Online research in philosophy

ABSTRACT: An introduction to Stoic logic. Stoic logic can in many respects be regarded as a fore-runner of modern propositional logic. I discuss: 1. the Stoic notion of sayables or meanings (lekta); the Stoic assertibles (axiomata) and their similarities and differences to modern propositions; the time-dependency of their truth; 2.-3. assertibles with demonstratives and quantified assertibles and their truth-conditions; truth-functionality of negations and conjunctions; non-truth-functionality of disjunctions and conditionals; language regimentation and ‘bracketing’ devices; (...) Stoic basic principles of propositional logic; 4. Stoic modal logic; 5. Stoic theory of arguments: two premisses requirement; validity and soundness; 6. Stoic syllogistic or theory of formally valid arguments: a reconstruction of the Stoic deductive system, which consisted of accounts of five types of indemonstrable syllogisms, which function as nullary argumental rules that identify indemonstrables or axioms of the system, and four deductive rules (themata) by which certain complex arguments can be reduced to indemonstrables and thus shown to be formally valid themselves; 7. arguments that were considered as non-syllogistically valid (subsyllogistic and unmethodically concluding arguments). Their validity was explained by recourse to formally valid arguments. (shrink)

ABSTRACT: Part 1 discusses the Stoic notion of propositions (assertibles, axiomata): their definition; their truth-criteria; the relation between sentence and proposition; propositions that perish; propositions that change their truth-value; the temporal dependency of propositions; the temporal dependency of the Stoic notion of truth; pseudo-dates in propositions. Part 2 discusses Stoic modal logic: the Stoic definitions of their modal notions (possibility, impossibility, necessity, non-necessity); the logical relations between the modalities; modalities as properties of propositions; contingent propositions; (...) the relation between the Stoic modal notions and those of Diodorus Cronus and Philo of Megara; the role of ‘external hindrances’ for the modalities; the temporal dependency of the modalities; propositions that change their modalities; the principle that something possible can follow from something impossible; the interpretations of the Stoic modal system by B. Mates, M. Kneale, M. Frede, J. Vuillemin and M. Mignucci are evaluated. -/- For a much shorter English version of Part 1 of the book see my ‘Stoic Logic’, in K. Algra et al. (eds), The Cambridge History of Hellenistic Philosophy, Cambridge 1999, 92-157. For a shorter, updated, English version of Part 2 of the book see my 'Chrysippus' Modal Logic and its Relation to Philo and Diodorus', in K. Doering / Th. Ebert (eds) Dialektiker und Stoiker (Stuttgart 1993) 63-84. (shrink)

This book uncovers and examines the confusion in antiquity between Aristotle's hypothetical syllogistic and Stoic logic, and offers a fresh perspective on the ...

ABSTRACT: A detailed presentation of Stoic theory of arguments, including truth-value changes of arguments, Stoic syllogistic, Stoic indemonstrable arguments, Stoic inference rules (themata), including cut rules and antilogism, argumental deduction, elements of relevance logic in Stoic syllogistic, the question of completeness of Stoic logic, Stoic arguments valid in the specific sense, e.g. "Dio says it is day. But Dio speaks truly. Therefore it is day." A more formal and more detailed account (...) of the Stoic theory of deduction can be found in S. Bobzien, Stoic Syllogistic, OSAP 1996. (shrink)

ABSTRACT: A detailed presentation of Stoic logic, part one, including their theories of propositions (or assertibles, Greek: axiomata), demonstratives, temporal truth, simple propositions, non-simple propositions(conjunction, disjunction, conditional), quantified propositions, logical truths, modal logic, and general theory of arguments (including definition, validity, soundness, classification of invalid arguments).

reasons for the disappreciation as well as for the rehabilitation of Stoic logic; it is found in I. M. Bochenski's Ancient Formal Logic (Amsterdam, 1951), and it clearly portrays the difference in attitude of the logicians of the twentieth century towards the Stoic logical system.

ABSTRACT: The 3rd BCE Stoic logician "Chrysippus says that the number of conjunctions constructible from ten propositions exceeds one million. Hipparchus refuted this, demonstrating that the affirmative encompasses 103,049 conjunctions and the negative 310,952." After laying dormant for over 2000 years, the numbers in this Plutarch passage were recently identified as the 10th (and a derivative of the 11th) Schröder number, and F. Acerbi showed how the 2nd BCE astronomer Hipparchus could have calculated them. What remained unexplained is why (...) Hipparchus’ logic differed from Stoic logic, and consequently, whether Hipparchus actually refuted Chrysippus. This paper closes these explanatory gaps. (1) I reconstruct Hipparchus’ notions of conjunction and negation, and show how they differ from Stoic logic. (2) Based on evidence from Stoic logic, I reconstruct Chrysippus’ calculations, thereby (a) showing that Chrysippus’ claim of over a million conjunctions was correct; and (b) shedding new light on Stoic logic and – possibly – on 3rd century BCE combinatorics. (3) Using evidence about the developments in logic from the 3rd to the 2nd centuries, including the amalgamation of Peripatetic and Stoic theories, I explain why Hipparchus, in his calculations, used the logical notions he did, and why he may have thought they were Stoic. (shrink)

ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental (...) rules which establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out. (shrink)

ABSTRACT: Alexander of Aphrodisias’ commentaries on Aristotle’s Organon are valuable sources for both Stoic and early Peripatetic logic, and have often been used as such – in particular for early Peripatetic hypothetical syllogistic and Stoic propositional logic. By contrast, this paper explores the role Alexander himself played in the development and transmission of those theories. There are three areas in particular where he seems to have made a difference: First, he drew a connection between certain passages (...) from Aristotle’s Topics and Prior Analytics and the Stoic indemonstrable arguments, and, based on this connection, appropriated at least four kinds of Stoic indemonstrables as Aristotelian. Second, he developed and made use of a specifically Peripatetic terminology in which to describe and discuss those arguments – which facilitated the integration of the indemonstrables into Peripatetic logic. Third, he made some progress towards a solution to the problem of what place and interpretation the Stoic third indemonstrables should be given in a Peripatetic and Platonist setting. Overall, the picture emerges that Alexander persistently (if not always consistently) presented passages from Aristotle’s logical œuvre in a light that makes it appear as if Aristotle was in the possession of a Peripatetic correlate to the Stoic theory of indemonstrables. (shrink)

ABSTRACT: This paper traces the evidence in Galen's Introduction to Logic (Institutio Logica) for a hypothetical syllogistic which predates Stoic propositional logic. It emerges that Galen is one of our main witnesses for such a theory, whose authors are most likely Theophrastus and Eudemus. A reconstruction of this theory is offered which - among other things - allows to solve some apparent textual difficulties in the Institutio Logica.

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with a focus on issues that may be of interest to contemporary logicians and covering important topics in Post-Aristotelian logic that are frequently neglected (such as Peripatetic hypothetical syllogistic, the Stoic axiomatic system of propositional logic and various later ancient developments).

ABSTRACT: The modal systems of the Stoic logician Chrysippus and the two Hellenistic logicians Philo and Diodorus Cronus have survived in a fragmentary state in several sources. From these it is clear that Chrysippus was acquainted with Philo’s and Diodorus’ modal notions, and also that he developed his own in contrast of Diodorus’ and in some way incorporated Philo’s. The goal of this paper is to reconstruct the three modal systems, including their modal definitions and modal theorems, and to (...) make clear the exact relations between them; moreover, to elucidate the philosophical reasons that may have led Chrysippus to modify his predessors’ modal concept in the way he did. It becomes apparent that Chrysippus skillfully combined Philo’s and Diodorus’ modal notions, with making only a minimal change to Diodorus’ concept of possibility; and that he thus obtained a modal system of modalities (logical and physical) which fit perfectly fit into Stoic philosophy. (shrink)

Logic as a discipline starts with the transition from the more or less unreflective use of logical methods and argument patterns to the reflection on and inquiry into these and their elements, including the syntax and semantics of sentences. In Greek and Roman antiquity, discussions of some elements of logic and a focus on methods of inference can be traced back to the late 5th century BCE. The Sophists, and later Plato (early 4th c.) displayed an interest in (...) sentence analysis, truth, and fallacies, and Eubulides of Miletus (mid-4th c.) is on record as the inventor of both the Liar and the Sorites paradox. But logic as a fully systematic discipline begins with Aristotle, who systematized much of the logical inquiry of his predecessors. His main achievements were his theory of the logical interrelation of affirmative and negative existential and universal statements and, based on this theory, his syllogistic, which can be interpreted as a system of deductive inference. Aristotle's logic is known as term-logic, since it is concerned with the logical relations between terms, such as ‘human being’, ‘animal’, ‘white’. It shares elements with both set theory and predicate logic. Aristotle's successors in his school, the Peripatos, notably Theophrastus and Eudemus, widened the scope of deductive inference and improved some aspects of Aristotle's logic. (shrink)

ABSTRACT: A very brief summary presentation of western ancient logic for the non-specialized reader, from the beginnings to Boethius. For a much more detailed presentation see my "Ancient Logic" in the Stanford Encyclopedia of Philosopy (also on PhilPapers).

ABSTRACT: In this paper I argue (i) that the hypothetical arguments about which the Stoic Chrysippus wrote numerous books (DL 7.196) are not to be confused with the so-called "hypothetical syllogisms", but are the same hypothetical arguments as those mentioned five times in Epictetus (e.g. Diss. 1.25.11-12); and (ii) that these hypothetical arguments are formed by replacing in a non-hypothetical argument one (or more) of the premisses by a Stoic "hypothesis" or supposition. Such "hypotheses" or suppositions differ from (...) propositions in that they have a specific logical form and no truth-value. The reason for the introduction of a distinct class of hypothetical arguments can be found in the context of dialectical argumentation. The paper concludes with the discussion of some evidence for the use of Stoic hypothetical arguments in ancient texts. (shrink)

ABSTRACT: This paper discusses ancient versions of paradoxes today classified as paradoxes of presupposition and how their ancient solutions compare with contemporary ones. Sections 1-4 air ancient evidence for the Fallacy of Complex Question and suggested solutions, introduce the Horn Paradox, consider its authorship and contemporary solutions. Section 5 reconstructs the Stoic solution, suggesting the Stoics produced a Russellian-type solution based on a hidden scope ambiguity of negation. The difference to Russell’s explanation of definite descriptions is that in the (...) Horn Paradox the Stoics uncovered a hidden conjunction rather than a hidden existential sentence. Sections 6 and 7 investigate hidden ambiguities in “to have” and “to lose” (including inalienable and alienable possession) and ambiguities of quantification based on substitution of indefinite plural expressions for indefinite or anaphoric pronouns, and Stoic awareness of these. Section 8 considers metaphorical readings and allusions that add further spice to the paradox. (shrink)

In this paper I present the text, a translation, and a commentary of a long anonymous scholium to Aristotle’s Analytics which is a Greek parallel to Boethius’ De Hypotheticis Syllogismis, but has so far not been recognized as such. The scholium discusses hypothetical syllogisms of the types modus ponens and modus tollens and hypothetical syllogisms constructed from three conditionals (‘wholly hypothetical syllogisms’). It is Peripatetic, and not Stoic, in its theoretical approach as well as its terminology. There are several (...) elements of early Peripatetic hypothetical syllogistic preserved in it, and there is a large number of close parallels to Boethius’ De Hypotheticis Syllogismis which we find in no other source. It is very likely that there was a Greek source from which both the scholium and large parts of Boethius’ De Hypotheticis Syllogismis are ultimately derived. (shrink)

ABSTRACT: Recently a bold and admirable interpretation of Chrysippus’ position on the Sorites has been presented, suggesting that Chrysippus offered a solution to the Sorites by (i) taking an epistemicist position1 which (ii) made allowances for higher-order vagueness.2 In this paper I argue (i) that Chrysippus did not take an epistemicist position, but − if any − a non-epistemic one which denies truth-values to some cases in a Sorites-series, and (ii) that it is uncertain whether and how he made allowances (...) for higher-order vagueness, but if he did, this was not grounded on an epistemicist position. (shrink)

The specialized essays in this collection study whether non-Aristotelian traditions of ancient logic had a role for medieval logicians. Special attention is given to Stoic logic and semantics, and to Neoplatonism.

Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. (...) Further volumes will follow, including Mediaeval and Renaissance Logic and Logic: A History of its Central. In designing the Handbook of the History of Logic, the Editors have taken the view that the history of logic holds more than an antiquarian interest, and that a knowledge of logic's rich and sophisticated development is, in various respects, relevant to the research programmes of the present day. Ancient logic is no exception. The present volume attests to the distant origins of some of modern logic's most important features, such as can be found in the claim by the authors of the chapter on Aristotle's early logic that, from its infancy, the theory of the syllogism is an example of an intuitionistic, non-monotonic, relevantly paraconsistent logic. Similarly, in addition to its comparative earliness, what is striking about the best of the Megarian and Stoic traditions is their sophistication and originality. Logic is an indispensably important pivot of the Western intellectual tradition. But, as the chapters on Indian and Arabic logic make clear, logic's parentage extends more widely than any direct line from the Greek city states. It is hardly surprising, therefore, that for centuries logic has been an unfetteredly international enterprise, whose research programmes reach to every corner of the learned world. Like its companion volumes, Greek, Indian and Arabic Logic is the result of a design that gives to its distinguished authors as much space as would be needed to produce highly authoritative chapters, rich in detail and interpretative reach. The aim of the Editors is to have placed before the relevant intellectual communities a research tool of indispensable value. Together with the other volumes, Greek, Indian and Arabic Logic, will be essential reading for everyone with a curiosity about logic's long development, especially researchers, graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the history of philosophy, and the history of ideas. (shrink)

Literature on the Stoa has recently concentrated on historical accounts of the development of the school and on Stoicism as a social movement. Professor Rist’s approach is to examine in detail a series of philosophical problems discussed by leading members of the Stoic school. He is not concerned with social history or with the influence of Stoicism on popular beliefs in the Ancient world, but with such questions as the relation between Stoicism and the thought of Aristotle, the meaning (...) and purpose of such Stoic paradoxes as, ‘all sins are equal’, and the philosophical interrelation of Stoic physics and ethics. There are chapters on aspects of Stoic logic and on the thought of particular thinkers such as Panaetius and Posidonius, but ethical problems occupy the centre of the stage. (shrink)

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated (...) by Herbrand’s Induction-Axiom Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

ABSTRACT: This paper collects the evidence in Ammonius' surviving works for elements of a propositional logic, coming to the conclusion that Ammonius had a theory of hypothetical syllogisms in the tradition of Aristotle and the Peripatetics, with Platonic elements mixed in, and using some Stoic elements, but not a propositional logic in the narrower sense as we find it in Stoic logic.

This book collects a series of important new studies on one of the richest and most influential intellectual traditions of antiquity. Leading scholars combine careful analytical attention to the original texts with historical sensitivity and philosophical acuity to point the way to a better understanding of Stoic ethics, political theory, logic, and science.

Although Kant envisaged a prominent role for logic in the argumentative structure of his Critique of pure reason, logicians and philosophers have generally judged Kant's logic negatively. What Kant called `general' or `formal' logic has been dismissed as a fairly arbitrary subsystem of first order logic, and what he called `transcendental logic' is considered to be not a logic at all: no syntax, no semantics, no definition of validity. Against this, we argue that Kant's (...) `transcendental logic' is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first order logic. The main technical application of the formalism developed here is a formal proof that Kant's Table of Judgements in §9 of the Critique of pure reason, is indeed, as Kant claimed, complete for the kind of semantics he had in mind. This result implies that Kant's 'general' logic is after all a distinguished subsystem of first order logic, namely what is known as geometric logic. (shrink)

Much of the last fifty years of scholarship on Aristotle’s syllogistic suggests a conceptual framework under which the syllogistic is a logic, a system of inferential reasoning, only if it is not a theory or formal ontology, a system concerned with general features of the world. In this paper, I will argue that this a misleading interpretative framework. The syllogistic is something sui generis: by our lights, it is neither clearly a logic, nor clearly a theory, but rather (...) exhibits certain characteristic marks of logics and certain characteristic marks of theories. In what follows, I will present a debate between a theoretical and a logical interpretation of the syllogistic. The debate centers on the interpretation of syllogisms as either implications or inferences. But the significance of this question has been taken to concern the nature and subject-matter of the syllogistic, and how it ought to be represented by modern techniques. For one might think that, if syllogisms are implications, propositions with conditional form, then the syllogistic, in so far as it is a systematic taxonomy of syllogisms, is a theory or a body of knowledge concerned with general features of the world. Furthermore, if the syllogistic is a theory, then it ought to be represented by an axiomatic system, a system deriving propositional theorems from axioms. On the other hand, if syllogisms are inferences, then the syllogistic is a logic, a system of inferential reasoning. And furthermore, it ought to be represented as a natural deduction system, a system deriving valid arguments by means of intuitively valid inferences. I will argue that one can disentangle these questions—are syllogisms inferences or implications, is the syllogistic a logic or a theory, is the syllogistic a body of worldly knowledge or a system of inferential reasoning, and ought we to represent the syllogistic as a natural deduction system or an axiomatic system—and that we must if we are to have a historically accurate understanding of Aristotle. (shrink)

This chapter begins with a discussion of Kant's theory of judgment-forms. It argues that it is not true in Kant's logic that assertoric or apodeictic judgments imply problematic ones, in the manner in which necessity and truth imply possibility in even the weakest systems of modern modal logic. The chapter then discusses theories of judgment-form after Kant, the theory of quantification, Frege's Begriffsschrift, C. I. Lewis and the beginnings of modern modal logic, the proof-theoretic approach to modal (...) logic, possible world semantics, correspondence theory, and modality and quantification. (shrink)

This collection, nearly all chosen by Boolos himself shortly before his death, includes thirty papers on set theory, second-order logic, and plural quantifiers; ...

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)

Rabern and Rabern (Analysis 68:105–112 2 ) and Uzquiano (Analysis 70:39–44 4 ) have each presented increasingly harder versions of ‘the hardest logic puzzle ever’ (Boolos The Harvard Review of Philosophy 6:62–65 1 ), and each has provided a two-question solution to his predecessor’s puzzle. But Uzquiano’s puzzle is different from the original and different from Rabern and Rabern’s in at least one important respect: it cannot be solved in less than three questions. In this paper we solve Uzquiano’s (...) puzzle in three questions and show why there is no solution in two. Finally, to cement a tradition, we introduce a puzzle of our own. (shrink)

Even among those philosophers who hold particular aspects of Hegel's philosophy in high regard, there have been few since the 19th century who have found Hegel's "metaphysics" plausible, and just as few not sceptical about the coherency of the "logical" project on which it is meant to be based. Indeed, against the type of work characteristic of the late nineteenth-century logical revolution which issued in modern analytic philosophy, it is often difficult to see exactly how Hegel's "logical" writings can be (...) read as a contribution to logic at all. Furthermore, any tendency toward skepticism here can only have been reinforced by the well-known views of Bertrand Russell about the logical inadequacy of the "Hegelian" approach of his predecessors. (shrink)

Friends, welcome to the first page of Logic in India. It is for Indian students prepared for first paper entitled Principles of Logic in Diploma-in-Reasoning course of Department of Philosophy, Kurukshetra University, Kurukshetra, where I taught four years. It is also beneficial for graduate students who have elementary logic course in their syllabus. Basically I used both printed books and internet sources to prepare it. You can find the course syllabus in my post “Philosophy is Nothing without (...) Logic” at The Positive Philosophy page and also in the side links of this page. This is only a draft, kindly send your suggestions and ideas to dr.sirswal@gmail.com or niyamak.drs@gmail.com, I shall be highly thankful to you. A short list of reference books are mentioned below of the Table of Contents and reference sites are linked with this page. This page introduces the basic conceptions of formal logic, informal logic and also Symbolic logic. (shrink)

ABSTRACT: ‘Aristotelian logic’, as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as ‘hypothetical syllogisms’. However, Aristotle did not discuss such arguments, nor did he call any arguments ‘hypothetical syllogisms’. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them ‘hypothetical (...) syllogisms’; nor did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle’s logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called ‘hypothetical syllogisms’? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle’s dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle’s logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories. (shrink)

In this paper, I first trace the course of Prior's struggles with the concepts and phenomena of modality and the reasoning that led him to his own rather peculiar modal logic Q. I find myself in almost complete agreement with Prior's intuitions and the arguments that rest upon them. However, I will argue that those intuitions do not of themselves lead to Q, but that one must also accept a certain picture of what it is for a proposition to (...) be possible. That picture, though, is not inevitable. Rather, implicit in Prior's own account is an alternative picture that has already appeared in various guises, most prominently in the work of Adams, Fine, Deutsch, and Almog. I, too, will opt for this alternative, though I will spell it out rather differently than these philosophers. I will then show that, starting with the alternative picture, Prior's intuitions can lead instead to a much happier and more standard quantified modal logic than Q. The last section of the paper is devoted to the formal development of the logic and its metatheory. (shrink)

ABSTRACT: 'Aristotelian logic', as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as 'hypothetical syllogisms'. However, Aristotle did not discuss such arguments, nor did he call any arguments 'hypothetical syllogisms'. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them 'hypothetical (...) syllogisms'; nor did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle's logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called 'hypothetical syllogisms'? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle's dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle's logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories. (shrink)

ABSTRACT: This paper discusses the Stoic treatment of fallacies that are based on lexical ambiguities. It provides a detailed analysis of the relevant passages, lays bare textual and interpretative difficulties, explores what the Stoic view on the matter implies for their theory of language, and compares their view with Aristotle’s. In the paper I aim to show that, for the Stoics, fallacies of ambiguity are complexes of propositions and sentences and thus straddle the realms of meaning (which is (...) the domain of logic) and of linguistic expressions (which is the domain of linguistics), but also involve a pragmatic element; that the Stoics believe that the premises of the fallacies, when uttered, have only one meaning and are true, and thus should be conceded; that hence there is no need for a mental process of disambiguation in the listeners; that Aristotle, by contrast, appears to assume that the premises always have all their meanings, and accordingly recommends that the listeners explicitly disambiguate them, which presupposes a process of mental disambiguation. I proffer two readings of the Stoic advice that we ‘be silent’ when confronted with a fallacy of ambiguity in dialectical discourse, and explicate how each leads to an overall consistent interpretation of the textual evidence. Finally, I demonstrate that the method advocated by the Stoics works in all cases of fallacies of lexical ambiguity. (shrink)

I propose a new semantics for intuitionistic logic, which is a cross between the construction-oriented semantics of Brouwer-Heyting-Kolmogorov and the condition-oriented semantics of Kripke. The new semantics shows how there might be a common semantical underpinning for intuitionistic and classical logic and how intuitionistic logic might thereby be tied to a realist conception of the relationship between language and the world.

We study a range of issues connected with the idea of replacing one formula by another in a fixed (linguistic) context. The replacement core of a consequence relation ⊢ is the relation holding between a set of formulas { A 1 , ..., A m , ...} and a formula B when for every context C (·), we have C ( A 1 ), ..., C ( A m ), ... ⊢ C ( B ). Section 1 looks at some (...) differences between which inferences are lost on passing to the replacement cores of the classical and intuitionistic consequence relations. For example, we find that while the inference from A and B to , sanctioned by both these initial consequence relations, is retained on passage to the replacement core in the classical case, it is lost in the intuitionistic case. Further discussion of these two (and some other) logics occupies Sections 3 and 4. Section 2 looks at the m = 1 case, describing A as replaceable by B according to ⊢ when B is a consequence of A by the replacement core of ⊢, and inquiring as to which choices of ⊢ render this induced replaceability relation symmetric. Section 5 investigates further conceptual refinements— such as a contrast between horizontal and vertical replaceability—suggested by some work of R. B. Angell and R. Harrop (and a comment on the latter by T. J. Smiley) in the 1950s and 1960s. Appendix 1 examines a related aspect of term-for-term replacement in connection with identity in predicate logic. Appendix 2 is a repository for proofs which would otherwise clutter up Section 3. (shrink)

This paper explores the question of what logic is not. It argues against the wide spread assumptions that logic is: a model of reason; a model of correct reason; the laws of thought, or indeed is related to reason at all such that the essential nature of the two are crucially or essentially co-illustrative. I note that due to such assumptions, our current understanding of the nature of logic itself is thoroughly entangled with the nature of reason. (...) I show that most arguments for the presence of any sort of essential re- lationship between logic and reason face intractable problems and demands, and fall well short of addressing them. These arguments include those for the notion that logic is normative for reason (or that logic and correct reason are in some way the same thing), that logic is some sort of description of correct reason and that logic is an abstracted or idealised version of correct reason. A strong version of logical realism is put forward as an alternative view, and is briefly explored. (shrink)

In this work we propose an encoding of Reiter’s Situation Calculus solution to the frame problem into the framework of a simple multimodal logic of actions. In particular we present the modal counterpart of the regression technique. This gives us a theorem proving method for a relevant fragment of our modal logic.

We present the inconsistency-adaptive deontic logic DP r , a nonmonotonic logic for dealing with conflicts between normative statements. On the one hand, this logic does not lead to explosion in view of normative conflicts such as O A ∧ O ∼A, O A ∧ P ∼A or even O A ∧ ∼O A. On the other hand, DP r still verifies all intuitively reliable inferences valid in Standard Deontic Logic (SDL). DP r interprets a given (...) premise set ‘as normally as possible’ with respect to SDL. Whereas some SDL-rules are verified unconditionally by DP r , others are verified conditionally. The latter are applicable unless they rely on formulas that turn out to behave inconsistently in view of the premises. This dynamic process is mirrored by the proof theory of DP r. (shrink)

: Convinced that logic has a history and that its history always manages to surprise the philosophers, Claude Imbert has devoted much of her work to the study of the Stoic school and of the late-nineteenth-century German logician Gottlob Frege. In the fifth chapter of her book Pour une histoire de la logique, she examines the trajectory of Frege's awareness of what his new logic entails, in particular the way it subverts the project of Kant.

This paper is chiefly aimed at individuating some deep, but as yet almost unnoticed, similarities between Aristotle's syllogistic and the Stoic doctrine of conditionals, notably between Aristotle's metasyllogistic equimodality condition (as stated at APr. I 24, 41b27–31) and truth-conditions for third type (Chrysippean) conditionals (as they can be inferred from, say, S.E. P. II 111 and 189). In fact, as is shown in §1, Aristotle's condition amounts to introducing in his (propositional) metasyllogistic a non-truthfunctional implicational arrow '', the truth-conditions (...) of which turn out to be logically equivalent to truth-conditions of third type conditionals, according to which only the impossible (and not the possible) follows from the impossible. Moreover, Aristotle is given precisely this non-Scotian conditional logic in two so far overlooked passages of (Latin and Hebraic translations of) Themistius' Paraphrasis of De Caelo (CAG V 4, 71.8–13 and 47.8–10 Landauer). Some further consequences of Aristotle's equimodality condition on his logic, and notably on his syllogistic (no <span class='Hi'>matter</span> whether modal or not), are pointed out and discussed at length. A (possibly Chrysippean) extension of Aristotle's condition is also discussed, along with a full characterization of truth-conditions of fourth type conditionals. (shrink)

We introduce a substructural propositional calculus of Sequential Dynamic Logic that subsumes a propositional part of dynamic predicate logic, and is shown to be expressively equivalent to propositional dynamic logic. Completeness of the calculus with respect to the intended relational semantics is established.

We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindström, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle quantifier scope dependencies of generalized quantifiers in a satisfying way the dependence atom in Dependence logic is not well suited and that the multivalued dependence atom is a better choice. This atom is in (...) fact definably equivalent to the independence atom recently introduced by Väänänen and Grädel. (shrink)

Here I revisit Bolzano's criticisms of Kant on the nature of logic. I argue that while Bolzano is correct in taking Kant to conceive of the traditional logic as a science of the activity of thinking rather than the content of thought, he is wrong to charge Kant with a failure to identify and examine this content itself within logic as such. This neglects Kant's own insistence that traditional logic does not exhaust logic as such, (...) since it must be supplemented by a transcendental logic that will in fact study nothing other than thought's content. Once this feature of Kant's views is brought to light, a much deeper accord emerges between the two thinkers than has hitherto been appreciated, on both the nature of the content that is at issue in logic and the sense of logic's generality and formality. (shrink)

We study a new formal logic LD introduced by Prof. Grzegorczyk. The logic is based on so-called descriptive equivalence, corresponding to the idea of shared meaning rather than shared truth value. We construct a semantics for LD based on a new type of algebras and prove its soundness and complete- ness. We further show several examples of classical laws that hold for LD as well as laws that fail. Finally, we list a number of open problems.

What is informal logic, is it ``logic" at all? Main contemporary approaches are briefly presented and critically commented. If the notion of consequence is at the heart of logic, does it make sense to speak about ``informal" consequence? A valid inference is truth preserving, if the premises are true, so is the conclusion. According to Prawitz two further conditions must also be satisfied in the case of classical logical consequence: (i) it is because of the logical form (...) of the sentences involved and not because of their specific content that the inference is truth preserving; (ii) it is necessary that if the premises are true, then so is the conclusion. According to the prevalent criteria of informal logic an argument is cogent if and only if (i) its premises are rationally Acceptable, (ii) its premises are Relevant to its conclusion and (iii) its premises constitute Grounds adequate for accepting the conclusion (the ``ARG" conditions according to Govier). The ARG criteria characterize a certain broad kind of consequence relation. We do not (in general) have truth preservence in cogent arguments but if the premises are acceptable and other criteria are met, then so is the conclusion. We can speak about form in a loose sense and finally, there is rational necessity of the grounding or support relation. So a certain broad notion of logical consequence emerges from this comparison. The norms of ARG are norms of elementary scientific methodology in which argument is seen as embodying reasoning within a process of inquiry or of belief formation in subject areas accessible to every informed intellectual. (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)

Epistemic closure has been a central issue in epistemology over the last forty years. According to versions of the relevant alternatives and subjunctivist theories of knowledge, epistemic closure can fail: an agent who knows some propositions can fail to know a logical consequence of those propositions, even if the agent explicitly believes the consequence (having “competently deduced” it from the known propositions). In this sense, the claim that epistemic closure can fail must be distinguished from the fact that agents do (...) not always believe, let alone know, the consequences of what they know—a fact that raises the “problem of logical omniscience” that has been central in epistemic logic. -/- This paper, part I of II, is a study of epistemic closure from the perspective of epistemic logic. First, I introduce models for epistemic logic, based on Lewis’s models for counterfactuals, that correspond closely to the pictures of the relevant alternatives and subjunctivist theories of knowledge in epistemology. Second, I give an exact characterization of the closure properties of knowledge according to these theories, as formalized. Finally, I consider the relation between closure and higher-order knowledge. The philosophical repercussions of these results and results from part II, which prompt a reassessment of the issue of closure in epistemology, are discussed further in companion papers. -/- As a contribution to modal logic, this paper demonstrates an alternative approach to proving modal completeness theorems, without the standard canonical model construction. By “modal decomposition” I obtain completeness and other results for two non-normal modal logics with respect to new semantics. One of these logics, dubbed the logic of ranked relevant alternatives, appears not to have been previously identified in the modal logic literature. More broadly, the paper presents epistemology as a rich area for logical study. (shrink)

The present chapter describes a probabilistic framework of human reasoning. It is based on probability logic. While there are several approaches to probability logic, we adopt the coherence based approach.

In a paper from the 1980s, Byrd claims that the logic of "eventual permanence" for linear time is KD5. In this note we take up Byrd's novel argument for this and, treating the problem as one concerning translational embeddings, show that rather than KD5 the correct logic of "eventual permanence" is KD45.

The purpose of the present note is to advertise an interesting conjecture concerning a well-known translation in modal logic, by confirming a (highly restricted) special case of the conjecture.

This paper develops a formal system, consisting of a language and semantics, called serial logic ( SL ). In rough outline, SL permits quantification over, and reference to, some finite number of things in an order , in an ordinary everyday sense of the word “order,” and superplural quantification over things thus ordered. Before we discuss SL itself, some mention should be made of an issue in philosophical logic which provides the background to the development of SL , (...) and with respect to which I wish to contend that the system permits progress. (shrink)

We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\varphi$ and $(\varphi\odot\psi)$ also satisfies $\psi$ , except trivially. We derive this as a corollary of a more general result concerning variable sharing.

This is part I of a two-part essay introducing case-intensional first order logic (CIFOL), an easy-to-use, uniform, powerful, and useful combination of first-order logic with modal logic resulting from philosophical and technical modifications of Bressan’s General interpreted modal calculus (Yale University Press 1972 ). CIFOL starts with a set of cases; each expression has an extension in each case and an intension, which is the function from the cases to the respective case-relative extensions. Predication is intensional; identity (...) is extensional. Definite descriptions are context-independent terms, and lambda-predicates and -operators can be introduced without constraints. These logical resources allow one to define, within CIFOL, important properties of properties, viz., extensionality (whether the property applies, depends only on an extension in one case) and absoluteness, Bressan’s chief innovation that allows tracing an individual across cases without recourse to any notion of “rigid designation” or “trans-world identity.” Thereby CIFOL abstains from incorporating any metaphysical principles into the quantificational machinery, unlike extant frameworks of quantified modal logic. We claim that this neutrality makes CIFOL a useful tool for discussing both metaphysical and scientific arguments involving modality and quantification, and we illustrate by discussing in diagrammatic detail a number of such arguments involving the extensional identification of individuals via absolute (substance) properties, essential properties, de re vs. de dicto , and the results of possible tests. (shrink)

In a number of publications A.N. Prior considered the use of what he called ‘metric tense logic’. This is a tense logic in which the past and future operators P and F have an index representing a temporal distance, so that Pnα means that α was true n -much ago, and Fn α means that α will be true n -much hence. The paper investigates the use of metric predicate tense logic in formalising phenomena ormally treated by (...) such devices as multiple indexing or quantification over times. (shrink)

Modeling a complex phenomena such as the mind presents tremendous computational complexity challenges. Modeling field theory (MFT) addresses these challenges in a non-traditional way. The main idea behind MFT is to match levels of uncertainty of the model (also, a problem or some theory) with levels of uncertainty of the evaluation criterion used to identify that model. When a model becomes more certain, then the evaluation criterion is adjusted dynamically to match that change to the model. This process is called (...) the Dynamic Logic of Phenomena (DLP) for model construction and it mimics processes of the mind and natural evolution. This paper provides a formal description of DLP by specifying its syntax, semantics, and reasoning system. We also outline links between DLP and other logical approaches. Computational complexity issues that motivate this work are presented using an example of polynomial models. (shrink)

We are used to the idea that computers operate on numbers, yet another kind of data is equally important: the syntax of formal languages, with variables, binding, and alpha-equivalence. The original application of nominal techniques, and the one with greatest prominence in this paper, is to reasoning on formal syntax with variables and binding. Variables can be modelled in many ways: for instance as numbers (since we usually take countably many of them); as links (since they may `point' to a (...) binding site in the term, where they are bound); or as functions (since they often, though not always, represent `an unknown'). None of these models is perfect. In every case for the models above, problems arise when trying to use them as a basis for a fully formal mechanical treatment of formal language. The problems are practical—but their underlying cause may be mathematical. The issue is not whether formal syntax exists, since clearly it does, so much as what kind of mathematical structure it is. To illustrate this point by a parody, logical derivations can be modelled using a Gödel encoding (i.e., injected into the natural numbers). It would be false to conclude from this that proof-theory is a branch of number theory and can be understood in terms of, say, Peano's axioms. Similarly, as it turns out, it is false to conclude from the fact that variables can be encoded e.g., as numbers, that the theory of syntax-with-binding can be understood in terms of the theory of syntax-without-binding, plus the theory of numbers (or, taking this to a logical extreme, purely in terms of the theory of numbers). It cannot; something else is going on. What that something else is, has not yet been fully understood. In nominal techniques, variables are an instance of names, and names are data. We model names using urelemente with properties that, pleasingly enough, turn out to have been investigated by Fraenkel and Mostowski in the first half of the 20th century for a completely different purpose than modelling formal language. What makes this model really interesting is that it gives names distinctive properties which can be related to useful logic and programming principles for formal syntax. Since the initial publications, advances in the mathematics and presentation have been introduced piecemeal in the literature. This paper provides in a single accessible document an updated development of the foundations of nominal techniques. This gives the reader easy access to updated results and new proofs which they would otherwise have to search across two or more papers to find, and full proofs that in other publications may have been elided. We also include some new material not appearing elsewhere. (shrink)

We examine the transitions between sets of possible worlds described by the compositional semantics of Modal Dependence Logic, and we use them as the basis for a dynamic version of this logic. We give a game theoretic semantics, a (compositional) transition semantics and a power game semantics for this new variant of modal Dependence Logic, and we prove their equivalence; and furthermore, we examine a few of the properties of this formalism and show that Modal Dependence (...) class='Hi'>Logic can be recovered from it by reasoning in terms of reachability. Then we show how we can generalize this approach to a very general formalism for reasoning about transformations between pointed Kripke models. (shrink)

A family of symmetries of polyadic inductive logic are described which in turn give rise to the purportedly rational Permutation Invariance Principle stating that a rational assignment of probabilities should respect these symmetries. An equivalent, and more practical, version of this principle is then derived.

In this paper, I want to substantiate three related claims regarding causal discovery from non-experimental data. Firstly, in scientific practice, the problem of ignorance is ubiquitous, persistent, and far-reaching. Intuitively, the problem of ignorance bears upon the following situation. A set of random variables V is studied but only partly tested for (conditional) independencies; i.e. for some variables A and B it is not known whether they are (conditionally) independent. Secondly, Judea Pearl’s most meritorious and influential algorithm for causal discovery (...) (the IC algorithm) cannot be applied in cases of ignorance. It presupposes that a full list of (conditional) independence relations is on hand and it would lead to unsatisfactory results when applied to partial lists. Finally, the problem of ignorance is successfully treated by means of ALIC, the adaptive logic for causal discovery presented in this paper. (shrink)

This paper shows that the basic logic induced by the parallel recurrence $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ of computability logic (i.e., the one in the signature $\{\neg,$\wedge$,\vee,\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}},\hspace {-2pt}\mbox {\raisebox {0.12cm}{\@setfontsize \small {7}{8}$\vee$}\hspace {-3.6pt}\raisebox {0.02cm}{\tiny $\mid$}\hspace {2pt}}\}$ ) is a proper superset of the basic logic induced by the branching recurrence $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$ (i.e., the one in the signature $\{\neg,$\wedge$,\vee,\mbox (...) {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}},\mbox {\raisebox {0.12cm}{$\circ$}\hspace {-0.115cm}\raisebox {0.02cm}{\tiny $\mid$}\hspace {2pt}}\}$ ). The latter is known to be precisely captured by the cirquent calculus system CL15 , conjectured by Japaridze to remain sound—but not complete—with $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ instead of $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$ . The present result is obtained by positively verifying that conjecture. A secondary result of the paper is showing that $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ is strictly weaker than $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$ in the sense that, while $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}F$ logically implies $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}F$ , the reverse does not hold. (shrink)

We develop a semantics for independence logic with respect to what we will call general models. We then introduce a simpler entailment semantics for the same logic, and we reduce the validity problem in the former to the validity problem in the latter. Then we build a proof system for independence logic and prove its soundness and completeness with respect to entailment semantics.

We present a reading of the traditional syllogistics in a fragment of the propositional intuitionistic multiplicative linear logic and prove that with respect to a diagrammatic logical calculus that we introduced in a previous paper, a syllogism is provable in such a fragment if and only if it is diagrammatically provable. We extend this result to syllogistics with complemented terms à la De Morgan, with respect to a suitable extension of the diagrammatic reasoning system for the traditional case and (...) a corresponding reading of such De Morgan style syllogistics in the previously referred to fragment of linear logic. (shrink)

Łukasiewicz three-valued logic Ł3 is often understood as the set of all 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic Ł3 is axiomatized as an extension (...) of Routley and Meyer’s basic positive logic following Brady’s strategy for axiomatizing many-valued logics by employing two-valued under-determined or over-determined interpretations. Finally, it is proved that “well determined” Łukasiewicz logics are paraconsistent. (shrink)

First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in the aforementioned proof (...) systems when capacity limits are put on the cognitive resources. Finally, we investigate the correlation between a number of mathematical complexity measures defined on graphs and sentences and some psychological complexity measures that were recorded in the experiment. (shrink)

Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. A distinction therefore arises between the logic and its substitution core, the set of formulas all of whose substitution instances are valid. The classic example of a non-uniform dynamic epistemic logic is Public Announcement Logic (PAL), and a well-known open problem is to axiomatize the substitution core of PAL. In this paper we solve this problem for PAL over the class of all relational (...) models with infinitely many agents, PAL-K_omega, as well as standard extensions thereof, e.g., PAL-T_omega, PAL-S4_omega, and PAL-S5_omega. We introduce a new Uniform Public Announcement Logic (UPAL), prove completeness of a deductive system with respect to UPAL semantics, and show that this system axiomatizes the substitution core of PAL. (shrink)

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬BOXp, read as “p is true but you do not know p.” Not only is (...) the Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas. (shrink)

This article challenges the common view that improvements in critical thinking are best pursued by investigations in informal logic. From the perspective of research in psychology and neuroscience, hu-man inference is a process that is multimodal, parallel, and often emo-tional, which makes it unlike the linguistic, serial, and narrowly cog-nitive structure of arguments. At-tempts to improve inferential prac-tice need to consider psychological error tendencies, which are patterns of thinking that are natural for peo-ple but frequently lead to mistakes in (...) judgment. This article discusses two important but neglected error ten-dencies: motivated inference and fear-driven inference. (shrink)

Statistical Default Logic (SDL) is an expansion of classical (i.e., Reiter) default logic that allows us to model common inference patterns found in standard inferential statistics, e.g., hypothesis testing and the estimation of a population‘s mean, variance and proportions. This paper presents an embedding of an important subset of SDL theories, called literal statistical default theories, into stable model semantics. The embedding is designed to compute the signature set of literals that uniquely distinguishes each extension on a statistical (...) default theory at a pre-assigned error-bound probability. (shrink)

Quantum theory is a probabilistic theory that embodies notoriously striking correlations, stronger than any that classical theories allow but not as strong as those of hypothetical ‘super-quantum’ theories. This raises the question ‘Why the quantum?’—whether there is a handful of principles that account for the character of quantum probability. We ask what quantum-logical notions correspond to this investigation. This project isn’t meant to compete with the many beautiful results that information-theoretic approaches have yielded but rather aims to complement that work.

Qualitative Reasoning (QR) is an area of research within Artificial Intelligence that automates reasoning and problem solving about the physical world. QR research aims to deal with representation and reasoning about continuous aspects of entities without the kind of precise quantitative information needed by conventional numerical analysis techniques. Order-of-magnitude Reasoning (OMR) is an approach in QR concerned with the analysis of physical systems in terms of relative magnitudes. In this paper we consider the logic OMR_N for order-of-magnitude reasoning with (...) the bidirectional negligibility relation. It is a multi-modal logic given by a Hilbert-style axiomatization that reflects properties and interactions of two basic accessibility relations (strict linear order and bidirectional negligibility). Although the logic was studied in many papers, nothing was known about its decidability. In the paper we prove decidability of OMR N by showing that the logic has the strong finite model property. (shrink)

This paper surveys important work done in relevant logic in the past 10 years.

This book deals with questions everyone should become acquainted with when studying logic. It, however, has nothing in common with current introductions to logic, which are actually introductions to a particular logic paradigm, mathematical logic. There is nothing wrong with this, except that at present such paradigm is a problematic one. For mathematical logic, on the one hand, is inadequate for the use for which it was originally designed – to give mathematics the most secure (...) foundation – and, on the other hand, has found no crucial alternative use. This fact is almost invariably passed over in silence in current introductions to logic. This is as it could be expected, for people working within a given paradigm tend to consider it as the only possible one and cannot conceive of any alternative. But to read only such introductions will give a rather narrow view of the subject. In this book mathematical logic is presented as being not ‘The Logic’ but rather a particular logic paradigm, with some basic limitations. An alternative logic paradigm is outlined, meant to remove such limitations, in which logic is supposed to be a logic of discovery and justification a part of discovery. With respect to mathematical logic, the alternative paradigm involves a different view of the relation of logic with nature. Logic is a continuation of the problem solving procedures with which biological evolution has endowed humans and all organisms generally. The alternative paradigm also involves a different view of the relation of logic with method. Method is the source of logic. To implement the alternative paradigm, a number of basic discovery procedures are discussed. By their very nature, discovery procedures do not form a closed set, given once for all, but rather an open set, which can always be expanded. Those considered in this book, however, are especially important. This book is not intended to replace any introduction to mathematical logic but rather to be read parallel to it. Its aim is, on the one hand, to put mathematical logic into perspective, on the other hand, to show that an alternative paradigm is possible and to outline it. I hope it will give the reader a better feel of what logic really is. (shrink)

This paper contributes to an increasing literature strengthening the connection between epistemic logic and epistemology (Van Benthem, Hendricks). I give a survey of the most important applications of epistemic logic in epistemology. I show how it is used in the history of philosophy (Steiner's reconstruction of Descartes' sceptical argument), in solutions to Moore's paradox (Hintikka), in discussions about the relation between knowledge and belief (Lenzen) and in an alleged refutation of verificationism (Fitch) and I examine an early argument (...) about the (im)possibility of epistemic logic (Hocutt). Subsequently, I deal with interpretive questions about epistemic logic that, although implicitly, already appeared in the first section. I contend that a conception of epistemic logic as a theory of knowledge assertions is incoherent, and I argue that it does not make sense to adopt a normative interpretation of epistemic logic. Finally, I show ways to extend epistemic logic with other branches of philosophical logic so as to make it useful for some epistemological questions. Conditional logics and logics of public announcement are used to understand causal theories of knowledge and versions of reliabilism. Temporal logic helps understand some dynamic aspects of knowledge as well as the verificationist thesis. (shrink)

We present the simplest solution ever to 'the hardest logic puzzle ever'. We then modify the puzzle to make it even harder and give a simple solution to the modified puzzle. The final sections investigate exploding god-heads and a two-question solution to the original puzzle.

Rabern and Rabern (2008) have noted the need to modify `the hardest logic puzzle ever’ as presented in Boolos 1996 in order to avoid trivialization. Their paper ends with a two-question solution to the original puzzle, which does not carry over to the amended puzzle. The purpose of this note is to offer a two-question solution to the latter puzzle, which is, after all, the one with a claim to being the hardest logic puzzle ever.

The purpose of this paper is to examine the status of logic from a metaphysical point of view – what is logic grounded in and what is its relationship with metaphysics. There are three general lines that we can take. 1) Logic and metaphysics are not continuous, neither discipline has no bearing on the other one. This seems to be a rather popular approach, at least implicitly, as philosophers often skip the question altogether and go about their (...) business, be it logic or metaphysics. However, it is not a particularly plausible view and it is very hard to maintain consistently, as we will see. 2) Logic is prior to metaphysics and has metaphysical implications. The extreme example of this kind of approach is the Dummettian one, according to which metaphysical questions are reducible to the question of which logic to adopt. 3) Metaphysics is prior to logic, and your logic should be compatible with your metaphysics. This approach suggests an answer to the question of what logic is grounded in, namely, metaphysics. Here I will defend the third option. (shrink)

Kaplan (1989a) insists that natural languages do not contain displacing devices that operate on character—such displacing devices are called monsters. This thesis has recently faced various empirical challenges (e.g., Schlenker 2003; Anand and Nevins 2004). In this note, the thesis is challenged on grounds of a more theoretical nature. It is argued that the standard compositional semantics of variable binding employs monstrous operations. As a dramatic first example, Kaplan’s formal language, the Logic of Demonstratives, is shown to contain monsters. (...) For similar reasons, the orthodox lambda-calculus-based semantics for variable binding is argued to be monstrous. This technical point promises to provide some far-reaching implications for our understanding of semantic theory and content. The theoretical upshot of the discussion is at least threefold: (i) the Kaplanian thesis that “directly referential” terms are not shiftable/bindable is unmotivated, (ii) since monsters operate on something distinct from the assertoric content of their operands, we must distinguish ingredient sense from assertoric content (cf. Dummett 1973; Evans 1979; Stanley 1997), and (iii) since the case of variable binding provides a paradigm of semantic shift that differs from the other types, it is plausible to think that indexicals—which are standardly treated by means of the assignment function—might undergo the same kind of shift. (shrink)

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

Traditionally transcendental logic has been set apart from formal logic. Transcendental logic had to deal with the conditions of possibility of judgements, which were presupposed by formal logic. Defined as a purely philosophical enterprise transcendental logic was considered as being a priori delivering either analytic or even synthetic a priori results. In this paper it is argued that this separation from the (empirical) cognitive sciences should be given up. Transcendental logic should be understood as (...) focusing on specific questions. These do not, as some recent analytic philosophy has it, include a refutation of scepticism. And they are not to be separated from meta-logical investigations. Transcendental logic properly understood, and redefined along these theses, should concern itself with the (formal) re-construction of the presupposed necessary conditions and rules of linguistic communication in general. It aims at universality and reflexive closure. (shrink)

Pace Necessitism – roughly, the view that existence is not contingent – essential properties provide necessary conditions for the existence of objects. Sufficiency properties, by contrast, provide sufficient conditions, and individual essences provide necessary and sufficient conditions. This paper explains how these kinds of properties can be used to illuminate the ontological status of merely possible objects and to construct a respectable possibilist ontology. The paper also reviews two points of interaction between essentialism and modal logic. First, we will (...) briefly see the challenge that arises against S4 from flexible essential properties; as well as the moves available to block it. After this, the emphasis is put on the Barcan Formula (BF), and on why it is problematic for essentialists. As we will see, Necessitism can accommodate both (BF) and essential properties. What necessitists cannot do at the same time is to continue to understanding essential properties as providing necessary conditions for the existence of individuals; against what might be for some a truism. (shrink)

Truth, etc. is a wide-ranging study of ancient logic based upon the John Locke lectures given by the eminent philosopher Jonathan Barnes in Oxford. The book presupposes no knowledge of logic and no skill in ancient languages: all ancient texts are cited in English translation; and logical symbols and logical jargon are avoided so far as possible. Anyone interested in ancient philosophy, or in logic and its history, will find much to learn and enjoy here.

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive (...) notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)

This text is a short introduction to logic that was primarily used for accompanying an introductory course in Logic for Linguists held at the New University of Lisbon (UNL) in fall 2010. The main idea of this course was to give students the formal background and skills in order to later assess literature in logic, semantics, and related fields and perhaps even use logic on their own for the purpose of doing truth-conditional semantics. This course in (...) logic does not replace a proper introduction to semantics and is not intended as such, although parts of Chapter 1 and 4 could be used to supplement an introductory course in semantics. In contrast to other introductions it has a certain focus on ‘writing things down correctly.’ Proofs of metatheorems are omitted, though. -/- This is work in progress. Please send suggestions and corrigenda to erich@snafu.de. Have fun! (shrink)