This unique book provides a gentle introduction to an increasingly important type of formal logic called CLAUSAL FORM LOGIC (CFL). Having evolved out of human reasoning, CFL represents the ideal to which all computer-based systems must approximate. Most present day computational logic systems are based on it, including the well-know artificial intelligence language PROLOG.

Hence, this book's provocative claim: today's so-called logical empiricism owes much more to Kant's notion of science than to Hume's.

We investigate a construction schema for first-order logical systems, called “form logic”. Form logic allows us to overcome the dualistic commitment of predicate logic to individual constants and predicates. Dualism is replaced by a pluralism of terms of different “logical forms”. Individual form-logical systems are generated by the determination of a range of logical forms and of the formbased syntax rules for combining terms into formulas. We develop a generic syntax and semantics for such (...) systems and provide a completeness proof for them. To illustrate the idea of form logic, and the possibilities it facilitates, we discuss three particular systems, one of which is the form-logical reconstruction of standard first-order predicate logic. (shrink)

A common reaction to Hegel’s suggestion that we collapse Kant’s distinction between form and content is that, since such a move would also deprive us of any way of distinguishing the merely logical from the real possibility of our concepts, it is incoherent and ought to be rejected. It is true that these two distinctions are intimately related in Kant, such that if one goes, the other does as well. But it is less obvious that giving them up as (...) Kant conceives them is as incoherent a proposal as many of Hegel’s critics think. It has been the point of a recent account of Hegel’s idealism to demonstrate that his critique of Kant’s dichotomy between form and content or concept and intuition does not commit him to the view that human cognition is materially creative of its content in the manner of a God-like or intuitive intellect—does not, in other words, signify a return to a pre-Critical metaphysics. This is the interpretative stance I adopt here as well, in hope of giving defenders of Kant reasons for taking Hegel more seriously. (shrink)

_Questions of Form _was first published in 1989. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. In _Questions on Form_, Joelle Proust traces the concept of the analytic proposition from Kant's development of the notion down to its place in the work of Rudolf Carnap, a founder of logical empiricism and a key figure in contemporary analytic philosophy. Using a method known in (...) France as _topique comparative_,she provides a rigorous exposition of analyticity, situating it within four major philosophical systems—those of Kant, Bolzano, Frege, and Carnap—and clearly delineating its development from one system to the next. Proust takes as her point of departure Kant's distinction between analytic and synthetic judgments. Though she makes clear that Kant drew on Locke, Hume, and Leibniz, she argues that his notion of analyticity was innovative, not simply an elaboration of something already found in their work. She shows that the analytic proposition unexpectedly came to play an important part in efforts to convert problems considered "transcendental" into questions of belonging to formal logic. Ultimately, her comparison of their systems reveals that the concept of the analytic, however specific its rile in each, remains linked to a foundationalist strategy—in effect, to the transcendentalist questions Kant used when he reinterpreted the findings of his empiricist predecessors. Hence, this book's provocative claim: today's so-called logical empiricism owes much more to Kant's notion of science than to Hume's. (shrink)

Logical form has always been a prime concern for philosophers belonging to the analytic tradition. For at least one century, the study of logical form has been widely adopted as a method of investigation, relying on its capacity to reveal the structure of thoughts or the constitution of facts. This book focuses on the very idea of logical form, which is directly relevant to any principled reflection on that method. Its central thesis is that there is no (...) such thing as a correct answer to the question of what is logical form: two significantly different notions of logical form are needed to fulfil two major theoretical roles that pertain respectively to logic and to semantics. This thesis has a negative and a positive side. The negative side is that a deeply rooted presumption about logical form turns out to be overly optimistic: there is no unique notion of logical form that can play both roles. The positive side is that the distinction between two notions of logical form, once properly spelled out, sheds light on some fundamental issues concerning the relation between logic and language. (shrink)

The Joint International Conference on Logic Programming, sponsored by the Association for Logic Programming, is a major forum for presentations of research, applications, and implementations in this important area of computer science. Logic programming is one of the most promising steps toward declarative programming and forms the theoretical basis of the programming language Prolog and its various extensions. Logic programming is also fundamental to work in artificial intelligence, where it has been used for nonmonotonic and commonsense (...) reasoning, expert systems implementation, deductive databases, and applications such as computer-aided manufacturing. Krzysztof R. Apt is project leader at the Centre for Mathematics and Computer Science in Amsterdam and part-time Professor in Computer Science at the University of Amsterdam. Topics covered: Theory Foundations. Programming Languages. Implementation. Programming Methodologies and Tools. Applications. Deductive Databases. Artificial Intelligence. Parallelism. (shrink)

Forms of thought are involved whenever we name, describe, or identify things, and whenever we distinguish between what is, might be, or must be the case. It appears to be a distinctive feature of human thought that we can have modal thoughts, about what might be possible or necessary, and conditional thoughts, about what would or might be the case if something else were the case. Even the simplest thoughts are structured like sentences, containing referential and predicative elements, and studying (...) these structures is the main task of philosophical logic. This clear and accessible book investigates the forms of thought, drawing out and focusing on the central logical notions of reference, predication, identity, modality and conditionality. It will be useful to students and other interested readers in epistemology and metaphysics, philosophy of mind and language, and philosophical logic. (shrink)

Chapter. 1. Logical. Form. as. a. Level. of. Linguistic. Representation. What is the relation of a sentence's syntactic form to its logical form? This issue has been of central concern in modern inquiry into the semantic properties of natural ...

We extend the relational algebra of Chin and Tarski so that it is multisorted or, as we prefer, typed. Each type supports a local Boolean algebra outfitted with a converse operator. From Lyndon, we know that relation algebras cannot be represented as proper relation algebras where a proper relation algebra has binary relations as elements and the algebra is singly-typed. Here, the intensional conjunction, which was to represent relational composition in Chin and Tarski, spans three different local algebras, thus the (...) term distributed in the title. Since we do not rely on proper relation algebras, we are free to re-express the algebras as typed. In doing so, we allow many different intensional conjunction operators. We construct a typed logic over these algebras, also known as heterogeneous algebras of Birkhoff and Lipson. The logic can be seen as a form of relevance logic with a classical negation connective where the Routley-Meyer star operator is reified as a converse connective in the logic. Relevance logic itself is not typed but our work shows how it can be made so. Some of the properties of classical relevance logic are weakened from Routley-Meyer’s version which is too strong for a logic over relation algebras. (shrink)

Theories of adverbial modification can be roughly distinguished into two sorts. One kind of theory takes logical form to follow surface grammatical form. Adverbs are treated as unanalyzable logical operators that turn a predicate or sentence into a different predicate or sentence respectively. And new rules of logic are stated for these operators. -/- A different kind of theory does not suppose that logical form must parallel surface grammatical form. It allows that logical form (...) may have more to do with deeper structures that might be studied in transformational grammar. Adverbs are treated as surface forms of the underlying predicates represented by corresponding adjectives and verbs. 'Slowly' is derived from 'slow'; 'intentionally' from 'intentional' or 'intend'; etc. And new rules of logic are avoided where they can be. -/- In this paper I attempt to state some of the advantages of the second sort of theory. My procedure will be this. First, I will try to say in outline what theories of logical form are. Then I will state five principles for evaluating such theories. Next, I will sketch the sorts of analyses acceptance of principles (1)-(5) leads to. In particular I will talk about adverbial phrases (e.g. locatives) that are best analyzed in terms of implicit references to events, relative modifiers (like 'large') which relate something to a comparison class, and 'that' clauses taken as names of propositions. By appealing to principles (1)-(5) I will defend these analyses against certain others, one that appeals to many logical operators, a second that treats all sentences as names of propositions, and a third that sees implicit reference to possible worlds in the language being analyzed. Finally, I will offer a pragmatic defense of my approach in terms of principles (1)-(5) as against a different approach that appeals to possible world semantics. (shrink)

Some apparently valid arguments crucially rely on context change. To take a kind of example first discussed by Frege, ‘Tomorrow, it’ll be sunny’ taken on a day seems to entail ‘Today, it’s sunny’ taken on the next day, but the first sentence taken on a day sadly does not seem to entail the second sentence taken on the second next day. Mid-argument context change has not been accounted for by the tradition that has extensively studied the distinctive logical properties of (...) context-dependent languages, for that tradition has focussed on arguments whose premises and conclusions are taken at the same context. I first argue for the desiderability of having a logic that accounts for mid-argument context change and I explain how one can informally understand such context change in a standard framework in which the relation of logical consequence holds among sentences. I then propose a family of simple temporal “intercontextual” logics that adequately model the validity of certain arguments in which the context changes. In particular, such logics validate the apparently valid argument in the Fregean example. The logics lack many traditional structural properties (reflexivity, contraction, commutativity etc.) as a consequence of the logical significance acquired by the sequence structure of premises and conclusions. The logics are however strong enough to capture in the form of logical truths all the valid arguments of both classical logic and Kaplan-style “intracontextual” logic. Finally, I extend the framework by introducing new operations into the object language, such as intercontextual conjunction, disjunction and implication, which, contrary to intracontextual conjunction, disjunction and implication, perfectly match the metalinguistic, intercontextual notions of premise combination, conclusion combination and logical consequence by representing their respective two operands as taken at different contexts. (shrink)

This is a first course in propositional modal logic, suitable for mathematicians, computer scientists and philosophers. Emphasis is placed on semantic aspects, in the form of labelled transition structures, rather than on proof theory. The book covers all the basic material - propositional languages, semantics and correspondence results, proof systems and completeness results - as well as some topics not usually covered in a modal logic course. It is written from a mathematical standpoint. To help the reader, (...) the material is covered in short chapters, each concentrating on one topic. These are arranged into five parts, each with a common theme. An important feature of the book is the many exercises and an extensive set of solutions is provided. (shrink)

Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments. The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably construed, in accordance with (...) an abundant or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational. (shrink)

Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truth-conditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the `logicality of language', accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter-examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired (...) with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is `blind' to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non-classical---indeed quite exotic---kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis-a-vis its open class terms and employs a deductive system that is basically classical. (shrink)

The paper offers an interpretation of thesis 6.01. The treatment touches upon variables, identity, elementary propositions, internal relations. Klammerausdrücke, and operations. Wittenstein's notations are found not to cover the particular form of definition by induction that is used at 6 and 6.01. It is concluded that Wittgenstein's ability to design of a formal system of logic does not match his outstanding logico-philosophical insight.

The word ‘logic’ has many senses. Here we will understand it as meaning an account of what follows from what and why. With contemporary methodology, logic in this sense – though it may not always have been thought of in this way – is a branch of applied mathematics. This has various implications for how one understands a number of issues concerning validity. In this paper I will explain this perspective of logic, and explore some of its (...) consequences with respect to the notion of logical form. (shrink)

Logical Forms explains both the detailed problems involved in finding logical forms and also the theoretical underpinnings of philosophical logic. In this revised edition, exercises are integrated throughout the book. The result is a genuinely interactive introduction which engages the reader in developing the argument. Each chapter concludes with updated notes to guide further reading.

This paper outlines a truth-conditional view of logical form, that is, a view according to which logical form is essentially a matter of truth-conditions. The main motivation for the view is a fact that seems crucial to logic. As _§_1 suggests, fundamental logical relations such as entailment or contradiction can formally be explained only if truth-conditions are formally represented.§2 spells out the view. _§_3 dwells on its anity with a conception of logical form that has been (...) defended in the past. _§§_4-6 draw attention to its impact on three major issues that concern, respectively, the extension of the domain of formal explanation, the semantics of tensed discourse, and the analysis of quantication. (shrink)

Antoine Arnauld and Pierre Nicole were philosophers and theologians associated with Port-Royal Abbey, a centre of the Catholic Jansenist movement in seventeenth-century France. Their enormously influential Logic or the Art of Thinking, which went through five editions in their lifetimes, treats topics in logic, language, theory of knowledge and metaphysics, and also articulates the response of 'heretical' Jansenist Catholicism to orthodox Catholic and Protestant views on grace, free will and the sacraments. In attempting to combine the categorical theory (...) of the proposition with a Cartesian account of knowledge, their Logic represents the classical view of judgment which inspired the modern transformation in logic and semantic theory by Frege, Russell, Wittgenstein and recent philosophers. This edition presents a new translation of the text, together with a historical introduction and suggestions for further reading. (shrink)

My first paper on the Is/Ought issue. The young Arthur Prior endorsed the Autonomy of Ethics, in the form of Hume’s No-Ought-From-Is (NOFI) but the later Prior developed a seemingly devastating counter-argument. I defend Prior's earlier logical thesis (albeit in a modified form) against his later self. However it is important to distinguish between three versions of the Autonomy of Ethics: Ontological, Semantic and Ontological. Ontological Autonomy is the thesis that moral judgments, to be true, must answer to (...) a realm of sui generis non-natural PROPERTIES. Semantic autonomy insists on a realm of sui generis non-natural PREDICATES which do not mean the same as any natural counterparts. Logical Autonomy maintains that moral conclusions cannot be derived from non-moral premises.-moral premises with the aid of logic alone. Logical Autonomy does not entail Semantic Autonomy and Semantic Autonomy does not entail Ontological Autonomy. But, given some plausible assumptions Ontological Autonomy entails Semantic Autonomy and given the conservativeness of logic – the idea that in a valid argument you don’t get out what you haven’t put in – Semantic Autonomy entails Logical Autonomy. So if Logical Autonomy is false – as Prior appears to prove – then Semantic and Ontological Autonomy would appear to be false too! I develop a version of Logical Autonomy (or NOFI) and vindicate it against Prior’s counterexamples, which are also counterexamples to the conservativeness of logic as traditionally conceived. The key concept here is an idea derived in part from Quine - that of INFERENCE-RELATIVE VACUITY. I prove that you cannot derive conclusions in which the moral terms appear non-vacuously from premises from which they are absent. But this is because you cannot derive conclusions in which ANY (non-logical) terms appear non-vacuously from premises from which they are absent Thus NOFI or Logical Autonomy comes out as an instance of the conservativeness of logic. This means that the reverse entailment that I have suggested turns out to be a mistake. The falsehood of Logical Autonomy would not entail either the falsehood Semantic Autonomy or the falsehood of Ontological Autonomy, since Semantic Autonomy only entails Logical Autonomy with the aid of the conservativeness of logic of which Logical Autonomy is simply an instance. Thus NOFI or Logical Autonomy is vindicated, but it turns out to be a less world-shattering thesis than some have supposed. It provides no support for either non-cognitivism or non-naturalism. (shrink)

The paper presents a demarcation of a “minimalistic” concept of logical form, which nevertheless largely agrees with the way the term “logical form” is commonly used in contemporary logic and philosophy of logic. We see logical forms as formulas of formal languages assigned to (compounds of) sentences of a natural language (perhaps modulo notational variance). We thus reject the views of logical forms as underlying structures of thoughts or of the material reality that surrounds us. The (...) assignment of the forms, we claim, aims at envisaging the logical (especially inferential) properties of the analyzed sentences, arguments, or other texts, as the analyzing formulas wear them, as it were, on their sleeves. Hence we suggest an “expressivist” and a pragmatic understanding of logical forms - they are used to expose (and fix) logical properties of sentences in textual contexts and the way of their employment is determined by the goals of the particular studies. (shrink)

This paper explores the relation between the philosophical idea that logic is a science studying logical forms, and a mathematical feature of logical systems called the principle of uniform substitution, which is often regarded as a technical counterpart of the philosophical idea. We argue that at least in one interesting sense the principle of uniform substitution does not capture adequately the requirement that logic is a matter of form and that logical truths are formal truths. We show (...) that some specific logical expressions can produce propositions of different kinds and the resulting diversity of informational types can lead to a justified failure of uniform substitution without undermining the view that logic is a purely formal discipline. (shrink)

Formalizations in first-order logic are standardly used to represent logical forms of sentences and to show the validity of ordinary-language arguments. Since every sentence admits of a variety of formalizations, a challenge arises: why should one valid formalization suffice to show validity even if there are other, invalid, formalizations? This paper suggests an explanation with reference to criteria of adequacy which ensure that formalizations are related in a hierarchy of more or less specific formalizations. This proposal is then compared (...) with stronger criteria and assumptions, especially the idea that sentences essentially have just one logical form. (shrink)

Much work in MKM depends on the application of formal logic to mathematics. However, much mathematical knowledge is informal. Luckily, formal logic only represents one tradition in logic, specifically the modeling of inference in terms of logical form. Many inferences cannot be captured in this manner. The study of such inferences is still within the domain of logic, and is sometimes called informal logic. This paper explores some of the benefits informal logic may (...) have for the management of informal mathematical knowledge. (shrink)

Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truth‐conditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the ‘logicality of language’, accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter‐examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired (...) with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is ‘blind’ to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non‐classical—indeed quite exotic—kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis‐á‐vis its open class terms and employs a deductive system that is basically classical. (shrink)

The standard view of logical form is that logical forms are synthetic structures which are the forms of sentences and of other linguistic entities. This is often associated with a more general linguistic view of logic which is articulated in different ways by various authors. This paper contains a critical discussion of such linguistic approaches to logical form, with special emphasis on Quine’s formulation of a logical grammar in Philosophy of Logic. An account of logical forms (...) as higher-order properties, which essentially builds on Frege’s analysis of quantification as higher-order predication, is suggested at the end. (shrink)

Logical Forms explains both the detailed problems involved in finding logical forms and also the theoretical underpinnings of philosophical logic. In this revised edition, exercises are integrated throughout the book. The result is a genuinely interactive introduction which engages the reader in developing the argument. Each chapter concludes with updated notes to guide further reading.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. (...) Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

This work contains Peter Long's important essay, Logic, Form and Grammar , which resolves many difficulties for the logical form of an argument where the reasoning is hypothetical. Also included are two essays on classical problems in philosophical logic, relating to logical form and formal relations.

Donald Davidson contributed to the discussion of logical form in two ways. On the one hand, he made several influential suggestions on how to give the logical forms of certain constructions of natural language. His account of adverbial modification and so called action-sentences is nowadays, in some form or other, widely employed in linguistics (Harman (forthcoming) calls it "the standard view"). Davidson's approaches to indirect discourse and quotation, while not as influential, also still attract attention today. On the (...) other hand, Davidson provided a general account of what logical form is. This paper is concerned with this general account. Its foremost aim is to give a faithful and detailed picture of what, according to Davidson, it means to give the logical form of a sentence. The structure of the paper is as follows. (1) I will first informally introduce a notion of logical form as the form that matters in certain kinds of entailments, and indicate why philosophers have taken an interest in such a notion. (2) The second section develops constraints that we should arguably abide by in giving an account of logical form. (3) I then turn to Davidson’s view of what is involved in giving such an account. To this end, I will try to reconstruct Davidson’s view of the connection between an assignment of logical forms, a truth theory and a meaning theory. (4) Finally, I will briefly discuss possible problems of Davidson’s account as developed in this paper. (shrink)

This book, one of the handful of truly pathbreaking works in twentieth-century philosophical theology, presents Hartshorne's persuasive rehabilitation of Anselm's Ontological Argument, recast in neoclassical form as "the Modal Proof.".

This new translation of The Science of Logic (also known as 'Greater Logic') includes the revised Book I (1832), Book II (1813), and Book III (1816). Recent research has given us a detailed picture of the process that led Hegel to his final conception of the System and of the place of the Logic within it. We now understand how and why Hegel distanced himself from Schelling, how radical this break with his early mentor was, and to (...) what extent it entailed a return (but with a difference) to Fichte and Kant. In the introduction to the volume, George di Giovanni presents in synoptic form the results of recent scholarship on the subject, and, while recognizing the fault lines in Hegel's System that allow opposite interpretations, argues that the Logic marks the end of classical metaphysics. The translation is accompanied by a full apparatus of historical and explanatory notes. (shrink)

This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This paper gives (...) outlines of two arguments that jointly show that this is the case. The first is intended to show that the logic is informally sound, in the sense that all of its theorems are informally valid. The second is intended to show that it is informally complete, in the sense that all informal validities are among its theorems. In order to give these arguments, a number of independently interesting results concerning the logic are proven. In particular, the soundness and completeness of two proof systems with respect to the semantics is proven (Theorems 2.11 and 2.15), as well as a normal form theorem (Theorem 3.2), an elimination theorem for the actuality operator (Corollary 3.6), and the decidability of the logic (Corollary 3.7). It turns out that the logic invalidates a plausible principle concerning the interaction of apriority and necessity; consequently, a variant semantics is briefly explored on which this principle is valid. The paper concludes by assessing the implications of these results for epistemic two-dimensional semantics. (shrink)

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge. Building on Stalnaker’s core insights, we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and what is knowable; we argue that the foundational axioms of Stalnaker’s system rely intuitively on both of these notions. More precisely, we argue that the plausibility of (...) the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an “evidence-in-hand” conception of knowledge and a weaker “evidence-out-there” notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker’s postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics. (shrink)

This paper raises some questions about the formalization of sentences containing ‘if’ or similar expressions. In particular, it focuses on three kinds of sentences that resemble conditionals in some respects but exhibit distinctive logical features that deserve separate consideration: whether-or-not sentences, biscuit conditionals, and concessive conditionals. As will be suggested, the examples discussed show in different ways that an adequate formalization of a sentence must take into account the content expressed by the sentence. This upshot is arguably what one should (...) expect on the view that logical form is determined by truth conditions. (shrink)

In this article we present a p-adic valued probabilistic logic equation image which is a complete and decidable extension of classical propositional logic. The key feature of equation image lies in ability to formally express boundaries of probability values of classical formulas in the field equation image of p-adic numbers via classical connectives and modal-like operators of the form Kr, ρ. Namely, equation image is designed in such a way that the elementary probability sentences Kr, ρα actually (...) do have their intended meaning—the probability of propositional formula α is in the equation image-ball with the center r and the radius ρ. Due to modal nature of the operators Kr, ρ, it was natural to use the probability Kripke like models as equation image-structures, provided that probability functions range over equation image instead of equation image or equation image. (shrink)

Those who conceive logic as a science have generally favoured one of two alternative conceptions as to what the subject-matter of this science ought to be. On the one hand is the nowadays somewhat old-fashioned-seeming view of logic as the science of judgment, or of thinking or reasoning activities in general. On the other hand is the view of logic as a science of ideal meanings, 'thoughts', or 'propositions in themselves'. There is, however, a third alternative conception, (...) which enjoyed only a brief flowering in the years leading up to the first World War, but whose lingering presence can be detected in the background of more recent ontologising trends in logic, as for example in the 'situation semantics' of Barwise and Perry. This third conception sees logic as a science of special objects called 'Sachverhalte' or 'states of affairs'. A view of this sort is present in simplified form in the works of Meinong, but it received its definitive formulation in the writings of Adolf Reinach, a student of Husserl who is otherwise noteworthy for having anticipated, in a monograph of 1913, large chunks of what later became known as the theory of speech acts.(1). (shrink)

Monists say that the nature of truth is invariant, whichever sentence you consider; pluralists say that the nature of truth varies between different sets of sentences. The orthodoxy is that logic and logical form favour monism: there must be a single property that is preserved in any valid inference; and any truth-functional complex must be true in the same way as its components. The orthodoxy, I argue, is mistaken. Logic and logical form impose only structural constraints (...) on a metaphysics of truth. Monistic theories are not guaranteed to satisfy these constraints, and there is a pluralistic theory that does so. (shrink)

This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. (...) Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness. (shrink)

The central thesis of the article is that there are two quite distinct concepts of logical form. Theories of logical form employing one of these concepts are different both in method of justification and in philosophical and psychological implications from theories employing the other concept.

First order logic does not distinguish between different forms of universal generalization; in this paper I argue that lawlike and accidental generalizations (broadly construed) have a different logical form, and that this distinction is syntactically marked in English. I then consider the relevance of this broader conception of lawlikeness to the philosophy of science.

Published in 1946, this volume does not purpose to be a treatise on logic. The author's contributions to the substance of logical doctrine have been made in his other works. What he has attempted in the studies that form this volume is an exploration of the periphery of logic, the relation of logic to the rest of the universe, the philosophical presuppositions which give logic its meaning and the applications which give it importance. It is (...) his belief that formal logic is the heart of philosophy, since the subject matter of logic is the formal aspect of all being. From this standpoint he explores the relation of logic to such fields of study as natural science, ethics, history and general philosophy. (shrink)

This paper reflects on the limits of logical form set by a novel criterion of logicality proposed in (Bonnay and Speitel, 2021). The interest stems from the fact that the delineation of logical terms according to the criterion exceeds the boundaries of standard first-order logic. Among ‘novel’ logical terms is the quantifier “there are infinitely many”. Since the structure of the natural numbers is categorically characterisable in a language including this quantifier we ask: does this imply that arithmetical (...) forms have been reduced to logical forms? And, in general, what other conditions need to be satisfied for a form to qualify as “fully logical”? We survey answers to these questions. (shrink)

Axel Honneth reconstructs Hegel’s social and political philosophy on the basis of the concept of recognition. For Honneth, recognition is a constitutive relation between individuals that is in principle symmetrical. By conceiving recognition through symmetry, Honneth effectively bans the inclusion of power within recognitive relation. He thus regards the relations of power as cases of non-recognition or misrecognition. In this paper, I develop an alternative theory of the constitutive relation between individuals for Hegel, one that is based on the asymmetrical (...) relation of power. To this aim, I focus on the chapter of “determinations of reflection” in the Science of Logic. Through a close analysis of Hegel’s Logic, I argue that the most fundamental form of relation between individuals is the relation of opposition, that individuals are solely constituted in and through the relation of opposition, and that the relation of opposition is essentially asymmetrical. Together, these claims establish... (shrink)