin Red Line theory. Our agenda transcends the semantics, however. We conclude that semantic considerations do not threaten the Thin Red Line theory and that the proper debate should be carried in the domain of metaphysics. (shrink)

Can Quine’s criterion for ontological commitment be comparatively applied across different logics? If so, how? Cross-logical evaluations of discourses are central to contemporary philosophy of mathematics and metaphysics. The focus here is on the influential and important arguments of George Boolos and David Lewis that second-order logic and plural quantification don’t incur additional ontological commitments over and above those incurred by first-order quantifiers. These arguments are challenged by the exhibition of a technical tool—the truncation-model construction of notational equivalents—that compares the (...) ontological role and increased expressive strength of non-first-order ideology to first-order ideology. (shrink)

A comparison is given between two conditions used to define logical constants: Belnap's uniqueness and Hacking's deducibility of identicals. It is shown that, in spite of some surface similarities, there is a deep difference between them. On the one hand, deducibility of identicals turns out to be a weaker and less demanding condition than uniqueness. On the other hand, deducibility of identicals is shown to be more faithful to the inferentialist perspective, permitting definition of genuinely proof-theoretical concepts. This kind of (...) analysis is driven by exploiting the Curry–Howard correspondence. In particular, deducibility of identicals is shown to correspond to the computational property of eta expansion, which is essential in the characterization of propositional identity. (shrink)

We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new fine-structure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and conditional logics. Throughout the modal walk through space, expressive power is analyzed in terms of language design, bisimulations, and correspondence phenomena. The result is both unification across the areas visited, and the uncovering of interesting new questions.

In this paper we study natural deduction for the intuitionistic and classical (normal) modal logics obtained from the combinations of the axioms T, B, 4 and 5. In this context we introduce a new multi-contextual structure, called T-sequent, that allows to design simple labelfree natural deduction systems for these logics. After proving that they are sound and complete we show that they satisfy the normalization property and consequently the subformula property in the intuitionistic case.

The paper presents a semantics for quantified modal logic which has a weaker axiomatization than the usual Kripke semantics. In particular, the Barcan Formula and its converse are not valid with the proposed semantics. Subclasses of models which validate BF and other interesting formulas are presented. A completeness theorem is proved, and the relation between this result and completeness with respect to Kripke models is investigated.

A complete axiomatic system CTL$_{rp}$ is introduced for a temporal logic for finitely branching $\omega^+$-trees in a temporal language extended with so called reference pointers. Syntactic and semantic interpretations are constructed for the branching time computation tree logic CTL$^{*}$ into CTL$_{rp}$. In particular, that yields a complete axiomatization for the translations of all valid CTL$^{*}$-formulae. Thus, the temporal logic with reference pointers is brought forward as a simpler (with no path quantifiers), but in a way more expressive medium for reasoning (...) about branching time. (shrink)

Every kind of ‘ought’ implies some kind of ‘can’ – but there are many kinds of ‘ought’ and even more kinds of ‘can’. In this essay, I shall focus on a particular kind of ‘ought’ – specifically, on what I shall call the “rational ‘ought’”. On every occasion of use, this kind of ‘ought’ is focused on the situation of a particular agent at a particular time; but this kind of ‘ought’ is concerned, not with how that agent acts at (...) that time, but with what beliefs or intentions the agent has at the time, or with the sort of reasoning by means of which the agent at that time forms or revises those beliefs or intentions. (shrink)

The French philosopher Jean Buridan's work on the logical paradoxes is currently attracting more attention than it has for several centuries. In part this is due to a general resurgence of interest in the paradoxes, but the immediate occasion is the recent publication of G. E. Hughes's edition, translation, and commentary on the chapter of Buridan's Sophismata most immediately concerned with the paradoxes. It is worth noting, therefore, that Buridan's theory fails, and in a way that makes it seem unlikely (...) that it can be developed into a serious rival of more modern theories. (shrink)

Moral philosophy has long been preoccupied by a supposed dichotomy between the "good" and the "right". This dichotomy has been taken to define certain allegedly central issues for ethics. How are the good and the right related to each other? For example, is one of the two "prior" to the other? If so, is the good prior to the right, or is the right prior to the good?

The logical independence of two statements is tantamount to their probabilistic independence, the latter understood in a sense that derives from stochastic independence. And analogous logical and probabilistic senses of having the same factual content similarly coincide. These results are extended to notions of non-symmetrical independence and independence among more than two statements.

John Corcoran’s natural deduction system for Aristotle’s syllogistic is reconsidered.Though Corcoran is no doubt right in interpreting Aristotle as viewing syllogisms as arguments and in rejecting Lukasiewicz’s treatment in terms of conditional sentences, it is argued that Corcoran is wrong in thinking that the only alternative is to construe Barbara and Celarent as deduction rules in a natural deduction system.An alternative is presented that is technically more elegant and equally compatible with the texts.The abstract role assigned by tradition and Lukasiewicz (...) to Barbara and Celarent is retained.The two ‘ perfect syllogisms’ serve as ‘basic elements’ in the construction of an inductively defined set of valid syllogisms.The proposal departs from Lukasiewicz, and follows Corcoran, however, in construing the construction as one in natural deduction.The result is a sequent system with fewer rules and in which Barbara and Celarent serve as basic deductions.To compare the theory to Corcoran’s, his original is reformulated in current terms and generalized.It is shown to be equivalent to the proposed sequent system, and several variations are discussed.For all systems mentioned, a method of Henkin‐style completeness proofs is given that is more direct and intuitive than Corcoran’s original. (shrink)

The situation calculus is one of the most established formalisms for reasoning about action and change. In this paper we will review the basics of Reiter’s version of the situation calculus, show how knowledge and time have been addressed in this framework, and point to some of the weaknesses of the situation calculus with respect to time. We then present a modal version of the situation calculus where these problems can be overcome with relative ease and without sacrificing the advantages (...) of the original. (shrink)

The aim of this paper is to show what sorts of logics are required by externalist and internalist accounts of the meanings of natural kind nouns. These logics give us a new perspective from which to evaluate the respective positions in the externalist-internalist debate about the meanings of such nouns. The two main claims of the paper are the following: first, that adequate logics for internalism and externalism about natural kind nouns are second-order logics; second, that an internalist second-order logic (...) is a free logic—a second order logic free of existential commitments for natural kind nouns, while an externalist second-order logic is not free of existential commitments for natural kind nouns—it is existentially committed. (shrink)

It it tempting to think that if an outcome had some probability of not occurring, then we cannot explain why that outcome in fact occurred. Despite this intuition, most philosophers of science have come to admit the possibility of indeterministic explanation. Yet some of them continue to hold that if an outcome was not determined, it cannot be explained why that outcome rather than some other occurred. I argue that this is an untenable compromise: if indeterministic explanation is possible, then (...) indeterministic contrastive explanation is possible too. In order to defend this conclusion, I develop an account of contrastive explanation. (shrink)

Tense logics formulated in the bimodal propositional language are investigated with respect to Kripke-completeness (completeness) and decidability. It is proved that all minimal tense extensions of modal logics of finite width (in the sense of K. Kine) as well as all minimal tense extensions of cofinal subframe logics (in the sense of M. Zakharyaschev) are complete. The decidability of all finitely axiomatizable minimal tense extensions of cofinal subframe logics is shown. A number of variations and extensions of these results are (...) also presented. (shrink)

We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language L with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which (...) may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987. (shrink)

This paper addresses the question whether future contingents are knowable, that is, whether one can know that things will go a certain way even though it is possible that things will not go that way. First I will consider a long-established view that implies a negative answer, and draw attention to some endemic problems that affect its credibility. Then I will sketch an alternative line of thought that prompts a positive answer: future contingents are knowable, although our epistemic access of (...) them is limited in some important respects. (shrink)

In addressing the problem of the (in)compatibility of divine foreknowledge and human freedom, philosophers of religion encounter problems regarding the metaphysics and structure of time. Some models of temporal logic developed for completely independent reasons have proved especially appropriate for representing the temporal structure of the world as Molinism conceives it. In particular, some models of the Thin Red Line ( $$\mathsf {TRL}$$ ) seem to imply that conditionals of freedom are true or false, as Molinists maintain. Noting the resemblance (...) between Molinism and $$\mathsf {TRL}$$ models, Restall (Molinism and the thin red line. In: Perszyk K (ed) Molinism: the contemporary debate, pp 227–239, 2011) has advanced some criticisms of Molinism that have also been leveled against $$\mathsf {TRL}$$ models. In particular, Restall believes that the implication $$p \rightarrow \mathbf {HF}p$$ is not true in $$\mathsf {TRL}$$ models. Because Molinists must also accept that this implication is not true, this is a problem for them. We will show that Restall’s criticism is wide of the mark. Firstly, it will be demonstrated that in many open future models (not just $$\mathsf {TRL}$$ ) the implication $$p \rightarrow \mathbf {HF}p$$ is invalid. Secondly, while it is possible to account for this implication, some modifications are required in respect of the branching time semantics. In proposing one such modification, we show that this new semantics can be adopted by advocates of the $$\mathsf {TRL}$$ and, as a consequence, by Molinists as well. We conclude that the principle stated by Restall is either a problem for many open future models (not just for Molinists) or can be accounted for by these models and so is not a problem for Molinists either. (shrink)

In this paper we use “generic submodels” to prove that each normal extension of G.3 (K4.3W) has the finite model property, by which we establish that each proper normal extension of G.3 is G.3Altn for some n≥0.

In this work we propose an encoding of Reiter’s Situation Calculus solution to the frame problem into the framework of a simple multimodal logic of actions. In particular we present the modal counterpart of the regression technique. This gives us a theorem proving method for a relevant fragment of our modal logic.

Here, I first prove that certain families of k-valued clones have the Gupta-Belnap fixed-point property. This essentially means that all propositional languages that are interpreted with operators belonging to those clones are such that any net of self-referential sentences in the language can be consistently evaluated. I then focus on two four-valued generalisations of the Kleene propositional operators that generalise the strong and weak Kleene operators: Belnap’s clone and Fitting’s clone, respectively. I apply the theorems from the initial part of (...) the paper to analyse the fixed-point property of Belnap’s and Fitting’s clones when some special operators that reflect the semantics are added. The conclusion of the paper is that Fitting’s clone is better suited than Belnap’s to provide self-referential languages with highly expressive resources. (shrink)

The axiomatic presentation of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generalizations of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective.

RésuméOn défend ici l'idée que la définition des notions sémantiques à l'aide des fonctions de probabilité devrait être vue non pas comme une généralisation de la sémantique standard en termes d'assignations de valeurs de vérité, mais plutôt comme une généralisation aux degrés de conséquence logique, de la caractérisation de la relation de conséquence que l'on retrouve dans le calcul des séquents de Gentzen.

Frege’s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel’s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ‘complete’ it is clear from Dedekind’s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical or (...) complete, there are logical extensions of these theories into second-order and by the addition of generalized quantifiers which are categorical. Frege’s project really found success through Gödel’s completeness theorem of 1930 and the subsequent development of first- and higher-order model theory. (shrink)

I advance a new theory of causal relevance, according to which causal claims convey information about conditional probability functions. This theory is motivated by the problem of disjunctive factors, which haunts existing probabilistic theories of causation. After some introductory remarks, I present in Section 3 a sketch of Eells's (1991) probabilistic theory of causation, which provides the framework for much of the discussion. Section 4 explains how the problem of disjunctive factors arises within this framework. After rejecting three proposed solutions, (...) I offer in Section 6 a new approach to causation that avoids the problem. Decision-theoretic considerations also support the new approach. Section 8 develops the consequences of the new theory for causal explanation. The resulting theory of causal explanation incorporates the new insights while respecting important work on scientific explanation by Salmon (1971), Railton (1981), and Humphreys (1989). My conclusions are enumerated in Section 9. (shrink)

It is well known that Tarski proved a result which can be stated roughly as: no sufficiently rich, consistent, classical language can contain its own truth definition. Tarski's way around this problem is to deal with two languages at a time, an object language for which we are defining truth and a metalanguage in which the definition occurs. An obvious question then is: under what conditions can we construct a definition of truth for a given object language. Tarski claims that (...) it is necessary and sufficient that the metalanguage be "essentially richer". Our contention, put bluntly, is that this claim deserves more scrutiny from philosophers than it usually gets and in fact is false unless "essentially richer" means nothing else than "sufficient to contain a truth definition for the object language.". (shrink)

It is demonstrated that Hamblin's 15 tense theorem does not only hold for temporal logic with linear time but also for branching time. Furthermore three other theorems with finitely many tenses are shown.

The problem of Common Knowledge will be considered in two classes of models: a class K.* of Kripke models and a class S of Scott models. Two modal logic systems will be defined. Those systems, KC and MC, include an axiomatisation of Common Knowledge. We prove determination of each system by the corresponding class of models. MSC: 03B45, 68T25.

The properties of belief revision operators are known to have an informal semantics which relates them to the axioms of conditional logic. The purpose of this paper is to make this connection precise via the model theory of conditional logic. A semantics for conditional logic is presented, which is expressed in terms of algebraic models constructed ultimately out of revision operators. In addition, it is shown that each algebraic model determines both a revision operator and a logic, that are related (...) by virtue of the stable Ramsey test. (shrink)

The paper develops a set of quantified temporal alethic boulesic doxastic systems. Every system in this set consists of five parts: a ‘quantified’ part, a temporal part, a modal (alethic) part, a boulesic part and a doxastic part. There are no systems in the literature that combine all of these branches of logic. Hence, all systems in this paper are new. Every system is defined both semantically and proof-theoretically. The semantic apparatus consists of a kind of$$T \times W$$T×Wmodels, and the (...) proof-theoretical apparatus of semantic tableaux. The ‘quantified part’ of the systems includes relational predicates and the identity symbol. The quantifiers are, in effect, a kind of possibilist quantifiers that vary over every object in the domain. The tableaux rules are classical. The alethic part contains two types of modal operators for absolute and historical necessity and possibility. According to ‘boulesic logic’ (the logic of the will), ‘willing’ (‘consenting’, ‘rejecting’, ‘indifference’ and ‘non-indifference’) is a kind of modal operator. Doxastic logic is the logic of beliefs; it treats ‘believing’ (and ‘conceiving’) as a kind of modal operator. I will explore some possible relationships between these different parts, and investigate some principles that include more than one type of logical expression. I will show that every tableau system in the paper is sound and complete with respect to its semantics. Finally, I consider an example of a valid argument and an example of an invalid sentence. I show how one can use semantic tableaux to establish validity and invalidity and read off countermodels. These examples illustrate the philosophical usefulness of the systems that are introduced in this paper. (shrink)

Temporal logic is one of the many areas in which a possible world semantics is adopted. Prior's Ockhamist and Peircean semantics for branching-time, though, depart from the genuine Kripke semantics in that they involve a quantification over histories, which is a second-order quantification over sets of possible worlds. In the paper, variants of the original Prior's semantics will be considered and it will be shown that all of them can be viewed as first-order counterparts of the original semantics.

A cut-free Gentzen sequent calculus for Ewald’s intuitionistic tense logic \ is established. By the proof-theoretic method, we prove that, for every set of strictly positive implications S, the classical tense logic \ is embedded into its intuitionistic analogue \ via Kolmogorov, Gödel–Genzten and Kuroda translations respectively. A sufficient and necessary condition for Glivenko type theorem in tense logics is established.

RésuméOn montre d'abord que la technique de révision des probabilités appelée « imagerie », qui a été introduite par Lewis pour la logique des conditionnels de Stalnaker, peut être généralisée à la sémantique des systèmes de sphères de Lewis si l'on permet aux énoncés conditionnels d'avoir des valeurs de vérité fractionnaires. Un système est proposé.

This paper is dedicated to Alonzo Church, who died in August 1995 after a long life devoted to logic. To Church we owe lambda calculus, the thesis bearing his name and the solution to the Entscheidungsproblem.His well-known book Introduction to Mathematical LogicI, defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed. Church was the creator of the Journal of Symbolic Logicthe best-known journal of the area, which he edited for several decades This (...) paper is in three sections. The first is written in journalistic style:the story of the life of AlonzoChurch is told, including some of the many anecdotes I have collected from different sources. The secondpart is devoted to his work, but is far from being exhaustive. The last part is more original; in it I attempto show that Church’s great discovery was lambda calculus and that his remaining contributions weremainly inspired afterthoughts in the sense that most of his contributions as well as some of his pupils derivefrom that initial achievement. Included are Kleene’s Recursion Theory and the completeness proof ofHenkin. I have added an appendix in which is presented the typed lambda calculus and a proof of theundecidability of first-order logic. (shrink)

We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by combining L with a pure propositional temporal logic T (such as linear temporal logic with Since and Until) in a special way. We refer to this method as adding a temporal dimension to L or just temporalising L. We show that the logic system T(L) preserves several properties of the original temporal logic (...) like soundness, completeness, decidability, conservativeness and separation over linear flows of time. We then focus on the temporalisation of first-order logic, and a comparison is make with other first-order approaches to the handling of time. (shrink)

A conjectural inference is proposed, aimed at producing conjectural theorems from formal conjectures assumed as axioms, as well as admitting contradictory statements as conjectural theorems. To this end, we employ Paraconsistent Informational Logic, which provides a formal setting where the notion of conjecture formulated by an epistemic agent can be defined. The paraconsistent systems on which conjectural deduction is based are sequent formulations of the C-systems presented in Carnielli-Marcos [CAR 02b]. Thus, conjectural deduction may also be considered to be a (...) tool for investigating the properties of paraconsistency in general. (shrink)

Developing some suggestions of Ramsey (1925), elementary logic is formulated with respect to an arbitrary categorial system rather than the categorial system of Logical Atomism which is retained in standard elementary logic. Among the many types of non-standard categorial systems allowed by this formalism, it is argued that elementary logic with predicates of variable degree occupies a distinguished position, both for formal reasons and because of its potential value for application of formal logic to natural language and natural science. This (...) is illustrated by use of such a logic to construct a theory of quantity which is argued to be scientifically superior to existing theories of quantity based on standard categorial systems, since it yields real-valued scales without the need for unrealistic existence assumptions. This provides empirical evidence for the hypothesis that the categorial structure of the physical world itself is non-standard in this sense. (shrink)

An old conjecture of modal logics states that every splitting of the major systems K4, S4, G and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof technique which will give a positive answer for large classes of splitting frames. The proof works by establishing a rather strong property of these splitting frames namely that (...) they preserve the finite model property in the following sense. Whenever an extension Λ has fmp so does the splitting Λ/f of Λ by f. Although we will also see that this method has its limitations because there are frames lacking this property, it has several desirable side effects. For example, properties such as compactness, decidability and others can be shown to be preserved in a similar way and effective bounds for the size of models can be given. Moreover, all methods and proofs are constructive. (shrink)