We identify a class of paradoxes that is neither set-theoretical nor semantical, but that seems to depend on intensionality. In particular, these paradoxes arise out of plausible properties of propositional attitudes and their objects. We try to explain why logicians have neglected these paradoxes, and to show that, like the Russell Paradox and the direct discourse Liar Paradox, these intensional paradoxes are recalcitrant and challenge logical analysis. Indeed, when we take these paradoxes seriously, we may need to rethink the commonly (...) accepted methods for dealing with the logical paradoxes. (shrink)

In this book, Scott Soames defends the revolution in philosophy led by Saul Kripke, Hilary Putnam, and David Kaplan against attack from those wishing to revive ..

Many of our concepts are introduced to us via, and seem only to be constrained by, roughand-ready explanations and some sample paradigm positive and negative applications. This happens even in informal logic and mathematics. Yet in some cases, the concepts in question – although only informally and vaguely characterized – in fact have, or appear to have, entirely determinate extensions. Here’s one familiar example. When we start learning computability theory, we are introduced to the idea of an algorithmically computable function (...) (from numbers to numbers) – i.e. one whose value for any given input can be determined by a step-by-step calculating procedure, where each step is fully determined by some antecedently given finite set of calculating rules. We are told that we are to abstract from practical considerations of how many steps will be needed and how much ink will be spilt in the process, so long as everything remains finite. We are also told that each step is to be ‘small’ and the rules governing it must be ‘simple’, available to a cognitively limited calculating agent: for we want an algorithmic procedure, step-by-minimal-step, to be idiot-proof. For a classic elucidation of this kind, see e.g. Rogers (1967, pp. 1–5). Church’s Thesis, in one form, then claims this informally explicated concept in fact has a perfectly precise extension, the set of recursive functions. Church’s Thesis can be supported in a quasi-empirical way, by the failure of our searches for counterexamples. It can be supported too in a more principled way, by the observation that different appealing ways of sharpening up the informal chararacterization of algorithmic computability end up specifying the same set of recursive functions. But such considerations fall short of a demonstration of the Thesis. So is there a different argumentative strategy we could use, one that could lead to a proof? Sometimes it is claimed that there just can’t be, because you can never really prove results involving an informal concept like algorithmic computability.. (shrink)

This is a book that every enthusiast for Gödel’s proofs of his incompleteness theorems will want to own. It gives an up-to-date account of connections between systems of modal logic and results on provability in formal systems for arithmetic, analysis, and set theory.

The dogma that the propositional logic of metaphysical modality is S5 is rebutted. The author exposes fallacies in standard arguments supporting S5, arguing that propositional metaphysical modal logic is weaker even than both S4 and B, and is instead the minimal and weak metaphysical-modal logic T.

The two-dimensional modal logic of Davies and Humberstone [3] is an important aid to our understanding the relationship between actuality, necessity and a priori knowability. I show how a cut-free hypersequent calculus for 2D modal logic not only captures the logic precisely, but may be used to address issues in the epistemology and metaphysics of our modal concepts. I will explain how the use of our concepts motivates the inference rules of the sequent calculus, and then show that the completeness (...) of the calculus for Davies–Humberstone models explains why those concepts have the structure described by those models. The result is yet another application of the completeness theorem. (shrink)

The question, "Which modal logic is the right one for logical necessity?," divides into two questions, one about model-theoretic validity, the other about proof-theoretic demonstrability. The arguments of Halldén and others that the right validity argument is S5, and the right demonstrability logic includes S4, are reviewed, and certain common objections are argued to be fallacious. A new argument, based on work of Supecki and Bryll, is presented for the claim that the right demonstrability logic must be contained in S5, (...) and a more speculative argument for the claim that it does not include S4.2 is also presented. (shrink)

This paper recalls some applications of two-dimensional modal logic from the 1980s, including work on the logic of Actually and on a somewhat idealized version of the indicative/subjunctive distinction, as well as on absolute and relative necessity. There is some discussion of reactions this material has aroused in commentators since. We also survey related work by Leslie Tharp from roughly the same period.

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality (...) operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5. (shrink)

The traditional view that all logical truths are metaphysically necessary has come under attack in recent years. The contrary claim is prominent in David Kaplan’s work on demonstratives, and Edward Zalta has argued that logical truths that are not necessary appear in modal languages supplemented only with some device for making reference to the actual world (and thus independently of whether demonstratives like ‘I’, ‘here’, and ‘now’ are present). If this latter claim can be sustained, it strikes close to the (...) heart of the traditional view. I begin this paper by discussing and refuting Zalta’s argument in the context of a language for propositional modal logic with an actuality connective (section 1). This involves showing that his argument in favor of real world validity his preferred explication of logical truth, is fallacious. Next (section 2) I argue for an alternative explication of logical truth called general validity. Since the rule of necessitation preserves general validity, the argument of section 2 provides a reason for affirming the traditional view. Finally (section 3) I show that the intuitive idea behind the discredited notion of real world validity finds legitimate expression in an object language connective for deep necessity. (shrink)

The paper studies many-dimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: p-morphisms, the finite depth method, normal forms, filtrations. Applications to first order predicate logics are considered too. The introduction and the conclusion contain a discussion of many related results and open problems in the area.

Epistemic two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. While this theory is usually presented in an informal manner, I take some steps in formalizing it in this paper. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of epistemic two-dimensional semantics. I also describe (...) some properties of the logic that are interesting from a philosophical perspective, and apply it to the so-called nesting problem. (shrink)

This volume succeeds the same authors' well-known An Introduction to Modal Logic and A Companion to Modal Logic. We designate the three books and their authors NIML, IML, CML and H&C respectively. Sadly, George Hughes died partway through the writing of NIML.

Graeme Forbes (2011) raises some problems for two-dimensional semantic theories. The problems concern nested environments: linguistic environments where sentences are nested under both modal and epistemic operators. Closely related problems involving nested environments have been raised by Scott Soames (2005) and Josh Dever (2007). Soames goes so far as to say that nested environments pose the “chief technical problem” for strong two-dimensionalism. We call the problem of handling nested environments within two-dimensional semantics “the nesting problem”. We show that the two-dimensional (...) semantics for attitude ascriptions developed in Chalmers (2011a) has no trouble accommodating certain forms of the nesting problem that involve factive verbs such as “know” or “establish”. A certain form of the nesting problem involving apriority and necessity operators does raise an interesting puzzle, but we show how a generalized version of the nesting problem arises independently of two-dimensional semantics—it arises, in fact, for anyone who accepts the contingent a priori. We, then, provide a two-dimensional treatment of the apriority operator that fits the two-dimensional treatment of attitude verbs and apply it to the generalized nesting problem. We conclude that two-dimensionalism is not seriously threatened by cases involving the nesting of epistemic and modal operators. (shrink)

When I say ‘Hesperus is Phosphorus’, I seem to express a proposition. And when I say ‘Joan believes that Hesperus is Phosphorus’, I seem to ascribe to Joan an attitude to the same proposition. But what are propositions? And what is involved in ascribing propositional attitudes?

We show that every proper normal extension of the bi-modal system S5 2 has the poly-size model property. In fact, to every proper normal extension L of S5 2 corresponds a natural number b(L) - the bound of L. For every L, there exists a polynomial P(·) of degree b(L) + 1 such that every L-consistent formula is satisfiable on an L-frame whose universe is bounded by P(||), where || denotes the number of subformulas of . It is shown that (...) this bound is optimal. (shrink)

We show that every proper normal extension of the bi-modal system S52 has the poly-size model property. In fact, to every proper normal extension L of S52 corresponds a natural number b(L) - the bound of L. For every L, there exists a polynomial P(·) of degree b(L) + 1 such that every L-consistent formula ϕ is satisfiable on an L-frame whose universe is bounded by P(|ϕ|), where |ϕ| denotes the number of subformulas of ϕ. It is shown that this (...) bound is optimal. (shrink)

We provide for the first time an exact translation into English of the Polish version of Alfred Tarski's classic 1936 paper, whose title we translate as ?On the Concept of Following Logically?. We also provide in footnotes an exact translation of all respects in which the German version, used as the basis of the previously published and rather inexact English translation, differs from the Polish. Although the two versions are basically identical, to an extent that is even uncanny, we note (...) more than 400 differences. Several dozen of these are substantive differences due to revisions by Tarski to the Polish version which he did not incorporate in the German version. With respect to these revisions the Polish version should be regarded as more authoritative than the German. Hence scholars limited to an English translation should use ours. (shrink)

Timothy Williamson gives an original and provocative treatment of deep metaphysical questions about existence, contingency, and change, using the latest resources of quantified modal logic. Contrary to the widespread assumption that logic and metaphysics are disjoint, he argues that modal logic provides a structural core for metaphysics.

The book is an extended study of the problem of consciousness. After setting up the problem, I argue that reductive explanation of consciousness is impossible , and that if one takes consciousness seriously, one has to go beyond a strict materialist framework. In the second half of the book, I move toward a positive theory of consciousness with fundamental laws linking the physical and the experiential in a systematic way. Finally, I use the ideas and arguments developed earlier to defend (...) a form of strong artificial intelligence and to analyze some problems in the foundations of quantum mechanics. (shrink)

It will be an essential resource for philosophers, mathematicians, computer scientists, linguists, or any scholar who finds connectives, and the conceptual issues surrounding them, to be a source of interest.This landmark work offers both ...

This long-awaited book replaces Hughes and Cresswell's two classic studies of modal logic: _An Introduction to Modal Logic_ and _A Companion to Modal Logic_. _A New Introduction to Modal Logic_ is an entirely new work, completely re-written by the authors. They have incorporated all the new developments that have taken place since 1968 in both modal propositional logic and modal predicate logic, without sacrificing tha clarity of exposition and approachability that were essential features of their earlier works. The book takes (...) readers from the most basic systems of modal propositional logic right up to systems of modal predicate with identity. It covers both technical developments such as completeness and incompleteness, and finite and infinite models, and their philosophical applications, especially in the area of modal predicate logic. (shrink)

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does (...) is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency. (shrink)

Why is two-dimensional semantics important? One can think of it as the most recent act in a drama involving three of the central concepts of philosophy: meaning, reason, and modality. First, Kant linked reason and modality, by suggesting that what is necessary is knowable a priori, and vice versa. Second, Frege linked reason and meaning, by proposing an aspect of meaning (sense) that is constitutively tied to cognitive signi?cance. Third, Carnap linked meaning and modality, by proposing an aspect of meaning (...) (intension) that is constitutively tied to possibility and necessity. (shrink)

Two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. Usually, this theory is presented in an informal manner. In this thesis, I take first steps in formalizing it, and use the formalization to present some considerations in favor of two-dimensional semantics. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that (...) captures the relevant ideas of two-dimensional semantics. I use this to show that some criticisms of two-dimensional semantics that claim that the theory is incoherent are not justified. I also axiomatize the logic, and compare it to the most important proposals in the literature that define similar logics. To indicate that two-dimensional semantics is a plausible semantic theory, I give an argument that shows that all theorems of the logic can be philosophically justified independently of two-dimensional semantics. (shrink)

Recent work on the philosophy of modality has tended to pass over questions about iterated modalities in favour of constructing ambitious metaphysical theories of possibility and necessity, despite the central importance of iterated modalities to modal logic. Yet there are numerous unresolved but fundamental issues involving iterated modalities: Chandler and Salmon have provided forceful arguments against the widespread assumption that all necessary truths are necessarily necessary, for example. The current paper examines a range of ways in which one might seek (...) to identify limited regions within which some of the most well-known principles featuring iterated modalities may safely be assumed. (shrink)