It is often argued that inferential role semantics (IRS) entails semantic holism as long as theorists fail to answer the question about which inferences, among the many, are meaning-constitutive. Since analyticity, as truth in virtue of meaning, is a widely dismissed notion in indicating which inferences determine meaning, it seems that holism follows. Semantic holism is often understood as facing problems with the stability of content and many usual explanations of communication. Thus, we should choose between giving up (...) IRS, to avoid these holistic entailments, and defending holism against this charge, to rescue IRS. I try to pursue the second goal by analyzing certain patterns of counterfactual reasoning. Wilfrid Sellars and Robert Brandom claim that, to defend IRS, content-constitutive inferences are those counterfactually robust. While it is difficult to assess the goodness of such a view, it nonetheless entails that counterfactually non-robust inferences (which I call “modally ruled out inferences”) are not content-constitutive. If this is true, and if we take certain remarks about the grasp of concepts on board, there is a way to restrict the scope of the holism entailed by IRS to the extent of reshaping problems with the stability of content. (shrink)

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings of (...) modal operators in terms of rules of inference. (shrink)

A conclusion follows from given premisses if and only if an acceptable counterfactual-supporting covering generalization of the argument rules out, either definitively or with some modal qualification, simultaneous acceptability of the premisses and non-accepta-bility of the conclusion, even though it does not rule out acceptability of the premisses and does not require acceptability of the conclusion independently of the premisses. Hence the reiterative associated conditional of an argument is true if and only it has such a covering generalization, and a (...) supposed unexpressed premiss supplied to make an argument formally valid should be a covering generalization. (shrink)

These lecture notes were composed while teaching a class at Stanford and studying the work of Brian Chellas (Modal Logic: An Introduction, Cambridge: Cambridge University Press, 1980), Robert Goldblatt (Logics of Time and Computation, Stanford: CSLI, 1987), George Hughes and Max Cresswell (An Introduction to Modal Logic, London: Methuen, 1968; A Companion to Modal Logic, London: Methuen, 1984), and E. J. Lemmon (An Introduction to Modal Logic, Oxford: Blackwell, 1977). The Chellas text influenced me the most, though the order of (...) presentation is inspired more by Goldblatt.2 My goal was to write a text for dedicated undergraduates with no previous experience in modal logic. The text had to meet the following desiderata: (1) the level of difficulty should depend on how much the student tries to prove on his or her own—it should be an easy text for those who look up all the proofs in the appendix, yet more difficult for those who try to prove everything themselves; (2) philosophers (i.e., colleagues) with a basic training in logic should be able to work through the text on their own; (3) graduate students should find it useful in preparing for a graduate course in modal logic; (4) the text should prepare people for reading advanced texts in modal logic, such as Goldblatt, Chellas, Hughes and Cresswell, and van Benthem, and in particular, it should help the student to see what motivated the choices in these texts; (5) it should link the two conceptions of logic, namely, the conception of a logic as an axiom system (in which the set of theorems is constructed from the bottom up through proof sequences) and the conception of a logic as a set containing initial ‘axioms’ and closed under ‘rules of inference’ (in which the set of theorems is constructed from the top down, by carving out the logic from the set of all formulas as the smallest set closed under the rules); finally, (6) the pace for the presentation of the completeness theorems should be moderate—the text should be intermediate between Goldblatt and Chellas in this regard (in Goldblatt, the completeness proofs come too quickly for the undergraduate, whereas in Chellas, too many unrelated.... (shrink)

A certain type of inference rules in modal logics, generalizing Gabbay's Irreflexivity rule, is introduced and some general completeness results about modal logics axiomatized with such rules are proved.

The presented paper takes up the attempt to analyse and specify the suspicion that some modal rules of inference are paralogical in application to non-logical reasonings (s.c. modal fallacy). The considerations have been limited to modal prepositional calculi: K and S5, which are intended to be a formal base of these non-logical reasonings - proofs of so called specific thesis on the grounds of the particular specific theories. Pointing out the properties of being permitted, being valid and being derivable in (...) case of inferences rules and also semantical relations of point, structure, frame and inferential consequence in standard semantics of possible worlds, enables to define two kinds of paralogism: point and structural. Justification of the suspicion of modal fallacy occurrence in the case of a given inference rule, depends on pointed metalogical properties of this rule and also on what kind of the notion of paralogism is being discussed. It appears that when a given rule is paralogical only pointly (and not structurally), the sufficient condition of avoiding modal fallacy is to consider the specific axioms of the given specific theory as the sentences which are structurally true (structural truth is of course not equivalent to logical truth). If we want to treat these axioms as sentences which are pointly true, we have to eliminate pointly paralogical rules. In this case it is enough to construct such axiomatisation of calculi K and S5, in which we use the notion of modal closure (it eliminates the primitive rule of Goedel and all rules derivable from it - rules which are structurally but not pointly correct). (shrink)

This paper discusses the account of alethic modality as presented by Wilfrid Sellars in his earlier work from 1947 to 1958. Its aim is twofold. First, I discuss Sellars' analysis by exploring its historical relationship to Carnap's account of modality. I argue that Carnap's early syntactic treatment of modality profoundly influenced Sellars' own so-called ‘regulist' account of modality in terms of rules of inference. Furthermore, it is suggested that Sellars' lesser-known possible worlds analysis was influenced by Carnap's later semantic account (...) of modality. The second aim of the paper is a critical one. I raise a number of objections to Sellars’ account of modality and argue that his account confronts some serious limitations. I also argue that Sellars does not provide a satisfying explanation of the relationship between his regulist account and his possible worlds analysis. I offer such an explanation by arguing that his regulist account should be understood as a pragmatic account of what is conveyed by the use of modal sentences, whereas his possible worlds analysis is a semantic account of the asserted content of modal statements. That is, the regulist account and the possible worlds analysis have different explanatory targets. (shrink)

This paper concerns modal logics of provability — Gödel-Löb systemGL and Solovay logicS — the smallest and the greatest representation of arithmetical theories in propositional logic respectively. We prove that the decision problem for admissibility of rules (with or without parameters) inGL andS is decidable. Then we get a positive solution to Friedman''s problem forGL andS. We also show that A. V. Kuznetsov''s problem of the existence of finite basis for admissible rules forGL andS has a negative solution. Afterwards we (...) give an algorithm deciding the solvability of logical equations inGL andS and constructing some solutions. (shrink)

I propose an account of indicative conditionals that combines features of minimal change semantics and information semantics. As in information semantics, conditionals are interpreted relative to an information state in accordance with the Ramsey test idea: “if p then q” is supported at a state s iff q is supported at the hypothetical state s[p] obtained by restricting s to the p-worlds. However, information states are not modeled as simple sets of worlds, but by means of a Lewisian system of (...) spheres. Worlds in the inner sphere are considered possible; worlds outside of it are ruled out, but to different degrees. In this way, even when a state supports “not p”, it is still possible to suppose p consistently. I argue that this account does better than its predecessors with respect to a set of desiderata concerning inferences with conditionals. In particular, it captures three important facts: that a conditional is logically independent from its antecedent; that a sequence of antecedents behaves like a single conjunctive antecedent ; and that conditionals restrict the quantification domain of epistemic modals. I also discuss two ways to construe the role of a premise, and propose a generalized notion of entailment that keeps the two apart. (shrink)

Two experiments (N1 = 141, N2 = 40) investigate two versions of Aristotle’s Thesis for the first time. Aristotle’s Thesis is a negated conditional, which consists of one propositional variable with a negation either in the antecedent (version 1) or in the consequent (version 2). This task allows us to infer if people interpret indicative conditionals as material conditionals or as conditional events. In the first experiment I investigate between-participants the two versions of Aristotle’s Thesis crossed with abstract versus concrete (...) task material. The modal response for all four groups is consistent with the conditional event and inconsistentwith the material conditional interpretation. This observation is replicated in the second experiment. Moreover, the second experiment rules out scope ambiguities of the negation of conditionals. Both experiments provide new evidence against thematerial conditional interpretation of conditionals and support the conditional event interpretation. Finally, I discuss implications formodeling indicative conditionals and the relevance of this work for experimental philosophy. (shrink)

Peter van Inwagen’s original formulation of the Consequence Argument employed an inference rule that was shown to be invalid given van Inwagen’s interpretation of the modal operators in the Consequence Argument. In response, van Inwagen recently suggested a revised interpretation of his modal operators. Following up on a debate between Blum and Schnieder, I analyze van Inwagen’s revised interpretation in terms of explanatory notions and I argue that van Inwagen faces a dilemma: he either has to admit that beta entails (...) fatalism, or he has to admit that a new counterexample invalidates beta. Either way, it seems reasonable to reject beta and to conclude that the Consequence Argument fails. Further, I argue that Widerker’s well-known substitute for rule beta is faced with a similar dilemma and, therefore, is bound to fail as well. I conclude that, if the modal operators are interpreted in terms of explanatory notions, neither van Inwagen’s nor Widerker’s rule of inference turns out to be valid. (shrink)

Research into the neural correlates of consciousness (NCC) aims to identify not just those brain areas that are NCC, but also those that are not. In the received method for ruling out a brain area from being an NCC, this is accomplished by showing a brain area’s content to be consistently absent from subjects’ reports about what they are experiencing. This paper points out how this same absence can be used to infer that the brain area’s content is cognitively inaccessible, (...) in which case we would expect its content to be absent from subjects’ reports whether its content is (phenomenally) conscious or not. If so, such reports cannot count as evidence against that brain area being an NCC, and the received method fails. An alternative method (one suggested in Block, 2007) is considered. (shrink)

Recent philosophy of language has been profoundly impacted by the idea that mainstream, model-theoretic semantics is somehow incompatible with deflationary accounts of truth and reference. The present article systematizes the case for incompatibilism, debunks circularity and “modal confusion” arguments familiar in the literature, and reconstructs the popular thought that truth-conditional semantics somehow “presupposes” a correspondence theory of truth as an inference to the best explanation. The case for compatibilism is closed by showing that this IBE argument fails to rule out (...) two kinds of deflationism: the position Field famously accused Tarski of having; and a less familiar version of the view that defines reference in terms of a deflated notion of truth. Finally, the distinction between unifying and constitutive explanation is used to forestall the response that correspondence theory is literally part of mainstream semantics. (shrink)

Philosophers are divided on whether the proof- or truth-theoretic approach to logic is more fruitful. The paper demonstrates the considerable explanatory power of a truth-based approach to logic by showing that and how it can provide (i) an explanatory characterization —both semantic and proof-theoretical—of logical inference, (ii) an explanatory criterion for logical constants and operators, (iii) an explanatory account of logic’s role (function) in knowledge, as well as explanations of (iv) the characteristic features of logic —formality, strong modal force, generality, (...) topic neutrality, basicness, and (quasi-)apriority, (v) the veridicality of logic and its applicability to science, (v) the normativity of logic, (vi) error, revision, and expansion in/of logic, and (vii) the relation between logic and mathematics. The high explanatory power of the truth-theoretic approach does not rule out an equal or even higher explanatory power of the proof-theoretic approach. But to the extent that the truth-theoretic approach is shown to be highly explanatory, it sets a standard for other approaches to logic, including the proof-theoretic approach. (shrink)

The “Game of the Rule” is easy enough: I give you the beginning of a sequence of numbers (say) and you have to figure out how the sequence continues, to uncover the rule by means of which the sequence is generated. The game depends on two obvious constraints, namely (1) that the initial segment uniquely identify the sequence, and (2) that the sequence be non-random. As it turns out, neither constraint can fully be met, among other reasons because the relevant (...) notion of randomness is either vacuous or undecidable. This may not be a problem when we play for fun. It is, however, a serious problem when it comes to playing the game for real, i.e., when the player to issue the initial segment is not one of us but the world out there, the sequence consisting not of numbers (say) but of the events that make up our history. Moreover, when we play for fun we know exactly what initial segment to focus on, but when we play for real we don’t even know that. This is the core difficulty in the philosophy of the inductive sciences. (shrink)

I contrast two modes of error-elimination relevant to evaluating evidence in accounts that emphasize frequentist reliability. The contrast corresponds to that between the use of of a reliable inference procedure and the critical scrutiny of a procedure with regard to its reliability, in light of what is and is not known about the setting in which the procedure is used. I propose a notion of security as a category of evidential assessment for the latter. In statistical settings, robustness theory and (...) misspecification testing exemplify two distinct strategies for securing statistical inferences. (shrink)

The main result of this paper is the following theorem: each modal logic extendingK4 having the branching property belowm and the effective m-drop point property is decidable with respect to admissibility. A similar result is obtained for intermediate intuitionistic logics with the branching property belowm and the strong effective m-drop point property. Thus, general algorithmic criteria which allow to recognize the admissibility of inference rules for modal and intermediate logics of the above kind are found. These criteria are applicable to (...) most modal logics for which decidability with respect to admissibility is known and to many others, for instance, to the modal logicsK4,K4.1,K4.2,K4.3,S4.1,S4.2,GL.2; to all smallest and greatest counterparts of intermediate Gabbay-De-Jong logicsD n; to all intermediate Gabbay-De-Jong logicsD n; to all finitely axiomatizable modal and intermediate logics of finite depth etc. Semantic criteria for recognizing admissibility for these logics are offered as well. (shrink)

ABSTRACT We1 study unification of formulas in modal logics and consider logics which are equivalent w.r.t. unification of formulas. A criteria is given for equivalence w.r.t. unification via existence or persistent formulas. A complete syntactic description of all formulas which are non-unifiable in wide classes of modal logics is given. Passive inference rules are considered, it is shown that in any modal logic over D4 there is a finite basis for passive rules.

We present a cut-admissible system for the modal logic S5 in a formalism that makes explicit and intensive use of deep inference. Deep inference is induced by the methods applied so far in conceptually pure systems for this logic. The system enjoys systematicity and modularity, two important properties that should be satisfied by modal systems. Furthermore, it enjoys a simple and direct design: the rules are few and the modal rules are in exact correspondence to the modal axioms.

The paper investigates admissible inference rules for the multi-modal logic LTK, which describes a combination of linear time and knowledge. This logic is semantically defined as the set of all ℒ.

The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on: admissible or permissible inference rules the derivability of the admissible inference rules the structural completeness of logics the bases for admissible and valid inference rules. There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered. The book is basically self-contained and special (...) attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included. The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook. (shrink)

We give a necessary and sufficient condition for any modal logic with fmp to inherit all inference rules admissible in S4. Using this condition we describe all tabular modal logics inheriting inference rules admissible for S4.

The aim of this paper is to look from the point of view of admissibility of inference rules at intermediate logics having the finite model property which extend Heyting's intuitionistic propositional logic H. A semantic description for logics with the finite model property preserving all admissible inference rules for H is given. It is shown that there are continuously many logics of this kind. Three special tabular intermediate logics λ, 1 ≥ i ≥ 3, are given which describe all tabular (...) logics preserving admissibility: a tabular logic λ preserves all admissible rules for H iff 7λ has width not more than 2 and is not included in each λ. MSC: 03B55, 03B20. (shrink)

Davidson’s remark is fairly conventional stuff in contemporary philosophy, but the argument that informs it is elusive. Is this a kind of unformulated transcendental argument, which amounts to the claim that the ‘sharing’ of ‘language,’ in some unspecified sense of these terms, is a condition of the possibility of ‘communication’ in some unspecified sense of this term? Or is it a kind of inference to the best explanation in which there are no real alternativesan inference, so to speak, to the (...) only explanation? There are good reasons to be suspicious of arguments of this form. Yet this general picture, of some sort of shared stuff at the basis of language, is highly appealing, and so is its extension to practices generally. The claim that there is some class of things that could not happen, were it not for the existence of some sort of shared practices, is a commonplace, despite, and perhaps because of its vagueness. (shrink)

What is an inference rule? This question does not have a unique answer. One usually finds two distinct standard answers in the literature; validity inference $(\sigma \vdash_\mathrm{v} \varphi$ if for every substitution $\tau$, the validity of $\tau \lbrack\sigma\rbrack$ entails the validity of $\tau\lbrack\varphi\rbrack)$, and truth inference $(\sigma \vdash_\mathrm{t} \varphi$ if for every substitution $\tau$, the truth of $\tau\lbrack\sigma\rbrack$ entails the truth of $\tau\lbrack\varphi\rbrack)$. In this paper we introduce a general semantic framework that allows us to investigate the notion of inference (...) more carefully. Validity inference and truth inference are in some sense the extremal points in our framework. We investigate the relationship between various types of inference in our general framework, and consider the complexity of deciding if an inference rule is sound, in the context of a number of logics of interest: classical propositional logic, a nonstandard propositional logic, various propositional modal logics, and first-order logic. (shrink)

In Meaning and Necessity (1947/1950), Carnap advances an intensional semantic framework on which modal claims are true in virtue of semantical rules alone, and so are a priori. In 'Empiricism, Semantics, and Ontology' (1950), Carnap advances an epistemic-ontological framework on which metaphysical claims are either trivial or meaningless, since lacking any means of substantive confirmation. Carnap carried out these projects two decades before Kripke influentially argued, in Naming and Necessity (1972/1980), that some modal claims are true a posteriori. How should (...) a neo-Carnapian respond to Kripke's results? Some (notably, Chalmers and Jackson, in their 2001) have suggested that an extension of intensional semantics along lines of "epistemic two-dimensionalism" can accommodate Kripke's results while largely preserving commitment to the semantics-based a priority of modal claims. Here we consider how best to implement this suggestion, and how the resulting semantics fits with Carnap's second project. We find that the most promising (and most Carnapian!) post-Kripke version of Carnap’s semantics---abductive two-dimensionalism---presupposes an epistemology which undermines Carnap's metaphysical anti-realism. (shrink)

In a range of contexts, one comes across processes resembling inference, but where input propositions are not in general included among outputs, and the operation is not in any way reversible. Examples arise in contexts of conditional obligations, goals, ideals, preferences, actions, and beliefs. Our purpose is to develop a theory of such input/output operations. Four are singled out: simple-minded, basic (making intelligent use of disjunctive inputs), simple-minded reusable (in which outputs may be recycled as inputs), and basic reusable. They (...) are defined semantically and characterised by derivation rules, as well as in terms of relabeling procedures and modal operators. Their behaviour is studied on both semantic and syntactic levels. (shrink)

In previous articles we presented a simple set of axioms named “Contexts, Systems and Modalities”, where the structure of quantum mechanics appears as a result of the interplay between the quantized number of modalities accessible to a quantum system, and the continuum of contexts that are required to define these modalities. In the present article we discuss further how to obtain Born’s rule within this framework. Our approach is compared with other former and recent derivations, and its strong links with (...) Gleason’s theorem are particularly emphasized. (shrink)

In a range of contexts, one comes across processes resembling inference, but where input propositions are not in general included among outputs, and the operation is not in any way reversible. Examples arise in contexts of conditional obligations, goals, ideals, preferences, actions, and beliefs. Our purpose is to develop a theory of such input/output operations. Four are singled out: simple-minded, basic (making intelligent use of disjunctive inputs), simple-minded reusable (in which outputs may be recycled as inputs), and basic reusable. They (...) are defined semantically and characterised by derivation rules, as well as in terms of relabeling procedures and modal operators. Their behaviour is studied on both semantic and syntactic levels. (shrink)

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for negations. The approach unifies in a philosophically motivated picture the (...) following results: nothing can be called a negation properly if it does not satisfy Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds. (shrink)

Modal logic is the study of modalities - expressions that qualify assertions about the truth of statements - like the ordinary language phrases necessarily, possibly, it is known/believed/ought to be, etc., and computationally or mathematically motivated expressions like provably, at the next state, or after the computation terminates. The study of modalities dates from antiquity, but has been most actively pursued in the last three decades, since the introduction of the methods of Kripke semantics, and now impacts on a wide (...) range of disciplines, including the philosophy of language and linguistics ('possible words' semantics for natural language), constructive mathematics (intuitionistic logic), theoretical computer science (dynamic logic, temporal and other logics for concurrency), and category theory (sheaf semantics). This volume collects together a number of the author's papers on modal logic, beginning with his work on the duality between algebraic and set-theoretic modals, and including two new articles, one on infinitary rules of inference, and the other about recent results on the relationship between modal logic and first-order logic. Another paper on the 'Henkin method' in completeness proofs has been substantially extended to give new applications. Additional articles are concerned with quantum logic, provability logic, the temporal logic of relativistic spacetime, modalities in topos theory, and the logic of programs. (shrink)

We find an explicit basis for all admissible rules of the modal logic S4. Our basis consists of an infinite sequence of rules which have compact and simple, readable form and depend on increasing set of variables. This gives a basis for all quasi-identities valid in the free modal algebra ℱS4 of countable rank.

In this paper I argue that Modal Realism, the thesis that there exist non-actual possible individuals and worlds, can be made compatible with Metaphysical Nihilism, the thesis that it is possible that nothing concrete exists. Modal Realism as developed by Lewis rules out the possibility of a world where nothing concrete exists and so conflicts with Metaphysical Nihilism. In the paper I argue that Modal Realism can be modified so as to be compatible with Metaphysical Nihilism. Such a modification makes (...) Modal Realism neither incur further theoretical costs nor lose its theoretical benefits. Thus such a modification constitutes an improvement of Modal Realism. (shrink)

Modal dispositionalism is the view that possibilities are a matter of the dispositions of individual objects: it is possible that p if and only if something has a disposition for p to be the case. We raise a problem for modal dispositionalism: nothing within the theory rules out that there could be necessary, perfect masks, which make the manifestation of a disposition impossible. Unless such necessary perfect masks are ruled out, modal dispositionalism runs the risk of failing to provide (...) a sufficient condition for possibility, and indeed of engendering contradictions. But to rule them out, modal dispositionalism would have to revise a crucial tenet of the view, its localism. (shrink)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity (...) problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning. (shrink)

What is our epistemic access to metaphysical modality? Timothy Williamson suggests that the epistemology of counterfactuals will provide the answer. This paper challenges Williamson's account and argues that certain elements of the epistemology of counterfactuals that he discusses, namely so called background knowledge and constitutive facts, are already saturated with modal content which his account fails to explain. Williamson's account will first be outlined and the role of background knowledge and constitutive facts analysed. Their key role is to restrict our (...) imagination to rule out irrelevant counterfactual suppositions. However, background knowledge turns out to be problematic in cases where we are dealing with metaphysically possible counterfactual suppositions that violate the actual laws of physics. As we will see, unless Williamson assumes that background knowledge corresponds with the actual, true laws of physics and that these laws are metaphysically necessary, it will be difficult to address this problem. Furthermore, Williamson's account fails to accommodate the distinction between conceivable yet metaphysically impossible scenarios, and conceivable and metaphysically possible scenarios. This is because background knowledge and constitutive facts are based strictly on our knowledge of the actual world. Williamson does attempt to address this concern with regard to metaphysical necessities – as they hold across all possible worlds – but we will see that even in this case the explanation is questionable. These problems, it will be suggested, cannot be addressed in a counterfactual account of the epistemology of modality. The paper finishes with an analysis of Williamson's possible rejoinders and some discussion about the prospects of an alternative account of modal epistemology. (shrink)

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on in our (...) field—a book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory. (shrink)

ABSTRACT We critically review two extant paradigms for understanding the systematic interaction between modality and tense, as well as their respective modifications designed to do justice to the contingency of time’s structure and composition. We show that on either type of theory, as well as their respective modifications, some principles prove logically valid whose truth might sensibly be questioned on metaphysical grounds. These considerations lead us to devise a more general logical framework that allows accommodation of those metaphysical views that (...) its predecessors rule out by fiat. (shrink)

We explore the consequences, for logical system-building, of taking seriously the aim of having irredundant rules of inference, and a preference for proofs of stronger results over proofs of weaker ones. This leads one to reconsider the structural rules of REFLEXIVITY, THINNING, and CUT. REFLEXIVITY survives in the minimally necessary form $\varphi:\varphi$. Proofs have to get started. CUT is subject to a CUT-elimination theorem, to the effect that one can always make do without applications of CUT. So CUT is redundant, (...) and should not be a rule of the system. CUT-elimination, however, in the context of the usual forms of logical rules, requires the presence, in the system, of THINNING. But THINNING, it turns out, is not really necessary. Provided only that one liberalizes the statement of certain logical rules in an appropriate way, one can make do without CUT or THINNING. From the methodological point of view of this study, the logical rules ought to be framed in this newly liberalized form. These liberalized logical rules determine the system of core logic. Given any intuitionistic Gentzen proof of $\Delta:\varphi$, one can determine from it a Core proof of some subsequent of $\Delta:\varphi$. Given any classical Gentzen proof of $\Delta:\varphi$, one can determine from it a classical Core proof of some subsequent of $\Delta:\varphi$. In both cases the Core proof is of a result at least as strong as that of the Gentzen proof; and the only structural rule used is $\varphi:\varphi$. (shrink)

Recently, there has been a shift away from traditional truth-conditional accounts of meaning towards non-truth-conditional ones, e.g., expressivism, relativism and certain forms of dynamic semantics. Fueling this trend is some puzzling behavior of modal discourse. One particularly surprising manifestation of such behavior is the alleged failure of some of the most entrenched classical rules of inference; viz., modus ponens and modus tollens. These revisionary, non-truth-conditional accounts tout these failures, and the alleged tension between the behavior of modal vocabulary and classical (...) logic, as data in support of their departure from tradition, since the revisionary semantics invalidate some of these patterns. I, instead, offer a semantics for modality with the resources to accommodate the puzzling data while preserving classical logic, thus affirming the tradition that modals express ordinary truth-conditional content. My account shows that the real lesson of the apparent counterexamples is not the one the critics draw, but rather one they missed: namely, that there are linguistic mechanisms, reflected in the logical form, that affect the interpretation of modal language in a context in a systematic and precise way, which have to be captured by any adequate semantic account of the interaction between discourse context and modal vocabulary. The semantic theory I develop specifies these mechanisms and captures precisely how they affect the interpretation of modals in a context, and do so in a way that both explains the appearance of the putative counterexamples and preserves classical logic. (shrink)

An attempt is made to include the axioms of Mackey for probabilities of experiments in quantum mechanics into the calculus x0 of ukasiewicz. The obtained calculusQ contains an additional modal signQ and four modal rules of inference. The propositionQx is read x is confirmed. The most specific rule of inference may be read: for comparable observations implication is equivalent to confirmation of material implication.The semantic truth ofQ is established by the interpretation with the help of physical objects obeying to the (...) rules of quantum mechanics. The embedding of the usual quantum propositional logic inQ is accomplished. (shrink)

Epistemic contradictions are now a well-known and often discussed phenomenon among those who study epistemic modals. These contradictions are expressed by sentences like ‘It is raining and it might not be raining’ whose oddness to the common ear demands an explanation. However, it has turned out to be a rather controversial enterprise to provide such an explanation in a sufficiently precise and general manner. According to pragmatic explanations, epistemic contradictions are semantically consistent but pragmatically defective. According to semantic explanations, one (...) should regard epistemic contradictions as plain semantic inconsistencies. Endorsing such a semantic solution is, however, tantamount to rejecting classical logic, which predicts the consistency of epistemic contradictions. After arguing that all existing solutions to this problem face as yet unmet challenges, I will present a new solution that adequately overcomes these challenges. I will propose to view epistemic contradictions as semantically consistent but epistemically defective sentences. The main thesis emerging will be that we need neither abandon the inference rules of classical logic nor the classical truth-conditional approach to semantics to deal adequately with epistemic contradictions. (shrink)

The interplay of philosophical ambitions and technical reality have given birth to rich and interesting approaches to explain the oft-claimed special character of mathematical and logical knowledge. Two projects stand out both for their audacity and their innovativeness. These are logicism and proof-theoretic semantics. This dissertation contains three chapters exploring the limits of these two projects. In both cases I find the formal results offer a mixed blessing to the philosophical projects. Chapter 1. Is a logicist bound to the claim (...) that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. I re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. I conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. Chapter 2. There have been several recent results bringing into focus the super-intuitionistic nature of most notions of proof-theoretic validity. But there has been very little work evaluating the consequences of these results. In this chapter, I explore the question of whether these results undermine the claim that proof-theoretic validity shows us which inferences follow from the meaning of the connectives when defined by their introduction rules. It is argued that the super-intuitionistic inferences are valid due to the correspondence between the treatment of the atomic formulas and more complex formulas. And so the goals of proof-theoretic validity are not undermined. Chapter 3. Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This chapter resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The chapter further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic.Abstract prepared by Will Stafford extracted partially from the dissertation. (shrink)

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. (shrink)

Open future is incompatible with realism about possible worlds. Since realistically conceived (concrete or abstract) possible worlds are maximal in the sense that they contain/represent the full history of a possible spacetime, past and future included, if such a world is actual now, the future is fully settled now, which rules out openness. The kind of metaphysical indeterminacy required for open future is incompatible with the kind of maximality which is built into the concept of possible worlds. The paper discusses (...) various modal realist responses and argues that they provide ersatz openness only, or they lead to incoherence, or they render the resulting theory inadequate as a theory of modality. The paper also considers various accounts of the open future, including rejection of bivalence, supervaluationism, and the ‘thin red line’ view (TRL), and claims that a version of (TRL) can avoid the incompatibility problem, but only at the cost of deflating the notion of openness. (shrink)

The purpose of this brief note is to prove a limitative theorem for a generalization of the deduction theorem. I discuss the relationship between the deduction theorem and rules of inference. Often when the deduction theorem is claimed to fail, particularly in the case of normal modal logics, it is the result of a confusion over what the deduction theorem is trying to show. The classic deduction theorem is trying to show that all so-called ‘derivable rules’ can be encoded into (...) the object language using the material conditional. The deduction theorem can be generalized in the sense that one can attempt to encode all types of rules into the object language. When a rule is encoded in this way I say that it is reflected in the object language. What I show, however, is that certain logics which reflect a certain kind of rule must be trivial. Therefore, my generalization of the deduction theorem does fail where the classic deduction theorem didn’t. (shrink)

Inferences are familiar movements of thought, but despite important recent work on the topic, we do not yet have a fully satisfying theory of inference. Here I provide a functionalist theory of inference. I argue that the functionalist framework allows us the flexibility to meet various demands on a theory of inference that have been proposed (such as that it must explain inferential Moorean phenomena and epistemological ‘taking’). While also allowing us to compare, contrast, adapt, and combine features of (...) extant theories of inference into one unified theory. In fleshing out the inference role, I also criticize the common assumption that inference requires rule-following. (shrink)