Unifiability of terms (and formulas) and structural completeness in the variety of relation algebras RA and in the products of modal logic S5 is investigated. Nonunifiable terms (formulas) which are satisfiable in varieties (in logics) are exhibited. Consequently, RA and products of S5 as well as representable diagonal-free n-dimensional cylindric algebras, RDfn, are almost structurally complete but not structurally complete. In case of S5n a basis for admissible rules and the form of all passive rules are provided.

We prove that every n-modal logic between K n and S5 n is undecidable, whenever n ≥ 3. We also show that each of these logics is non- finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov-Fine frame formulas with algebraic logic results of Halmos, (...) Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite relation algebras. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of (...) the representation problem of finite relation algebras. (shrink)

This book is an introductory logic text of moderate difficulty which contains added topics not usually found in an introductory book. The book has two parts--basic logic and advanced logic. The basic logic contains propositional logic through conditional proofs, syllogistic logic, the fundamentals of set theory and their application to both syllogistic and non-syllogistic inferences along with the use of Venn and Carroll diagrams, and concludes with predicate logic using the rules for Universal (...) Instantiation, Existential Instantiation, Universal Generalization, and Existential Generalization. The part on advanced logic begins by extending the predicate logic to identity, descriptions, and relations. A presentation of modal logic includes C. I. Lewis' modal systems S3, S4, and S5; modal logic with quantification; epistemic, doxastic, and deontic modalities; and mixed modalities. A chapter on the logic of ordinary language considers analytic statements, empty expressions, and equivalence and implication in ordinary language. The author also discusses "tools of analysis" such as definition, the theory of meaning, probability theory, and scientific inference. Each chapter concludes with a section entitled "Philosophical Applications" or "Philosophical Difficulties." One of these sections contains the author's version of Hartshorne's modal proof of Anselm's ontological argument. Answers are provided for the even-numbered exercises.--T. G. N. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of (...) the (undecidable) representation problem of finite relation algebras. (shrink)

A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical (...) consequence according to which α1, ..., αn entails β just in case □α1, ..., □αn entails □β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status. (shrink)

This is the first, out of two papers, in which we identify all logics between C1 and S5 having the same theses without iterated modalities. All these logics canbe divided into certain groups. Each such group depends only on which of thefollowing formulas are theses of all logics from this group:,,, ⌜∨ ☐q⌝,and for any n > 0 a formula ⌜ ∨ ⌝, where has not the atom ‘q’, and and have no common atom. We generalize Pollack’s result from [12],where (...) he proved that all modal logics between S1 and S5 have the same theseswhich does not involve iterated modalities. (shrink)

This is the second, out of two papers, in which we identify all logics between C1 and S5 having the same theses without iterated modalities. All these logics can be divided into certain groups. Each such group depends only on which of the following formulas are theses of all logics from this group:,,, ⌜∨☐q⌝, and for any n > 0 a formula ⌜ ∨ ⌝, where has not the atom ‘q’, and and have no common atom. We generalize Pollack’s result (...) from [1], where he proved that all modal logics between S1 and S5 have the same theses which does not involve iterated modalities. (shrink)

The simplest quantified modal logic combines classical quantification theory with the propositional modal logic K. The models of simple QML relativize predication to possible worlds and treat the quantifier as ranging over a single fixed domain of objects. But this simple QML has features that are objectionable to actualists. By contrast, Kripke-models, with their varying domains and restricted quantifiers, seem to eliminate these features. But in fact, Kripke-models also have features to which actualists object. Though these (...) philosophers have introduced variations on Kripke-models to eliminate their objectionable features, the most well-known variations all have difficulties of their own. The present authors reexamine simple QML and discover that, in addition to having a possibilist interpretation, it has an actualist interpretation as well. By introducing a new sort of existing abstract entity, the contingently nonconcrete, they show that the seeming drawbacks of the simplest QML are not drawbacks at all. Thus, simple QML is independent of certain metaphysical questions. (shrink)

We prove that the existence of a measurable cardinal is equivalent to the existence of a normal space whose modal logic coincides with the modal logic of the Kripke frame isomorphic to the powerset of a two element set.

We show, first, that the normalization procedure for S4 modal logic presented by Dag Prawitz in [5] does not work. We then develop a new natural deduction system for S4 classical modal logic that is logically equivalent to that of Prawitz, and we show that every derivation in this new system can be transformed into a normal derivation.

We show, first, that the normalization procedure for S4 modal logic presented by Dag Prawitz in [5] does not work. We then develop a new natural deduction system for S4 classical modal logic that is logically equivalent to that of Prawitz, and we show that every derivation in this new system can be transformed into a normal derivation.

Within ZFC, we develop a general technique to topologize trees that provides a uniform approach to topological completeness results in modal logic with respect to zero-dimensional Hausdorff spaces. Embeddings of these spaces into well-known extremally disconnected spaces then gives new completeness results for logics extending S4.2.

The relationship between formal logic and general philosophy is discussed under headings such as A Re-examination of Our Tense-Logical Postulates, Modal Logic in the Style of Frege, and Intentional Logic and Indeterminism.

PA is Peano Arithmetic. Pr(x) is the usual Σ1-formula representing provability in PA. A strong provability predicate is a formula which has the same properties as Pr(·) but is not Σ1. An example: Q is ω-provable if PA + ¬ Q is ω-inconsistent (Boolos [4]). In [5] Dzhaparidze introduced a joint provability logic for iterated ω-provability and obtained its arithmetical completeness. In this paper we prove some further modal properties of Dzhaparidze's logic, e.g., the fixed point property (...) and the Craig interpolation lemma. We also consider other examples of the strong provability predicates and their applications. (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)

The formulation of propositional modal logic is revised by interposing a domain of structured propositions between the modal language and the models. Interpretations of the language (i.e., ways of mapping the language into the domain of propositions) are distinguished from models of the domain of propositions (i.e., ways of assigning truth values to propositions at each world), and this contrasts with the traditional formulation. Truth and logical consequence are defined, in the first instance, as properties of, and (...) relations among, propositions. -/- These definitions have certain interesting consequences. One is that they resolve a question that cannot be answered by the traditional analysis, in which Kripke models directly interpret modal language. The question is, is the reason a modal sentence has a different truth value at other possible worlds due to the fact that the sentence has a different meaning at that world or due to a change in the world? The philosophical conception elaborates the second answer. Another interesting consequence is that modal language, properly speaking, is not intensional, at least according to the classic tests for, and definitions of, intensionality. (shrink)

This book was designed primarily as a textbook; though the author hopes that it will prove to be of interests to others beside logic students. Part I of this book covers the fundamentals of the subject the propositional calculus and the theory of quantification. Part II deals with the traditional formal logic and with the developments which have taken that as their starting-point. Part III deals with modal, three-valued, and extensional systems.

The author revises the formulation of propositional modal logic by interposing a domain of structured propositions between the modal language and the models. Interpretations of the language (i.e., ways of mapping the language into the domain of propositions) are distinguished from models of the domain of propositions (i.e., ways of assigning truth values to propositions at each world), and this contrasts with the traditional formulation. Truth and logical consequence are defined, in the first instance, as properties of, (...) and relations among, propositions. (shrink)

Surveys and extens work that has been done in the past two years on 'tense logic' and is a sequel to the author's book, Time and Modality.

In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability (...) semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus. (shrink)

But Findlay's remark, like so much that has been written on the subject of time in the present century, was provoked in the first place by McTaggart's ...

Pietruszczak (Bull Sect Log 38(3/4):163–171, 2009) proved that the normal logics K45 , KB4 (=KB5), KD45 are determined by suitable classes of simplified Kripke frames of the form ⟨W,A⟩ , where A⊆W. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of simplified frames if and only if it is a normal extension of K45. Furthermore, a modal logic is a normal extension of K45 (resp. (...) KD45; KB4; S5) if and only if it is determined by a set consisting of finite simplified frames (resp. such frames with A≠∅; such frames with A=W or A=∅; such frames with A=W). Secondly, for all normal extensions of K45, KB4, KD45 and S5, in particular for extensions obtained by adding the so-called “verum” axiom, Segerberg’s formulas and/or their T-versions, we prove certain versions of Nagle’s Fact (J Symbol Log 46(2):319–328, 1981) (which concerned normal extensions of K5). Thirdly, we show that these extensions are determined by certain classes of finite simplified frames generated by finite subsets of the set N of natural numbers. In the case of extensions with Segerberg’s formulas and/or their T-versions these classes are generated by certain finite subsets of N. (shrink)

Using a quantified propositional logic involving the operators it is known that and it is possible to know that, we formalize various interesting philosophical claims involved in the realism debate. We set out inferential rules for the epistemic modalities, ranging from ones that are obviously analytic, to ones that are epistemologically more substantive or even controversial. Then we investigate various aporias for the realism debate. These are constructively inconsistent triads of claims from our list: a claim expressing some sort (...) of common ground in the debate; a characteristically anti-realist thesis about truth and knowability; and a characteristically realist thesis about determinacy of truth value. Various patterns of reductio proof for these inconsistent triads are generated, so as to display their variety. The reductio proofs use only the inferential rules set out earlier. The philosophical utility of each aporia for the anti-realist is then assessed. This involves consideration of the acceptability of the premiss expressing common ground; the strength and plausibility of the anti-realist premiss; the strength of the realist premiss that is the ultimate target of the reductio; and the analytic status of the inferential rules applied within the proof. (shrink)

Hintikka, J. Knowing how, knowing that, and knowing what: observations on their relation in Plato and other Greek philosophers.--Hedenius, I. The concept of punishment.--Marc-Wogau, K. On the concept of dialectial development in Marxism.--Ekelöf, P. O. Definitions and concept formation in the law.--Hermerén, G. The existence of aesthetic qualities.--Regnéll, H. Explanation in analytical philosophy.--Furberg, M. On questions and pseudo-problems.--Moritz, M. Imperative implication and conditional imperatives.--Sosa, E. Standard conditions.--Danielsson, S. On the strength of commitments.--Aqvist, L. The emotive theory of ethics in the (...) light of recent developments in formal semantics and pragmatics.--Von Wright, G. H. Truth as modality. A contribution to the logic of sense and nonsense.--Hansson, B. and Gärdenfors, P. A guide to intentional semantics.--Kanger, S. Entailment.--Edman, M. Adding independent pieces of evidence.--Lindström, P. A characterization of elementary logic.--Woodruff, P. W. On constructive nonsense logic.--Segerberg, K. Halldén's theorem on Post completeness.--Philosophical works by Sören Halldén. (shrink)

We introduce a type of weighted modal logic with explicit weights both in the language and in the models. The framework has its applications in epistemic logic for reasoning about agents’ knowledge based on their capability, and in deontic logic for agents’ choices based on their deontic capability or utilities. We make use of weighted Kripke models with the weights understood epistemically as a similarity measure between states and deontically as a measure of expected utilities. We (...) present sound and complete axiomatizations for the logics, and discuss variants and possible extensions. (shrink)

This volume is a collection of my essays on philosophy of logic from a phenomenological perspective. They deal with the four kinds of logic I have been concerned with: formal logic, transcendental logic, speculative logic and hermeneutic logic. Of these, only one, the essay on Hegel, touches upon 'speculative logic', and two, those on Heidegger and Konig, are concerned with hermeneutic logic. The rest have to do with Husser! and Kant. I have (...) not tried to show that the four logics are compatible. I believe, they are--once they are given a phenomenological underpinning. The original plan of writing an Introduction in which the issues would have to be formulated, developed and brought together, was abandoned in favor of writing an Introductory Essay on the 'origin'- in the phenomenological sense -of logic. J.N.M. Philadelphia INTRODUCTION: THE ORIGIN OF LOGIC The question of the origin of logic may pertain to historical origin (When did it all begin? Who founded the science of logic?), psychological origin (When, in the course of its mental development, does the child learn logical operations?), cultural origin (What cultural - theological, metaphysical and linguisti- conditions make such a discipline as logic possible?), or transcendental constitutive origin (What sorts of acts and/or practices make logic possible?). (shrink)

This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of linear logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic linear logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of (...) Kripke resource models extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatisation and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the l.. (shrink)

Some recently-proposed counterexamples to the traditional definition of essential property do not require a separate logic of essence. Instead, the examples can be analysed in terms of the logic and theory of abstract objects. This theory distinguishes between abstract and ordinary objects, and provides a general analysis of the essential properties of both kinds of object. The claim ‘x has F necessarily’ becomes ambiguous in the case of abstract objects, and in the case of ordinary objects there are (...) various ways to make the definition of ‘F is essential to x’ more fine-grained. Consequently, the traditional definition of essential property for abstract objects in terms of modal notions is not correct, and for ordinary objects the relationship between essential properties and modality, once properly understood, addresses the counterexample. (shrink)