Here are considered the conditions under which the method of diagrams is liable to include non-classical logics, among which the spatial representation of non-bivalent negation. This will be done with two intended purposes, namely: a review of the main concepts involved in the definition of logical negation; an explanation of the epistemological obstacles against the introduction of non-classical negations within diagrammaticlogic.
It is argued, on the basis of ideas derived from Wittgenstein's Tractatus and Husserl's Logical Investigations, that the formal comprehends more than the logical. More specifically: that there exist certain formal-ontological constants (part, whole, overlapping, etc.) which do not fall within the province of logic. A two-dimensional directly depicting language is developed for the representation of the constants of formal ontology, and means are provided for the extension of this language to enable the representation of certain materially necessary relations. (...) The paper concludes with a discussion of the relationship between formal logic, formal ontology, and mathematics. (shrink)
A Mathematical Review by John Corcoran, SUNY/Buffalo -/- Macbeth, Danielle Diagrammatic reasoning in Frege's Begriffsschrift. Synthese 186 (2012), no. 1, 289–314. ABSTRACT This review begins with two quotations from the paper: its abstract and the first paragraph of the conclusion. The point of the quotations is to make clear by the “give-them-enough-rope” strategy how murky, incompetent, and badly written the paper is. I know I am asking a lot, but I have to ask you to read the quoted passages—aloud (...) if possible. Don’t miss the silly attempt to recycle Kant’s quip “Concepts without intuitions are empty; intuitions without concepts are blind”. What the paper was aiming at includes the absurdity: “Proofs without definitions are empty; definitions without proofs are, if not blind, then dumb.” But the author even bollixed this. The editor didn’t even notice. The copy-editor missed it. And the author’s proof-reading did not catch it. In order not to torment you I will quote the sentence as it appears: “In a slogan: proofs without definitions are empty, merely the aimless manipulation of signs according to rules; and definitions without proofs are, if no blind, then dumb.”[sic] The rest of my review discusses the paper’s astounding misattribution to contemporary logicians of the information-theoretic approach. This approach was cruelly trashed by Quine in his 1970 Philosophy of Logic, and thereafter ignored by every text I know of. The paper under review attributes generally to modern philosophers and logicians views that were never espoused by any of the prominent logicians—such as Hilbert, Gödel, Tarski, Church, and Quine—apparently in an attempt to distance them from Frege: the focus of the article. On page 310 we find the following paragraph. “In our logics it is assumed that inference potential is given by truth-conditions. Hence, we think, deduction can be nothing more than a matter of making explicit information that is already contained in one’s premises. If the deduction is valid then the information contained in the conclusion must be contained already in the premises; if that information is not contained already in the premises […], then the argument cannot be valid.” Although the paper is meticulous in citing supporting literature for less questionable points, no references are given for this. In fact, the view that deduction is the making explicit of information that is only implicit in premises has not been espoused by any standard symbolic logic books. It has only recently been articulated by a small number of philosophical logicians from a younger generation, for example, in the prize-winning essay by J. Sagüillo, Methodological practice and complementary concepts of logical consequence: Tarski’s model-theoretic consequence and Corcoran’s information-theoretic consequence, History and Philosophy of Logic, 30 (2009), pp. 21–48. The paper omits definitions of key terms including ‘ampliative’, ‘explicatory’, ‘inference potential’, ‘truth-condition’, and ‘information’. The definition of prime number on page 292 is as follows: “To say that a number is prime is to say that it is not divisible without remainder by another number”. This would make one be the only prime number. The paper being reviewed had the benefit of two anonymous referees who contributed “very helpful comments on an earlier draft”. Could these anonymous referees have read the paper? -/- J. Corcoran, U of Buffalo, SUNY -/- PS By the way, if anyone has a paper that has been turned down by other journals, any journal that would publish something like this might be worth trying. (shrink)
This book is written for those who wish to learn some basic principles of formal logic but more importantly learn some easy methods to unpick arguments and assess their value for truth and validity. -/- The first section explains the ideas behind traditional logic which was formed well over two thousand years ago by the ancient Greeks. Terms such as ‘categorical syllogism’, ‘premise’, ‘deduction’ and ‘validity’ may appear at first sight to be inscrutable but will easily be understood (...) with examples bringing the subjects to life. Traditionally, Venn diagrams have been employed to test arguments. These are very useful but their application is limited and they are not open to quantification. The mid-section of this book introduces a methodology that makes the analysis of arguments accessible with the use of a new form of diagram, modified from those of the mathematician Leonhard Euler. These new diagrammatic methods will be employed to demonstrate an addition to the basic form of syllogism. This includes a refined definition of the terms ‘most’ and ‘some’ within propositions. This may seem a little obscure at the moment but one will readily apprehend these new methods and principles of a more modern logic. (shrink)
In the visual representation of ontologies, in particular of part-whole relationships, it is customary to use graph theory as the representational background. We claim here that the standard graph-based approach has a number of limitations, and we propose instead a new representation of part-whole structures for ontologies, and describe the results of experiments designed to show the effectiveness of this new proposal especially as concerns reduction of visual complexity. The proposal is developed to serve visualization of ontologies conformant to the (...) Basic Formal Ontology. But it can be used also for more general applications, particularly in the biomedical domain. (shrink)
The present dissertation presents an examination of the Carrollian logic through the reconstruction of its syllogistic theory. Lewis Carroll was one of the main responsible for the dissemination of logic during the nineteenth century, but most of his logical writings remained unknown until a posthumous publication of 1977. The reconstruction of the Carrollian syllogistic theory was based on the comparison of the two books on author's logic, "The Game of Logic" and "Symbolic Logic". The analysis (...) of the Carrollian syllogistics starts from a study of the historical context of development of the logic and the developments of syllogistics previous to the contribution of the author. Situated in the historical period of algebraical logic, Carrollian syllogistics is characterized as a conservative extension of the Aristotelian syllogistics, the main innovation is the use of negative terms and the introduction of a diagrammatic method suitable for the representation of negative terms. The diagrammatic method of the Carrollian syllogistics presents advances in relation to the methods of Euler and Venn. The use of negative terms also requires a redefinition of the notion of syllogism, simplifying and expanding the amount of arguments amenable to logical treatment. Carroll does not use four, but only three categorical propositions in his syllogistic, with interpretation of existential presuppositions congruent with a syntactic-existential reading. Carrollian syllogistics uses some techniques found in the work of algebraists of logic and also made the same confusions between notions of "class" and "member" that were common in the period. Convinced of the social utility of logic and dedicated to popularize it, Carroll priorized a creation of new didactics for the teaching of logic in his works, where he can include his diagrammatic method of solving syllogisms. Carroll made only scant considerations of his conception of logic. Based on the small considerations found throughout the study and on the constant claim of the social utility of logic, it is suggested that Carroll is close to the so-called pragmatic position, which considers a logic as an instrument of regulation of discourse. (shrink)
Logicians commonly speak in a relatively undifferentiated way about pre-euler diagrams. The thesis of this paper, however, is that there were three periods in the early modern era in which euler-type diagrams (line diagrams as well as circle diagrams) were expansively used. Expansive periods are characterized by continuity, and regressive periods by discontinuity: While on the one hand an ongoing awareness of the use of euler-type diagrams occurred within an expansive period, after a subsequent phase of regression the entire knowledge (...) about the systematic application and the history of euler-type diagrams was lost. I will argue that the first expansive period lasted from Vives (1531) to Alsted (1614). The second period began around 1660 with Weigel and ended in 1712 with lange. The third period of expansion started around 1760 with the works of Ploucquet, euler and lambert. Finally, it is shown that euler-type diagrams became popular in the debate about intuition which took place in the 1790s between leibnizians and Kantians. The article is thus limited to the historical periodization between 1530 and 1800. (shrink)
This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn't arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the $\lambda$-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both (...) a diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the $\lambda$-calculus. (shrink)
Necessity is a touchstone issue in the thought of Charles Peirce, not least because his pragmatist account of meaning relies upon modal terms. We here offer an overview of Peirce’s highly original and multi-faceted take on the matter. We begin by considering how a self-avowed pragmatist and fallibilist can even talk about necessary truth. We then outline the source of Peirce’s theory of representation in his three categories of Firstness, Secondness and Thirdness, (monadic, dyadic and triadic relations). These have modal (...) purport insofar as the first category corresponds to possibility, the second to mechanical necessity and the third to a kind of semantic or intentional necessity. We then turn to Peirce’s explicit modal epistemology and show how it began as information-relative, with different modalities (e.g. logical, physical, practical) distinguished in terms of respective ‘designated states of information’, and shifted later in his life towards a more robust realism founded in direct perception of ideas in their relations. We then turn to Peirce’s formal logic, focusing on his diagrammatic system of Existential Graphs where he did his most serious logical research. Finally we discuss Peirce’s modal metaphysics and its implications for determinism and realism about universals. (shrink)
Charles Peirce's diagrammaticlogic — the Existential Graphs — is presented as a tool for illuminating how we know necessity, in answer to Benacerraf's famous challenge that most ‘semantics for mathematics’ do not ‘fit an acceptable epistemology’. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the particular structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not merely (...) to individual things but to how those things are related in larger structures, certain aspects of which relations force certain other aspects to be a certain way. (shrink)
There has been a recent surge of work on deontic modality within philosophy of language. This work has put the deontic logic tradition in contact with natural language semantics, resulting in significant increase in sophistication on both ends. This chapter surveys the main motivations, achievements, and prospects of this work.
(1) This paper is about how to build an account of the normativity of logic around the claim that logic is constitutive of thinking. I take the claim that logic is constitutive of thinking to mean that representational activity must tend to conform to logic to count as thinking. (2) I develop a natural line of thought about how to develop the constitutive position into an account of logical normativity by drawing on constitutivism in metaethics. (3) (...) I argue that, while this line of thought provides some insights, it is importantly incomplete, as it is unable to explain why we should think. I consider two attempts at rescuing the line of thought. The first, unsuccessful response is that it is self-defeating to ask why we ought to think. The second response is that we need to think. But this response secures normativity only if thinking has some connection to human flourishing. (4) I argue that thinking is necessary for human flourishing. Logic is normative because it is constitutive of this good. (5) I show that the resulting account deals nicely with problems that vex other accounts of logical normativity. (shrink)
We present epistemic multilateral logic, a general logical framework for reasoning involving epistemic modality. Standard bilateral systems use propositional formulae marked with signs for assertion and rejection. Epistemic multilateral logic extends standard bilateral systems with a sign for the speech act of weak assertion (Incurvati and Schlöder 2019) and an operator for epistemic modality. We prove that epistemic multilateral logic is sound and complete with respect to the modal logic S5 modulo an appropriate translation. The logical (...) framework developed provides the basis for a novel, proof-theoretic approach to the study of epistemic modality. To demonstrate the fruitfulness of the approach, we show how the framework allows us to reconcile classical logic with the contradictoriness of so-called Yalcin sentences and to distinguish between various inference patterns on the basis of the epistemic properties they preserve. (shrink)
I develop and defend a truthmaker semantics for the relevant logic R. The approach begins with a simple philosophical idea and develops it in various directions, so as to build a technically adequate relevant semantics. The central philosophical idea is that truths are true in virtue of specific states. Developing the idea formally results in a semantics on which truthmakers are relevant to what they make true. A very natural notion of conditionality is added, giving us relevant implication. I (...) then investigate ways to add conjunction, disjunction, and negation; and I discuss how to justify contraposition and excluded middle within a truthmaker semantics. (shrink)
In this paper I will develop a view about the semantics of imperatives, which I term Modal Noncognitivism, on which imperatives might be said to have truth conditions (dispositionally, anyway), but on which it does not make sense to see them as expressing propositions (hence does not make sense to ascribe to them truth or falsity). This view stands against “Cognitivist” accounts of the semantics of imperatives, on which imperatives are claimed to express propositions, which are then enlisted in explanations (...) of the relevant logico-semantic phenomena. It also stands against the major competitors to Cognitivist accounts—all of which are non-truth-conditional and, as a result, fail to provide satisfying explanations of the fundamental semantic characteristics of imperatives (or so I argue). The view of imperatives I defend here improves on various treatments of imperatives on the market in giving an empirically and theoretically adequate account of their semantics and logic. It yields explanations of a wide range of semantic and logical phenomena about imperatives—explanations that are, I argue, at least as satisfying as the sorts of explanations of semantic and logical phenomena familiar from truth-conditional semantics. But it accomplishes this while defending the notion—which is, I argue, substantially correct—that imperatives could not have propositions, or truth conditions, as their meanings. (shrink)
The aim of the paper is to argue that all—or almost all—logical rules have exceptions. In particular, it is argued that this is a moral that we should draw from the semantic paradoxes. The idea that we should respond to the paradoxes by revising logic in some way is familiar. But previous proposals advocate the replacement of classical logic with some alternative logic. That is, some alternative system of rules, where it is taken for granted that these (...) hold without exception. The present proposal is quite different. According to this, there is no such alternative logic. Rather, classical logic retains the status of the ‘one true logic’, but this status must be reconceived so as to be compatible with (almost) all of its rules admitting of exceptions. This would seem to have significant repercussions for a range of widely held views about logic: e.g. that it is a priori, or that it is necessary. Indeed, if the arguments of the paper succeed, then such views must be given up. (shrink)
The five English words—sentence, proposition, judgment, statement, and fact—are central to coherent discussion in logic. However, each is ambiguous in that logicians use each with multiple normal meanings. Several of their meanings are vague in the sense of admitting borderline cases. In the course of displaying and describing the phenomena discussed using these words, this paper juxtaposes, distinguishes, and analyzes several senses of these and related words, focusing on a constellation of recommended senses. One of the purposes of this (...) paper is to demonstrate that ordinary English properly used has the resources for intricate and philosophically sound investigation of rather deep issues in logic and philosophy of language. No mathematical, logical, or linguistic symbols are used. Meanings need to be identified and clarified before being expressed in symbols. We hope to establish that clarity is served by deferring the extensive use of formalized or logically perfect languages until a solid “informal” foundation has been established. Questions of “ontological status”—e.g., whether propositions or sentences, or for that matter characters, numbers, truth-values, or instants, are “real entities”, are “idealizations”, or are “theoretical constructs”—plays no role in this paper. As is suggested by the title, this paper is written to be read aloud. -/- I hope that reading this aloud in groups will unite people in the enjoyment of the humanistic spirit of analytic philosophy. (shrink)
Sentences containing definite descriptions, expressions of the form ‘The F’, can be formalised using a binary quantifier ι that forms a formula out of two predicates, where ιx[F, G] is read as ‘The F is G’. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INFι of intuitionist negative free logic extended by such a quantifier, which was presented (...) in (Kürbis 2019), INFι is first compared to a system of Tennant’s and an axiomatic treatment of a term forming ι operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INFι in which the G of ιx[F, G] is restricted to identity. INFι is then compared to an intuitionist version of a system of Lambert’s which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion. (shrink)
We formally introduce a novel, yet ubiquitous, category of norms: norms of instrumentality. Norms of this category describe which actions are obligatory, or prohibited, as instruments for certain purposes. We propose the Logic of Agency and Norms (LAN) that enables reasoning about actions, instrumentality, and normative principles in a multi-agent setting. Leveraging LAN , we formalize norms of instrumentality and compare them to two prevalent norm categories: norms to be and norms to do. Last, we pose principles relating the (...) three categories and evaluate their validity vis-à-vis notions of deliberative acting. On a technical note, the logic will be shown decidable via the finite model property. (shrink)
Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the (...) idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)
Weakly Aggregative Modal Logic (WAML) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. WAML has some interesting applications on epistemic logic and logic of games, so we study some basic model theoretical aspects of WAML in this paper. Specifically, we give a van Benthem-Rosen characterization theorem of WAML based on an intuitive notion of bisimulation and show that each basic WAML system Kn lacks Craig Interpolation.
Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages (...) when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness. (shrink)
The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few (...) rudimentary facts of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism. (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)
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)
We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justi cations and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a (...) class='Hi'>logic of assertions and conjectures: looking at the S4 modal translation, we give a de nition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is de ned and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, de ned as a hypothesis that in some situation the truth of p is epistemically necessary. (shrink)
An exact truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting notion (...) of entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for exact entailment and give soundness and completeness results. (shrink)
“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our (...) traditional intuitive logical framework and that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving radical repudiation of a part of our scientific tradition, a tradition that is defended by classical logicians. But it is also viewed as reactionary: as being analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been. (shrink)
In the paper we present a formal system motivated by a specific methodology of creating norms. According to the methodology, a norm-giver before establishing a set of norms should create a picture of the agent by creating his repertoire of actions. Then, knowing what the agent can do in particular situations, the norm-giver regulates these actions by assigning deontic qualifications to each of them. The set of norms created for each situation should respect (1) generally valid deontic principles being the (...) theses of our logic and (2) facts from the ontology of action whose relevance for the systems of norms we postulate. (shrink)
We investigate an enrichment of the propositional modal language L with a "universal" modality ■ having semantics x ⊧ ■φ iff ∀y(y ⊧ φ), and a countable set of "names" - a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒ $_{c}$ proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ(⍯) of ℒ, where ⍯ is an additional modality with the semantics x ⊧ ⍯φ (...) iff Vy(y ≠ x → y ⊧ φ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in ℒ $_{c}$ . Strong completeness of the normal ℒ $_{c}$ logics is proved with respect to models in which all worlds are named. Every ℒ $_{c}$ -logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from ℒ to ℒ $_{c}$ are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched. (shrink)
Epistemic logics based on the possible worlds semantics suffer from the problem of logical omniscience, whereby agents are described as knowing all logical consequences of what they know, including all tautologies. This problem is doubly challenging: on the one hand, agents should be treated as logically non-omniscient, and on the other hand, as moderately logically competent. Many responses to logical omniscience fail to meet this double challenge because the concepts of knowledge and reasoning are not properly separated. In this paper, (...) I present a dynamic logic of knowledge that models an agent’s epistemic state as it evolves over the course of reasoning. I show that the logic does not sacrifice logical competence on the altar of logical non- omniscience. (shrink)
In the present paper we propose a system of propositional logic for reasoning about justification, truthmaking, and the connection between justifiers and truthmakers. The logic of justification and truthmaking is developed according to the fundamental ideas introduced by Artemov. Justifiers and truthmakers are treated in a similar way, exploiting the intuition that justifiers provide epistemic grounds for propositions to be considered true, while truthmakers provide ontological grounds for propositions to be true. This system of logic is then (...) applied both for interpreting the notorious definition of knowledge as justified true belief and for advancing a new solution to Gettier counterexamples to this standard definition. (shrink)
This paper presents a way of formalising definite descriptions with a binary quantifier ι, where ιx[F, G] is read as ‘The F is G’. Introduction and elimination rules for ι in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form ιx[F, G] are given, and it is shown that deductions in the system can be brought into normal form.
A dynamic epistemic logic is presented in which the single agent can reason about his knowledge stages before and after announcements. The logic is generated by reinterpreting multi agent private announcements in a single agent environment. It is shown that a knowability principle is valid for such logic: any initially true ϕ can be known after a certain number of announcements.
Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's two-volume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning (...) showing by deductively evident steps that its conclusion is a consequence of its premises. In particular, a demonstration is a deduction whose premises are known to be true. Aristotle's general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of intermediate immediately evident steps that shows its final conclusion to follow logically from its premises. To illustrate his general theory of deduction, he presented an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. (shrink)
In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of the (...) system is deducible by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)
I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. -/- There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is (...) countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. -/- Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects. (shrink)
In this paper I introduce a sequent system for the propositional modal logic S5. Derivations of valid sequents in the system are shown to correspond to proofs in a novel natural deduction system of circuit proofs (reminiscient of proofnets in linear logic, or multiple-conclusion calculi for classical logic). -/- The sequent derivations and proofnets are both simple extensions of sequents and proofnets for classical propositional logic, in which the new machinery—to take account of the modal vocabulary—is (...) directly motivated in terms of the simple, universal Kripke semantics for S5. The sequent system is cut-free and the circuit proofs are normalising. (shrink)
George Boole emerged from the British tradition of the “New Analytic”, known for the view that the laws of logic are laws of thought. Logicians in the New Analytic tradition were influenced by the work of Immanuel Kant, and by the German logicians Wilhelm Traugott Krug and Wilhelm Esser, among others. In his 1854 work An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Boole argues that the laws of (...) thought acquire normative force when constrained to mathematical reasoning. Boole’s motivation is, first, to address issues in the foundations of mathematics, including the relationship between arithmetic and algebra, and the study and application of differential equations (Durand-Richard, van Evra, Panteki). Second, Boole intended to derive the laws of logic from the laws of the operation of the human mind, and to show that these laws were valid of algebra and of logic both, when applied to a restricted domain. Boole’s thorough and flexible work in these areas influenced the development of model theory (see Hodges, forthcoming), and has much in common with contemporary inferentialist approaches to logic (found in, e.g., Peregrin and Resnik). (shrink)
Priest has provided a simple tableau calculus for Chellas's conditional logic Ck. We provide rules which, when added to Priest's system, result in tableau calculi for Chellas's CK and Lewis's VC. Completeness of these tableaux, however, relies on the cut rule.
Analyzing the position of two philosophers whose views are recognizably divergent, W. O. Quine and M. Dummett, we intend to support a striking point of agreement between them: the idea that our logical principles constitute our principles about what there is, and therefore, that logic is metaphysics.
Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as (...) the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)
The aim of this paper is to introduce a system of dynamic deontic logic in which the main problems related to the de finition of deontic concepts, especially those emerging from a standard analysis of permission in terms of possibility of doing an action without incurring in a violation of the law, are solved. The basic idea is to introduce two crucial distinctions allowing us to differentiate (i) what is ideal with respect to a given code, which fixes the (...) types of action that are abstractly prescribed, and what is ideal with respect to the specific situation in which the agent acts, and (ii) the transitions associated with actions and the results of actions, which can obtain even without the action being performed. (shrink)
It is often said that ‘every logical truth is obvious’ (Quine 1970: 82), that the ‘axioms and rules of logic are true in an obvious way’ (Murawski 2014: 87), or that ‘logic is a theory of the obvious’ (Sher 1999: 207). In this chapter, I set out to test empirically how the idea that logic is obvious is reflected in the scholarly work of logicians and philosophers of logic. My approach is data-driven. That is to say, (...) I propose that systematically searching for patterns of usage in databases of scholarly works, such as JSTOR, can provide new insights into the ways in which the idea that logic is obvious is reflected in logical and philosophical practice, i.e., in the arguments that logicians and philosophers of logic actually make in their published work. (shrink)
(See also the separate entry for the volume itself.) This introduction has three parts. The first providing an overview of some main lines of research in deontic logic: the emergence of SDL, Chisholm's paradox and the development of dyadic deontic logics, various other puzzles/challenges and areas of development, along with philosophical applications. The second part focus on some actual and potential fruitful interactions between deontic logic, computer science and artificial intelligence. These include applications of deontic logic to (...) AI knowledge representation in legal systems, to modelling computer systems where it is expected that sub-ideal states will emerge and require countermeasures, to norm-governed human interactions with computer systems, and to the representation of some features of multi-agent systems where different agent-like computer systems interact with one another. The third and final part briefly groups and previews the papers in the anthology. (shrink)
It is well known that systems of action deontic logic emerging from a standard analysis of permission in terms of possibility of doing an action without incurring in a violation of the law are subject to paradoxes. In general, paradoxes are acknowledged as such if we have intuitions telling us that things should be different. The aim of this paper is to introduce a paradox-free deontic action system by (i) identifying the basic intuitions leading to the emergence of the (...) paradoxes and (ii) exploiting these intuitions in order to develop a consistent deontic framework, where it can be shown why some phenomena seem to be paradoxical and why they are not so if interpreted in a correct way. (shrink)
The purpose of this paper is to explore the question of how truthmaker theorists ought to think about their subject in relation to logic. Regarding logic and truthmaking, I defend the view that considerations drawn from advances in modal logic have little bearing on the legitimacy of truthmaker theory. To do so, I respond to objections Timothy Williamson has lodged against truthmaker theory. As for the logic of truthmaking, I show how the project of understanding the (...) logical features of the truthmaking relation has led to an apparent impasse. I offer a new perspective on the logic of truthmaking that both explains the problem and offers a way out. (shrink)
Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important (...) programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. (shrink)
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