2nd edition. The theory of logical consequence is central in modern logic and its applications. However, it is mostly dispersed in an abundance of often difficultly accessible papers, and rarely treated with applications in mind. This book collects the most fundamental aspects of this theory and offers the reader the basics of its applications in computer science, artificial intelligence, and cognitive science, to name but the most important fields where this notion finds its many applications.

Bilateralists, who accept that there are two primitive speech acts, assertion and denial, can offer an attractive definition of consequence: Y follows from X if and only if it is incoherent to assert all formulas X and to deny all formulas Y. The present paper argues that this definition has consequences many will find problematic, amongst them that truth coincides with assertibility. Philosophers who reject these consequences should therefore reject this definition of consequence.

An interesting question is whether deflationism about truth (and falsity) extends to related properties and relations on truthbearers. Lionel Shapiro (2011) answers affirmatively by arguing that a certain deflationism about truth is as plausible as an analogous version of deflationism about logical consequence. I argue that the argument fails on two counts. First, it trivializes to any relation between truthbearers, including substantive ones; in other words, his argument can be used to establish that deflationism about truth is as (...) plausible as deflationism about an arbitrary sentential relation. Second, the alleged analogy between the arguments for deflationism about truth and deflationism about consequence fails. Along the way I consider what implications the failure of the equiplausibility thesis has for deflationism about falsity. (shrink)

ABSTRACT: This 1974 paper builds on our 1969 paper (Corcoran-Weaver [2]). Here we present three (modal, sentential) logics which may be thought of as partial systematizations of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of these three logics coincide with one another and with those of standard formalizations of Lewis's S5. These logics, when regarded as logistic systems (cf. Corcoran [1], p. 154), are seen to be (...) equivalent; but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be as closely linked as previously thought. -/- This 1974 paper uses the linear notation for natural deduction presented in [2]: each two-dimensional deduction is represented by a unique one-dimensional string of characters. Thus obviating need for two-dimensional trees, tableaux, lists, and the like—thereby facilitating electronic communication of natural deductions. The 1969 paper presents a (modal, sentential) logic which may be thought of as a partial systematization of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of this logic coincides those of standard formalizations of Lewis’s S4. Among the paper's innovations is its treatment of modal logic in the setting of natural deduction systems--as opposed to axiomatic systems. The author’s apologize for the now obsolete terminology. For example, these papers speak of “a proof of a sentence from a set of premises” where today “a deduction of a sentence from a set of premises” would be preferable. 1. Corcoran, John. 1969. Three Logical Theories, Philosophy of Science 36, 153–77. J P R -/- 2. Corcoran, John and George Weaver. 1969. Logical Consequence in Modal Logic: Natural Deduction in S5 Notre Dame Journal of Formal Logic 10, 370–84. MR0249278 (40 #2524). 3. Weaver, George and John Corcoran. 1974. Logical Consequence in Modal Logic: Some Semantic Systems for S4, Notre Dame Journal of Formal Logic 15, 370–78. MR0351765 (50 #4253). (shrink)

Compare two conceptions of validity: under an example of a modal conception, an argument is valid just in case it is impossible for the premises to be true and the conclusion false; under an example of a topic-neutral conception, an argument is valid just in case there are no arguments of the same logical form with true premises and a false conclusion. This taxonomy of positions suggests a project in the philosophy of logic: the reductive analysis of the modal (...) conception of logical consequence to the topic-neutral conception. Such a project would dispel the alleged obscurity of the notion of necessity employed in the modal conception in favour of the clarity of an account of logical consequence given in terms of tractable notions of logical form, universal generalization and truth simpliciter. In a series of publications, John Etchemendy has characterized the model-theoretic definition of logical consequence as truth preservation in all models as intended to provide just such an analysis. In this paper, I will argue that Aristotle intends to provide an account of a modal conception of logical consequence in topic-neutral terms and so is engaged in a project comparable to the one described above. That Aristotle would be engaged in this sort of project is controversial. Under the standard reading of the Prior Analytics, Aristotle does not and cannot provide an account of logical consequence. Rather, he must take the validity of the first figure syllogisms (such as the syllogism known by its medieval mnemonic ‘Barbara’: A belongs to all B; B belongs to all C; so A belongs to all C) as obvious and not needing justification; he then establishes the validity of the other syllogisms by showing that they stand in a suitable relation to the first figure syllogisms. I will argue that Aristotle does attempt to provide an account of logical consequence—namely, by appeal to certain mereological theorems. For example, he defends the status of Barbara as a syllogism by appeal to the transitivity of mereological containment. There are, as I will discuss, reasons to doubt the success of this account. But the attempt is not implausible given certain theses Aristotle holds in semantics, mereology and the theory of relations. (shrink)

In a recent article, “Logical Consequence and Natural Language”, Michael Glanzberg claims that there is no relation of logical consequence in natural language (2015). The present paper counters that claim. I shall discuss Glanzberg’s arguments and show why they don’t hold. I further show how Glanzberg’s claims may be used to rather support the existence of logical consequence in natural language.

An essay review of Gila Sher's *Logical Consequence*.

Our standard model-theoretic definition of logical consequence is originally based on Alfred Tarski’s (1936) semantic definition, which, in turn, is based on Rudolf Carnap’s (1934) similar definition. In recent literature, Tarski’s definition is described as a conceptual analysis of the intuitive ‘everyday’ concept of consequence or as an explication of it, but the use of these terms is loose and largely unaccounted for. I argue that the definition is not an analysis but an explication, in the Carnapian (...) sense: the replacement of the inexact everyday concept with an exact one. Some everyday intuitions were thus brought into a precise form, others were ignored and forgotten. How exactly did the concept of logical consequence change in this process? I suggest that we could find some of the forgotten intuitions in Bernard Bolzano’s (1837) definition of ‘deducibility’, which is traditionally viewed as the main precursor of Tarski’s definition from a time before formalized languages. It turns out that Bolzano’s definition is subject to just the kind of natural features—paradoxicality of everyday language, Platonism about propositions, and dependence on the external world—that Tarski sought to tame by constructing an artificial concept for the special needs of mathematical logic. (shrink)

If an argument is valid, it is impossible for its premises to be true, and its conclusion false. But how should we understand these notions of truth and impossibility? Here, I present the answers given by John Buridan (ca. 1300-60), showing (i) how he understands truth in his anti-realist metaphysics, and (ii) how he understands modality in connection with causal powers. In short: if an argument exists and is valid, there does not exist a power capable of making the premises (...) true and, at the same time, making the conclusion false. (shrink)

When some P implies some Q, this should have some impact on what attitudes we take to P and Q. In other words: logical consequence has a normative import. I use this idea, recently explored by a number of scholars, as a stepping stone to a bolder view: that relations of logical consequence can be identified with norms on our propositional attitudes, or at least that our talk of logical consequence can be explained in (...) terms of such norms. I investigate the prospects of such a cognitive norm account of logical consequence. I go over the challenges involved in finding a plausible bridge principle connecting logical consequence to cognitive norms, in particular a biconditional principle that gives us not only necessary but sufficient conditions for logical consequence in terms of norms on propositional attitudes. Then, on the assumption that an adequate norm can be found, I consider what the philosophical merits of such a cognitive norm account would be, and what theoretical commitments it would generate. (shrink)

In a recent paper, “The Concept of Logical Consequence,” W. H. Hanson criticizes a formal-structural characterization of logical consequence in Tarski and Sher. Hanson accepts many principles of the formal-structural view. Relating to Sher 1991 and 1996a, he says.

In this work, we propose a definition of logical consequence based on the relation between the quantity of information present in a particular set of formulae and a particular formula. As a starting point, we use Shannon‟s quantitative notion of information, founded on the concepts of logarithmic function and probability value. We first consider some of the basic elements of an axiomatic probability theory, and then construct a probabilistic semantics for languages of classical propositional logic. We define the (...) quantity of information for the formulae of these languages and introduce the concept of informational logical consequence, identifying some important results, among them: certain arguments that have traditionally been considered valid, such as modus ponens, are not valid from the informational perspective; the logic underlying informational logical consequence is not classical, and is at the least paraconsistent sensu lato; informational logical consequence is not a Tarskian logical consequence. (shrink)

There is a profound, but frequently ignored relationship between logical consequence (formal implication) and material implication. The first repeats the patterns of the latter, but with a wider modal reach. It is argued that this kinship between formal and material implication simply means that they express the same kind of implication, but differ in scope. Formal implication is unrestricted material implication. This apparently innocuous observation has some significant corollaries: (1) conditionals are not connectives, but arguments; (2) the traditional (...) examples of valid argumentative forms are metalogical principles that express the properties of logical consequence; (3) formal logic is not a useful guide to detect valid arguments in the real world; (4) it is incoherent to propose alternatives to the material implication while accepting the classical properties of formal implication; (5) some of the counter-examples to classical argumentative forms and known conditional puzzles are unsound. (shrink)

This paper proposes substitutional definitions of logical truth and consequence in terms of relative interpretations that are extensionally equivalent to the model-theoretic definitions for any relational first-order language. Our philosophical motivation to consider substitutional definitions is based on the hope to simplify the meta-theory of logical consequence. We discuss to what extent our definitions can contribute to that.

Consequence rleations over sets of "judgments" are defined by using "overdetermined" as well as "underdetermined" valuations. Some of these relations are shown to be categorical. And generalized soundness and completeness results are given for both multiple and single conclusion consequence relations.

Logical consequence is typically construed as a metalinguistic relation between sentences. Deflationism is an account of logic that challenges this orthodoxy. In Williamson’s recent presentation of deflationism, logic’s primary concern is with universal generalizations over absolutely everything. As well as an interesting account of logic in its own right, deflationism has also been recruited to decide between competing logics in resolving semantic paradoxes. This paper defends deflationism from its most important challenge to date, due to Ole Hjortland. It (...) then presents two new problems for the view. Hjortland’s objection is that deflationism cannot discriminate between distinct logics. I show that his example of classical logic and supervaluationism depends on equivocating about whether the language includes a “definitely” operator. Moreover, I prove a result that blocks this line of objection no matter the choice of logics. I end by criticizing deflationism on two fronts. First, it cannot do the work it has been recruited to perform. That is, it cannot help adjudicate between competing logics. This is because a theory of logic cannot be as easily separated from a theory of truth as its proponents claim. Second, deflationism currently has no adequate answer to the following challenge: what does a sentence’s universal generalization have to do with its logical truth? I argue that the most promising, stipulative response on behalf of the deflationist amounts to an unwarranted change of subject. (shrink)

Intermediary metabolism molecules are orchestrated into logical pathways stemming from history (L-amino acids, D-sugars) and dynamic constraints (hydrolysis of pyrophosphate or amide groups is the driving force of anabolism). Beside essential metabolites, numerous variants derive from programmed or accidental changes. Broken down, variants enter standard pathways, producing further variants. Macromolecule modification alters enzyme reactions specificity. Metabolism conform thermodynamic laws, precluding strict accuracy. Hence, for each regular pathway, a wealth of variants inputs and produces metabolites that are similar to but (...) not the exact replicas of core metabolites. As corollary, a shadow, paralogous metabolism, is associated to standard metabolism. We focus on a logic of paralogous metabolism based on diversion of the core metabolic mimics into pathways where they are modified to minimize their input in the core pathways where they create havoc. We propose that a significant proportion of paralogues of well-characterized enzymes have evolved as the natural way to cope with paralogous metabolites. A second type of denouement uses a process where protecting/deprotecting unwanted metabolites - conceptually similar to the procedure used in the laboratory of an organic chemist - is used to enter a completely new catabolic pathway. (shrink)

In his new book, Logical Form, Andrea Iacona distinguishes between two different roles that have been ascribed to the notion of logical form: the logical role and the semantic role. These two roles entail a bifurcation of the notion of logical form. Both notions of logical form, according to Iacona, are descriptive, having to do with different features of natural language sentences. I agree that the notion of logical form bifurcates, but not that the (...) logical role is merely descriptive. In this paper, I focus on formalization, a process by which logical form, on its logical role, is attributed to natural language sentences. According to some, formalization is a form of explication, and it involves normative, pragmatic, as well as creative aspects. I present a view by which formalization involves explicit commitments on behalf of a reasoner or an interpreter, which serve the normative grounds for the evaluation of a given text. In previous work, I proposed the framework of semantic constraints for the explication of logical consequence. Here, I extend the framework to include formalization constraints. The various constraints then serve the role of commitments. I discuss specific issues raised by Iacona concerning univocality, co-reference and equivocation, and I show how our views on these matters diverge as a result of our different starting assumptions. (shrink)

This special issue collects together nine new essays on logical consequence :the relation obtaining between the premises and the conclusion of a logically valid argument. The present paper is a partial, and opinionated,introduction to the contemporary debate on the topic. We focus on two influential accounts of consequence, the model-theoretic and the proof-theoretic, and on the seeming platitude that valid arguments necessarilypreserve truth. We briefly discuss the main objections these accounts face, as well as Hartry Field’s contention (...) that such objections show consequenceto be a primitive, indefinable notion, and that we must reject the claim that valid arguments necessarily preserve truth. We suggest that the accountsin question have the resources to meet the objections standardly thought to herald their demise and make two main claims: (i) that consequence, as opposed to logical consequence, is the epistemologically significant relation philosophers should be mainly interested in; and (ii) that consequence is a paradoxical notion if truth is. (shrink)

I discuss Greg Restall’s attempt to generate an account of logical consequence from the incoherence of certain packages of assertions and denials. I take up his justification of the cut rule and argue that, in order to avoid counterexamples to cut, he needs, at least, to introduce a notion of logical form. I then suggest a few problems that will arise for his account if a notion of logical form is assumed. I close by sketching what (...) I take to be the most natural minimal way of distinguishing content and form and suggest further problems arising for this route. (shrink)

This paper explores the logical consequences of the the thesis that all of the essential properties of consciousness can be known introspectively (Completeness, called "Strong Transparency" in the paper, following D.M. Armstrong's older terminology). It is argued that it can be known introspectively that consciousness does not have complete access to its essential properties; and it is show how this undermines conceivability arguments for dualism.

Accounting for the epistemic contribution of deduction has been a pervasive problem for logicians interested in deduction, such as, among others, Jakko Hintikka. The problem arises because the conclusion validly deduced from a set of premises is said to be “contained” in that set; because of this containment relation, the conclusion would be known from the moment the premises are known. Assuming this, it is problematic to explain how we can gain knowledge by deducing a logical consequence implied (...) by a set of known premises. To address this problem, we offer an alternative account of the epistemic contribution of deduction as the process required to deduce a conclusion or a theorem, understanding such a process not only in terms of the number of steps in the derivation but also, more importantly, in terms of the reason for or justification for every step. That is, we do not know a proposition unless we have a justification or proof of that proposition. With this goal in mind, we develop a justification logic system which exhibits the epistemic contribution of a deductive derivation as the resulting justified formula. (shrink)

Edelcio G. de Souza is a Brazilian logician and philosopher who has researches in the domains of abstract logic, non-classical systems, philosophy of science and the foundations of mathematics. This book is in his honor with the purpose of celebrating his 60th birthday. It contains some articles connected with the above topics and other subjects in logical investigations.

This study concerns logical systems considered as theories. By searching for the problems which the traditionally given systems may reasonably be intended to solve, we clarify the rationales for the adequacy criteria commonly applied to logical systems. From this point of view there appear to be three basic types of logical systems: those concerned with logical truth; those concerned with logical truth and with logical consequence; and those concerned with deduction per se as (...) well as with logical truth and logical consequence. Adequacy criteria for systems of the first two types include: effectiveness, soundness, completeness, Post completeness, "strong soundness" and strong completeness. Consideration of a logical system as a theory of deduction leads us to attempt to formulate two adequacy criteria for systems of proofs. The first deals with the concept of rigor or "gaplessness" in proofs. The second is a completeness condition for a system of proofs. An historical note at the end of the paper suggests a remarkable parallel between the above hierarchy of systems and the actual historical development of this area of logic. (shrink)

By and large, the conditional connective in three-valued logic has two different functions. First, by means of a deduction theorem, it can express a specific relation of logical consequence in the logical language itself. Second, it can represent natural language structures such as "if/then'' or "implies''. This chapter surveys both approaches, shows why none of them will typically end up with a three-valued material conditional, and elaborates on connections to probabilistic reasoning.

Logical pluralism is the view that there is more than one correct logic. This very general characterization gives rise to a whole family of positions. I argue that not all of them are stable. The main argument in the paper is inspired by considerations known as the “collapse problem”, and it aims at the most popular form of logical pluralism advocated by JC Beall and Greg Restall. I argue that there is a more general argument available that challenges (...) all variants of logical pluralism that meet the following three conditions: that there are at least two correct logical systems characterized in terms of different consequence relations, that there is some sort of rivalry among the correct logics, and that logical consequence is normative. The hypothesis I argue for amounts to conditional claim: If a position satisfies all these conditions, then that position is unstable in the sense that it collapses into competing positions. (shrink)

We present a framework for epistemic logic, modeling the logical aspects of System 1 and System 2 cognitive processes, as per dual process theories of reasoning. The framework combines non-normal worlds semantics with the techniques of Dynamic Epistemic Logic. It models non-logically-omniscient, but moderately rational agents: their System 1 makes fast sense of incoming information by integrating it on the basis of their background knowledge and beliefs. Their System 2 allows them to slowly, step-wise unpack some of the (...) class='Hi'>logical consequences of such knowledge and beliefs, by paying a cognitive cost. The framework is applied to three instances of limited rationality, widely discussed in cognitive psychology: Stereotypical Thinking, the Framing Effect, and the Anchoring Effect. (shrink)

While the epistemic significance of disagreement has been a popular topic in epistemology for at least a decade, little attention has been paid to logical disagreement. This monograph is meant as a remedy. The text starts with an extensive literature review of the epistemology of (peer) disagreement and sets the stage for an epistemological study of logical disagreement. The guiding thread for the rest of the work is then three distinct readings of the ambiguous term ‘logical disagreement’. (...) Chapters 1 and 2 focus on the Ad Hoc Reading according to which logical disagreements occur when two subjects take incompatible doxastic attitudes toward a specific proposition in or about logic. Chapter 2 presents a new counterexample to the widely discussed Uniqueness Thesis. Chapters 3 and 4 focus on the Theory Choice Reading of ‘logical disagreement’. According to this interpretation, logical disagreements occur at the level of entire logical theories rather than individual entailment-claims. Chapter 4 concerns a key question from the philosophy of logic, viz., how we have epistemic justification for claims about logical consequence. In Chapters 5 and 6 we turn to the Akrasia Reading. On this reading, logical disagreements occur when there is a mismatch between the deductive strength of one’s background logic and the logical theory one prefers (officially). Chapter 6 introduces logical akrasia by analogy to epistemic akrasia and presents a novel dilemma. Chapter 7 revisits the epistemology of peer disagreement and argues that the epistemic significance of central principles from the literature are at best deflated in the context of logical disagreement. The chapter also develops a simple formal model of deep disagreement in Default Logic, relating this to our general discussion of logical disagreement. The monograph ends in an epilogue with some reflections on the potential epistemic significance of convergence in logical theorizing. (shrink)

I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism.

If you think, feel, and even give shape to something illogical, you automatically endow it with logic. And it magically ceases to be illogical, becoming merely a logical consequence of being contrasted with the logical. In other words, it becomes logical.

Supervaluationism is often described as the most popular semantic treatment of indeterminacy. There’s little consensus, however, about how to fill out the bare-bones idea to include a characterization of logical consequence. The paper explores one methodology for choosing between the logics: pick a logic thatnorms beliefas classical consequence is standardly thought to do. The main focus of the paper considers a variant of standard supervaluational, on which we can characterizedegrees of determinacy. It applies the methodology above to (...) focus ondegree logic. This is developed first in a basic, single-premise case; and then extended to the multipremise case, and to allowdegreesof consequence. The metatheoretic properties of degree logic are set out. On the positive side, the logic is supraclassical—all classical valid sequents are degree logic valid. Strikingly, metarules such as cut and conjunction introduction fail. (shrink)

In this paper I present and evaluate van Inwagen’s famous Consequence Argument, as presented in An Essay on Free Will. The grounds for the incompatibility of freewill and determinism, as argued by van Inwagen, is dependent on our actions being logical consequences of events outside of our control. Particularly, his arguments depend upon, in one guise or another, the transference of the modal property of not being possibly rendered false through the logical consequence relation, i.e. the (...) β-principle. I argue that, due to the symmetric nature of determinism, van Inwagen is exposed to what I call “reversibility arguments” in the literature. Such arguments reverse the β-principle and start from our apparent control over our own actions to our control over the initial conditions. Since van Inwagen does not endorse a particular theory of laws or logical consequence, he is open to such counterarguments. The plausibility of such reversibility arguments depends on what would be called a Wittgensteinian conception of logical consequence. In the _Remarks on the Foundation of Mathematics_, one of Wittgenstein’s main concerns is the normativity of logical inference i.e. proof. Such concerns with normativity and rule-following are generally a feature of his later philosophy. In the _Remarks_ Wittgenstein resists a conception of logical deduction which places the source of normativity outside of human practice. (shrink)

What is the connection between valid inference and true conditionals? Many conditional logics require that when A is a logical consequence of B, "if B then A" is true. Taking counterlogical conditionals seriously leads to systems that permit counterexamples to that general rule. However, this leaves those of us who endorse non-trivial accounts of counterpossible conditionals to explain what the connection between conditionals and consequence is. The explanation of the connection also answers a common line of objection (...) to non-trivial counterpossibles, which is based on a transition from valid arguments to the corresponding conditionals. It also contributes to the wider project of illuminating the connections between contexts of utterance, on the one hand, and the truth-conditions of conditionals uttered in those contexts, on the other. (shrink)

Contemporary semantic description of logic is based on the ontology of all possible interpretations, an insufficiently clear metaphysical concept. In this article, logic is described as the internal organization of language. Logical concepts -- logical constants, logical truths, and logical consequence -- are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which (...) the universal and existential quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language. (shrink)

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)

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The hyperintensional profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal and hyperintensional profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, (...) $\Omega$-logical validity is genuinely logical. Second, the foregoing provides a hyperintensional account of the interpretation of mathematical and metamathematical vocabulary. (shrink)

In this paper we prove the completeness of three logical systems I LI, IL2 and IL3. IL1 deals solely with identities {a = b), and its deductions are the direct deductions constructed with the three traditional rules: (T) from a = b and b = c infer a = c, (S) from a = b infer b = a and (A) infer a = a(from anything). IL2 deals solely with identities and inidentities {a ± b) and its deductions include (...) both the direct and the indirect deductions constructed with the three traditional rules. IL3 is a hybrid of IL1 and IL2: its deductions are all direct as in IL1 but it deals with identities and inidentities as in IL2. IL1 and IL2 have a high degree of naturalness. Although the hybrid system IL3 was constructed as an artifact useful in the mathematical study of IL1 and IL2, it nevertheless has some intrinsically interesting aspects. The main motivation for describing and studying such simple systems is pedagogical. In teaching beginning logic one would like to present a system of logic which has the following properties. First, it exemplifies the main ideas of logic: implication, deduction, non-implication, counterargument(or countermodel), logical truth, self-contradiction, consistency,satisfiability, etc. Second, it exemplifies the usual general metaprinciples of logic: contraposition and transitivity of implication, cut laws, completeness,soundness, etc. Third, it is simple enough to be thoroughly grasped by beginners. Fourth, it is obvious enough so that its rules do not appear to be arbitrary or purely conventional. Fifth, it does not invite confusions which must be unlearned later. Sixth, it involves a minimum of presuppositions which are no longer accepted in mainstream contemporary logic. (shrink)

There is a natural story about what logic is that sees it as tied up with two operations: a ‘throw things into a bag’ operation and a ‘closure’ operation. In a pair of recent papers, Jc Beall has fleshed out the account of logic this leaves us with in more detail. Using Beall’s exposition as a guide, this paper points out some problems with taking the second operation to be closure in the usual sense. After pointing out these problems, I (...) then turn to fixing them in a restricted case and modulo a few simplifying assumptions. In a followup paper, the simplifications and restrictions will be removed. (shrink)

The need to distinguish between logical and extra-logical varieties of inference, entailment, validity, and consistency has played a prominent role in meta-ethical debates between expressivists and descriptivists. But, to date, the importance that matters of logical form play in these distinctions has been overlooked. That’s a mistake given the foundational place that logical form plays in our understanding of the difference between the logical and the extra-logical. This essay argues that descriptivists are better positioned (...) than their expressivist rivals to provide the needed account of logical form, and so better able to capture the needed distinctions. This finding is significant for several reasons: First, it provides a new argument against expressivism. Second, it reveals that descriptivists can make use of this new argument only if they are willing to take a controversial—but plausible—stand on claims about the nature and foundations of logic. (shrink)

Reasoning over our knowledge bases and theories often requires non-deductive inferences, especially – but by no means only – when commonsense reasoning is the case, i.e. when practical agency is called for. This kind of reasoning can be adequately formalized via the notion of supraclassical consequence, a non-deductive consequence tightly associated with default and non-monotonic reasoning and featuring centrally in abductive, inductive, and probabilistic logical systems. In this paper, we analyze core concepts and problems of these systems (...) in the light of supraclassical consequence. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework (...) recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. (...) Her development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (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)

I explore the logic of ground. I first develop a logic of weak ground. This logic strengthens the logic of weak ground presented by Fine in his ‘Guide to Ground.’ This logic, I argue, generates many plausible principles which Fine’s system leaves out. I then derive from this a logic of strict ground. I argue that there is a strong abductive case for adopting this logic. It’s elegant, parsimonious and explanatorily powerful. Yet, so I suggest, adopting it has important consequences. (...) First, it means we should think of ground as a type of identity. Second, it means we should reject much of Fine’s logic of strict ground. I also show how the logic I develop connects to other systems in the literature. It is definitionally equivalent both to Angell’s logic of analytic containment and to Correia’s system G. (shrink)

Complex logic is a novel logical framework, which formalizes the semantics of the categories of matter, space, and time in a system of logic that operates with complex logical objects. A complex logical object represents a superposition of a logical statement and its logical negation positioning any statement co-relatively to its logical negation. In the system of logical notations, where S is a logical statement and Not S is its logical negation, (...) complex logic includes co-relative logical positions of S and Not S with the probabilities of their truth within the scale 0... 1, excluding the boundary values of 0 and 1, into a single logical superposition. Thus, the logical superposition, summing up the probabilistic positions of S and Not S, has an invariant truth value equal to 1. In this context, complex logic proves to be invariant to reality, offering a unified logical interpretation of reality, spanning from quantum to cosmological scales. By defining the nature of matter and reality as a complex two-format processual essence with corresponding logical consequences, complex logic enhances our concept and understanding of reality. (shrink)

‘Feminist logic’ may sound like an impossible, incoherent, or irrelevant project, but it is none of these. We begin by delineating three categories into which projects in feminist logic might fall: philosophical logic, philosophy of logic, and pedagogy. We then defuse two distinct objections to the very idea of feminist logic: the irrelevance argument and the independence argument. Having done so, we turn to a particular kind of project in feminist philosophy of logic: Valerie Plumwood's feminist argument for a relevance (...) logic. Plumwood's work serves as our primary case study as we turn to the project of considering three different ways we might understand her argument and revisionist arguments like it: as a priori theorizing, as ameliorative conceptual engineering, or as instances of anti-exceptionalist approaches to logic. After arguing that the anti-exceptionalist approach seems to provide the most promising means of understanding the kind of project undertaken in a feminist challenge to classical logic, we briefly address the consequences that this might have for logic instruction. Here, we argue for the perhaps unexpected conclusion that feminist programs ought to offer more, not less, instruction in logic for those who take interest. (shrink)

In some recent works, Crupi and Iacona have outlined an analysis of ‘if’ based on Chrysippus’ idea that a conditional holds whenever the negation of its consequent is incompatible with its antecedent. This paper presents a sound and complete system of conditional logic that accommodates their analysis. The soundness and completeness proofs that will be provided rely on a general method elaborated by Raidl, which applies to a wide range of systems of conditional logic.