Firstly I characterize Simple Partial Logic (SPL) as the generalization and extension of a certain two-valued logic. Based on the characterization I present two definitions of validity in SPL. Finally I show that given my characterization these two definitions are more appropriate than other definitions that have been prevalent, since both have some desirable semantic properties that the others lack.

Simple partial logic (=SPL) is, broadly speaking, an extensional logic which allows for the truth-value gap. First I give a system of propositional SPL by partializing classical logic, as well as extending it with several non-classical truth-functional operators. Second I show a way based on SPL to construct a system of tensed ontology, by representing tensed statements as two kinds of necessary statements in a linear model that consists of the present and future worlds. Finally I (...) compare that way with other two ways based on Łukasiewicz’s three-valued logic and branching temporal logic. (shrink)

Implicational tonoid logics and their extensions with abstract Galois properties have been introduced by Yang and Dunn. They introduced matrix semantics for the implicational tonoid logics but did not do for the extensions. Here we provide such semantics for implicational partial gaggle logics as one sort of such extensions. To this end, first we discuss implicational partial gaggle logics in Hilbert-style. We next introduce one kind of matrix semantics based on Lindenbaum– Tarski matrices for the logics and show (...) that those logics are complete with respect to the matrix semantics. Finally, we further introduce a slightly different kind of matrix semantics based on reduced models for the logics and show that those logics are complete with respect to this matrix semantics. (shrink)

We have to gain from recognizing a relation between epistemic agents and the parts of subject matters that play a role in their cognitive lives. I call this relation “grasping”. Namely, I zone in on one notion of having a partial grasp of a subject matter—that of agents grasping part of the subject matter that they are attending to—and characterize it. I propose that giving up the idealization that we fully grasp the subject matters we attend to allows one (...) to more realistically characterize the epistemic life of agents. To show this, I propose an epistemic logic with partial grasp that has in mind considerations from first-order aboutness theory with the aim of avoiding certain forms of logical omniscience, and which provides an alternative to immanent closure (Yablo Aboutness, Princeton University Press, 2014). (shrink)

In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of Non-Contradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent (...) logics once the right set-up is chosen. Our logic L4 has a semantics that also underlies Belnap’s [4] and is related to the logic of bilattices. L4 is in focus most of the time, but it is also shown how results obtained for L4 can be transferred to several variants. (shrink)

In this paper, we introduce the basics of what we shall call "partial model theory", which is an extension of traditional model theory to partial structures. These are a specific kind of structure developed within the partial structures approach, which is a view constituting the semantic approach of theories. And together with other related semantical concepts, like the concept of quasi-truth, partial structures have been used in contemporary philosophy of science for several purposes. Nonetheless, those uses (...) presuppose certain technical results, or could be improved by resorting to certain technical notions and results, that have not been put forward so far. Thus, in the present work, we intend to introduce part of this particular technical apparatus that is still lacking especially in logic, throughout the development of partial model theory. We begin by extending traditional notions of model theory for partial structures, like the notions of substructure and homomorphism, and by proving some results concerning the notion of quasi-truth. Posteriorly, we show how the content introduced can be used to improve a particular application of partial structures and quasi-truth in the philosophy of science. (shrink)

Recently, Bourne constructed a system of three-valued logic that he supposed to replace Łukasiewicz’s three-valued logic in view of the problems of future contingents. In this paper, I will show first that Bourne’s system makes no improvement to Łukasiewicz’s system. However, finding some good motivations and lessons in his attempt, next I will suggest a better way of achieving his original goal in some sense. The crucial part of my way lies in reconsidering the significance of the intermediate (...) truth-value so as to reconstruct Łukasiewicz’s three-valued logic as a kind of extensional modal logic based on partial logic. (shrink)

The article recapitulates what logic is about traditionally and works out two roles it has been playing in philosophy: the role of an instrument and of a philosophical discipline in its own right. Using Tarski’s philosophical-logical work as case study, it develops a logical reconstructionist methodology of philosophical logic that extends and refines Rudolf Carnap’s account of explication and rational reconstruction. The methodology overlaps with, but also partially diverges from, contemporary anti-exceptionalism about logic.

The logical interpretation of probability, or "objective Bayesianism'' – the theory that (some) probabilities are strictly logical degrees of partial implication – is defended. The main argument against it is that it requires the assignment of prior probabilities, and that any attempt to determine them by symmetry via a "principle of insufficient reason" inevitably leads to paradox. Three replies are advanced: that priors are imprecise or of little weight, so that disagreement about them does not matter, within limits; that (...) it is possible to distinguish reasonable from unreasonable priors on logical grounds; and that in real cases disagreement about priors can usually be explained by differences in the background information. It is argued also that proponents of alternative conceptions of probability, such as frequentists, Bayesians and Popperians, are unable to avoid committing themselves to the basic principles of logical probability. (shrink)

This is part one of a two-part paper, in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. This allows us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths of arithmetic (...) and show that the theory is a proof-theoretically conservative extension of the theory PT of positive truth. We construct models for the theory and draw some conclusions for the semantics of conceptualist ground. (shrink)

One response to the problem of logical omniscience in standard possible worlds models of belief is to extend the space of worlds so as to include impossible worlds. It is natural to think that essentially the same strategy can be applied to probabilistic models of partial belief, for which parallel problems also arise. In this paper, I note a difficulty with the inclusion of impossible worlds into probabilistic models. Under weak assumptions about the space of worlds, most of the (...) propositions which can be constructed from possible and impossible worlds are in an important sense inexpressible; leaving the probabilistic model committed to saying that agents in general have at least as many attitudes towards inexpressible propositions as they do towards expressible propositions. If it is reasonable to think that our attitudes are generally expressible, then a model with such commitments looks problematic. (shrink)

A general framework for translating various logical systems is presented, including a set of partial unary operators of affirmation and negation. Despite its usual reading, affirmation is not redundant in any domain of values and whenever it does not behave like a full mapping. After depicting the process of partial functions, a number of logics are translated through a variety of affirmations and a unique pair of negations. This relies upon two preconditions: a deconstruction of truth-values as ordered (...) and structured objects, unlike its mainstream presentation as a simple object; a redefinition of the Principle of Bivalence as a set of four independent properties, such that its definition does not equate with normality. (shrink)

The Logic of Causation: Definition, Induction and Deduction of Deterministic Causality is a treatise of formal logic and of aetiology. It is an original and wide-ranging investigation of the definition of causation (deterministic causality) in all its forms, and of the deduction and induction of such forms. The work was carried out in three phases over a dozen years (1998-2010), each phase introducing more sophisticated methods than the previous to solve outstanding problems. This study was intended as part (...) of a larger work on causal logic, which additionally treats volition and allied cause-effect relations (2004). The Logic of Causation deals with the main technicalities relating to reasoning about causation. Once all the deductive characteristics of causation in all its forms have been treated, and we have gained an understanding as to how it is induced, we are able to discuss more intelligently its epistemological and ontological status. In this context, past theories of causation are reviewed and evaluated (although some of the issues involved here can only be fully dealt with in a larger perspective, taking volition and other aspects of causality into consideration, as done in Volition and Allied Causal Concepts). Phase I: Macroanalysis. Starting with the paradigm of causation, its most obvious and strongest form, we can by abstraction of its defining components distinguish four genera of causation, or generic determinations, namely: complete, partial, necessary and contingent causation. When these genera and their negations are combined together in every which way, and tested for consistency, it is found that only four species of causation, or specific determinations, remain conceivable. The concept of causation thus gives rise to a number of positive and negative propositional forms, which can be studied in detail with relative ease because they are compounds of conjunctive and conditional propositions whose properties are already well known to logicians. The logical relations (oppositions) between the various determinations (and their negations) are investigated, as well as their respective implications (eductions). Thereafter, their interactions (in syllogistic reasoning) are treated in the most rigorous manner. The main question we try to answer here is: is (or when is) the cause of a cause of something itself a cause of that thing, and if so to what degree? The figures and moods of positive causative syllogism are listed exhaustively; and the resulting arguments validated or invalidated, as the case may be. In this context, a general and sure method of evaluation called ‘matricial analysis’ (macroanalysis) is introduced. Because this (initial) method is cumbersome, it is used as little as possible – the remaining cases being evaluated by means of reduction. Phase II: Microanalysis. Seeing various difficulties encountered in the first phase, and the fact that some issues were left unresolved in it, a more precise method is developed in the second phase, capable of systematically answering most outstanding questions. This improved matricial analysis (microanalysis) is based on tabular prediction of all logically conceivable combinations and permutations of conjunctions between two or more items and their negations (grand matrices). Each such possible combination is called a ‘modus’ and is assigned a permanent number within the framework concerned (for 2, 3, or more items). This allows us to identify each distinct (causative or other, positive or negative) propositional form with a number of alternative moduses. This technique greatly facilitates all work with causative and related forms, allowing us to systematically consider their eductions, oppositions, and syllogistic combinations. In fact, it constitutes a most radical approach not only to causative propositions and their derivatives, but perhaps more importantly to their constituent conditional propositions. Moreover, it is not limited to logical conditioning and causation, but is equally applicable to other modes of modality, including extensional, natural, temporal and spatial conditioning and causation. From the results obtained, we are able to settle with formal certainty most of the historically controversial issues relating to causation. Phase III: Software Assisted Analysis. The approach in the second phase was very ‘manual’ and time consuming; the third phase is intended to ‘mechanize’ much of the work involved by means of spreadsheets (to begin with). This increases reliability of calculations (though no errors were found, in fact) – but also allows for a wider scope. Indeed, we are now able to produce a larger, 4-item grand matrix, and on its basis find the moduses of causative and other forms needed to investigate 4-item syllogism. As well, now each modus can be interpreted with greater precision and causation can be more precisely defined and treated. In this latest phase, the research is brought to a successful finish! Its main ambition, to obtain a complete and reliable listing of all 3-item and 4-item causative syllogisms, being truly fulfilled. This was made technically feasible, in spite of limitations in computer software and hardware, by cutting up problems into smaller pieces. For every mood of the syllogism, it was thus possible to scan for conclusions ‘mechanically’ (using spreadsheets), testing all forms of causative and preventive conclusions. Until now, this job could only be done ‘manually’, and therefore not exhaustively and with certainty. It took over 72’000 pages of spreadsheets to generate the sought for conclusions. This is a historic breakthrough for causal logic and logic in general. Of course, not all conceivable issues are resolved. There is still some work that needs doing, notably with regard to 5-item causative syllogism. But what has been achieved solves the core problem. The method for the resolution of all outstanding issues has definitely now been found and proven. The only obstacle to solving most of them is the amount of labor needed to produce the remaining (less important) tables. As for 5-item syllogism, bigger computer resources are also needed. (shrink)

In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl’s ideas of pure grammar, Leśniewski-Ajukiewicz’s theory syntactic/semantic categories and in accordance with Frege’s ontological canons, Bocheński’s famous motto—syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its (...) cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science. (shrink)

In this paper, we make distinctions between Classical Logic (where the propositions are 100% true, or 100 false) and the Neutrosophic Logic (where one deals with partially true, partially indeterminate and partially false propositions) in order to respond to K. Georgiev’s criticism [1]. We recall that if an axiom is true in a classical logic system, it is not necessarily that the axiom be valid in a modern (fuzzy, intuitionistic fuzzy, neutrosophic etc.) logic system.

A analysis of some concepts of logic is proposed, around the work of Edelcio de Souza. Two of his related issues will be emphasized, namely: opposition, and quasi-truth. After a review of opposition between logical systems [2], its extension to many-valuedness is considered following a special semantics including partial operators [13]. Following this semantic framework, the concepts of antilogic and counterlogic are translated into opposition-forming operators [15] and specified as special cases of contradictoriness and contrariety. Then quasi-truth [5] (...) is introduced and equally translated as a product of two partial operators. Finally, the reflections proposed around opposition and quasi-truth lead to a third new logical concept: quasi-opposition, borrowing the central feature of partiality and opening the way to a potential field of new investigations into philosophical logic. (shrink)

Standpoint logic is a recently proposed formalism in the context of knowledge integration, which advocates a multi-perspective approach permitting reasoning with a selection of diverse and possibly conflicting standpoints rather than forcing their unification. In this paper, we introduce nested sequent calculi for propositional standpoint logics---proof systems that manipulate trees whose nodes are multisets of formulae---and show how to automate standpoint reasoning by means of non-deterministic proof-search algorithms. To obtain worst-case complexity-optimal proof-search, we introduce a novel technique in the (...) context of nested sequents, referred to as "coloring," which consists of taking a formula as input, guessing a certain coloring of its subformulae, and then running proof-search in a nested sequent calculus on the colored input. Our technique lets us decide the validity of standpoint formulae in CoNP since proof-search only produces a partial proof relative to each permitted coloring of the input. We show how all partial proofs can be fused together to construct a complete proof when the input is valid, and how certain partial proofs can be transformed into a counter-model when the input is invalid. These "certificates" (i.e. proofs and counter-models) serve as explanations of the (in)validity of the input. (shrink)

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our (...) theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to. (shrink)

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework (...) directly. We solve this problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. (shrink)

A Fortiori Logic: Innovations, History and Assessments is a wide-ranging and in-depth study of a fortiori reasoning, comprising a great many new theoretical insights into such argument, a history of its use and discussion from antiquity to the present day, and critical analyses of the main attempts at its elucidation. Its purpose is nothing less than to lay the foundations for a new branch of logic and greatly develop it; and thus to once and for all dispel the (...) many fallacious ideas circulating regarding the nature of a fortiori reasoning. -/- The work is divided into three parts. The first part, Formalities, presents the author’s largely original theory of a fortiori argument, in all its forms and varieties. Its four (or eight) principal moods are analyzed in great detail and formally validated, and secondary moods are derived from them. A crescendo argument is distinguished from purely a fortiori argument, and similarly analyzed and validated. These argument forms are clearly distinguished from the pro rata and analogical forms of argument. Moreover, we examine the wide range of a fortiori argument; the possibilities of quantifying it; the formal interrelationships of its various moods; and their relationships to syllogistic and analogical reasoning. Although a fortiori argument is shown to be deductive, inductive forms of it are acknowledged and explained. Although a fortiori argument is essentially ontical in character, more specifically logical-epistemic and ethical-legal variants of it are acknowledged. -/- The second part of the work, Ancient and Medieval History, looks into use and discussion of a fortiori argument in Greece and Rome, in the Talmud, among post-Talmudic rabbis, and in Christian, Moslem, Chinese and Indian sources. Aristotle’s approach to a fortiori argument is described and evaluated. There is a thorough analysis of the Mishnaic qal vachomer argument, and a reassessment of the dayo principle relating to it, as well as of the Gemara’s later take on these topics. The valuable contribution, much later, by Moshe Chaim Luzzatto is duly acknowledged. Lists are drawn up of the use of a fortiori argument in the Jewish Bible, the Mishna, the works of Plato and Aristotle, the Christian Bible and the Koran; and the specific moods used are identified. Moreover, there is a pilot study of the use of a fortiori argument in the Gemara, with reference to Rodkinson’s partial edition of the Babylonian Talmud, setting detailed methodological guidelines for a fuller study. There is also a novel, detailed study of logic in general in the Torah. -/- The third part of the present work, Modern and Contemporary Authors, describes and evaluates the work of numerous (some thirty) recent contributors to a fortiori logic, as well as the articles on the subject in certain lexicons. Here, we discover that whereas a few authors in the last century or so made some significant contributions to the field, most of them shot woefully off-target in various ways. The work of each author, whether famous or unknown, is examined in detail in a dedicated chapter, or at least in a section; and his ideas on the subject are carefully weighed. The variety of theories that have been proposed is impressive, and stands witness to the complexity and elusiveness of the subject, and to the crying need for the present critical and integrative study. But whatever the intrinsic value of each work, it must be realized that even errors and lacunae are interesting because they teach us how not to proceed. -/- This book also contains, in a final appendix, some valuable contributions to general logic, including new analyses of symbolization and axiomatization, existential import, the tetralemma, the Liar paradox and the Russell paradox. (shrink)

According to the preference-centric approach to understanding partial belief, the connection between partial beliefs and preferences is key to understanding what partial beliefs are and how they’re measured. As Ramsey put it, the ‘degree of a belief is a causal property of it, which we can express vaguely as the extent to which we are prepared to act on it’ The Foundations of Mathematics and Other Logical Essays, Routledge, Oxon, pp 156–198, 1931). But this idea is not (...) as popular as it once was. Nowadays, the preference-centric approach is frequently dismissed out-of-hand as behaviouristic, unpalatably anti-realist, and/or prone to devastating counterexamples. Cases like Eriksson and Hájek’s :183–213, 2007) preferenceless Zen monk and Christensen’s :356–376, 2001) other roles argument have suggested to many that any account of partial belief that ties them too closely to preferences is irretrievably flawed. In this paper I provide a defence of preference-centric accounts of partial belief. (shrink)

This paper uses a partially ordered set of syntactic categories to accommodate optionality and licensing in natural language syntax. A complex but well-studied data set pertaining to the syntax of quantifier scope and negative polarity licensing in Hungarian is used to illustrate the proposal. The presentation is geared towards both linguists and logicians. The paper highlights that the main ideas can be implemented in different grammar formalisms, and discusses in detail an implementation where the partial ordering on categories is (...) given by the derivability relation of a calculus with residuated and Galois-connected unary operators. (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)

The famous Cartesian Nicolas Malebranche (1638-1715) espoused the occasionalist doctrine that ‘there is only one true cause because there is only one true God; that the nature or power of each thing is nothing but the will of God; that all natural causes are not true causes but only occasional causes’ (LO, 448, original italics). One of Malebranche’s well-known arguments for occasionalism, known as, the ‘no necessary connection’ argument (or, NNC ) stems from the principle that ‘a true cause… is (...) one such that the mind perceives a necessary connection between it and its effect’ (LO, 450). The outline of this paper is as follows. I explicitly layout NNC and articulate some of its prima facie strengths (§1). I then critically discuss, what I take to be, the two main arguments against NNC of the Lee-Pyle interpretation (§2). The main conclusion from (§2) is that Malebranche did not abandon NNC in his later works given textual evidence from the Dialogues, contrary to the Lee-Pyle interpretation. In (§3) I discuss in what ways Suárez, Leibniz, Régis and Spinoza all accepted the main premise of NNC. Then, I rebut Steven Nadler’s influential and unchallenged criticism that Malebranche conflated causal and logical necessity, and provide a more accurate interpretation of Malebranche that only commits him to a partial reduction of causal to logical necessity (§4). (shrink)

Łukasiewicz has often been criticized for his motive for inventing his three-valued logic, namely the avoidance of determinism. First of all, I want to show that almost all of the critcism along this line was wrong. Second I will indicate that he made mistakes, however, in constructing his system, because he had other motives at the same time. Finally I will propose some modification of his system and its interpretation which can attain his original purpose in some sense.

I outline the theoretical framework of, and three research programs within American speculative philosophy of science during the period 1900-1931. One program applies verificationism to research in psychology, one investigates the methodology of research programs, and one analyses scientific explanation and other scientific concepts. The primary sources for my outline are works by Morris Raphael Cohen, Grace Andrus de Laguna, Theodore de Laguna, Edgar Arthur Singer Jr., Harold Robert Smart, and Marie Collins Swabey. I also use my outline to provide (...) a partial comparison of American speculative philosophy of science and 1930s logical positivism. My comparison suggests that logical positivism was a proposal for substantially narrowing down and winding back American philosophy of science and was based on positions that were already problematized in the American context. -/- . (shrink)

This article derives from a project attempting to show that Western formal logic, from Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. In the present article, I will first discuss, in light of Frege’s honorary role as founder of the philosophy of mathematics, Reuben Hersh’s What is Mathematics, Really? Second, I will explore how the infamous section of Frege’s 1924 diary (specifically the entries from March 10 to April 9) (...) supports Hersh's claim regarding the link between political conservatism and the (historically and currently) dominant school of the philosophy of mathematics (to which Frege undeniably belongs). Third, I will examine Frege’s attempt at a more reader-friendly introduction to his philosophy of mathematics, The Foundations of Arithmetic. And finally, I will briefly analyze Frege’s Begriffsschrift to see how questions of race arise even at the heights of his logical abstraction. (shrink)

This essay is part of larger project in which I attempt to show that Western formal logic, from its inception in Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. More specifically, (a) racist/quasi-racist/proto-racist political forces were part of the impetus for logic’s attempt to classify the world into mutually exclusive, hierarchically-valued categories in the first place; and (b) these classifications, in turn, have been deployed throughout history to justify (...) and empower racism/quasi-racism/proto-racism. In other words, (a) an important part of the chaos and messiness that so troubles logic has been the natural bio-cultural diversity of human beings as described by concepts such as race and ethnicity; and (b) once these concepts were historically in place, taken for granted, buttressed with pseudo-science, and had their origins forgotten, they became tools for the oppression of diverse human groups. The central thesis of this essay is that close readings of a variety of Leibniz’s writings (including several essays and one letter) will reveal a pattern of mutual dependency and creation in regard to logic and racism in Leibniz. My conclusion will be that there is a meaningful connection between logic and racism (or at least between logic and politics in general) in Leibniz such that one should be skeptical of his logic’s claim to offer transparent and neutral descriptions of the world as well as imperatives for action. Put differently, I want to ask the following specific question: how do Leibniz’s own racially-informed political goals, along with the emergence of race-centric European colonialism, both invest and disguise themselves within Leibniz’s mathematizing of logic? (shrink)

In the paper, various notions of the logical semiotic sense of linguistic expressions – namely, syntactic and semantic, intensional and extensional – are considered and formalised on the basis of a formal-logical conception of any language L characterised categorially in the spirit of certain Husserl's ideas of pure grammar, Leśniewski-Ajdukiewicz's theory of syntactic/semantic categories and, in accordance with Frege's ontological canons, Bocheński's and some of Suszko's ideas of language adequacy of expressions of L. The adequacy ensures their unambiguous syntactic and (...) semantic senses and mutual, syntactic and semantic correspondence guaranteed by the acceptance of a postulate of categorial compatibility of syntactic and semantic (extensional and intensional) categories of expressions of L. This postulate defines the unification of these three logical senses. There are three principles of compositionality which follow from this postulate: one syntactic and two semantic ones already known to Frege. They are treated as conditions of homomorphism of partial algebra of L into algebraic models of L: syntactic, intensional and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, obviously, an idealisation. The syntactic and semantic unambiguity of its expressions is not, of course, a feature of natural languages, but every syntactically and semantically ambiguous expression of such languages may be treated as a schema representing all of its interpretations that are unambiguous expressions. (shrink)

Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved - independently of Löwenheim and Skolem's earlier work - the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important (...) programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. (shrink)

A graph-theoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as a multi-graph (m-graph) where the nodes and the m-edges include the sorts and the constructors of the signatures at hand. Fibring of two models is a multi-graph (m-graph) where the nodes and the m-edges are the values and the operations in the models, respectively. Fibring of two deductive systems is an (...) m-graph whose nodes are language expressions and the m-edges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graph-theoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with non-deterministic semantics, logics with an algebraic semantics, logics with partial semantics and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided. (shrink)

In this paper we focus our attention on tableau methods for propositional interval temporal logics. These logics provide a natural framework for representing and reasoning about temporal properties in several areas of computer science. However, while various tableau methods have been developed for linear and branching time point-based temporal logics, not much work has been done on tableau methods for interval-based ones. We develop a general tableau method for Venema's \cdt\ logic interpreted over partial orders (\nsbcdt\ for short). (...) It combines features of the classical tableau method for first-order logic with those of explicit tableau methods for modal logics with constraint label management, and it can be easily tailored to most propositional interval temporal logics proposed in the literature. We prove its soundness and completeness, and we show how it has been implemented. (shrink)

This article informally presents a solution to the paradoxes of truth and shows how the solution solves classical paradoxes (such as the original Liar) as well as the paradoxes that were invented as counterarguments for various proposed solutions (“the revenge of the Liar”). This solution complements the classical procedure of determining the truth values of sentences by its own failure and, when the procedure fails, through an appropriate semantic shift allows us to express the failure in a classical two-valued language. (...) Formally speaking, the solution is a language with one meaning of symbols and two valuations of the truth values of sentences. The primary valuation is a classical valuation that is partial in the presence of the truth predicate. It enables us to determine the classical truth value of a sentence or leads to the failure of that determination. The language with the primary valuation is precisely the largest intrinsic fixed point of the strong Kleene three-valued semantics (LIFPSK3). The semantic shift that allows us to express the failure of the primary valuation is precisely the classical closure of LIFPSK3: it extends LIFPSK3 to a classical language in parts where LIFPSK3 is undetermined. Thus, this article provides an argumentation, which has not been present in contemporary debates so far, for the choice of LIFPSK3 and its classical closure as the right model for the truth predicate. In the end, an erroneous critique of Kripke-Feferman axiomatic theory of truth, which is present in contemporary literature, is pointed out. (shrink)

This article tries to present a possible way of proliferating bivalent, trivalent, etc. calculation. Beyond the controversies as to the effects that involve the presence versus the absence of the temporal factor, no significant attempts of adapting a bivalent logic to other pairs of values have been mentioned. Should the transformation of the alethiologic modalities into logic values be possible, five more bivalent calculations may be added to the classic bivalent calculation. All these new calculations are compatible with (...) the traditional one, to the extent that they submit to the conditions of non-contradictoriness and exhaustibility, and the connectors receive, even if partially, the corresponding value interpretations. (shrink)

A theorem due to Shoesmith and Smiley that axiomatizes two-valued multiple-conclusion logics is extended to partial logics.

Although it seems intuitively clear that acts of requesting are different from acts of commanding, it is not very easy to sate their differences precisely in dynamic terms. In this paper we show that it becomes possible to characterize, at least partially, the effects of acts of requesting and compare them with the effects of acts of commanding by combining dynamified deontic logic with epistemic logic. One interesting result is the following: each act of requesting is appropriately differentiated (...) from an act of commanding with the same content, but for each act of requesting, there is another act of commanding with much more complex content which updates models in exactly the same way as it does. We will also consider an application of our characterization of acts of requesting to acts of asking yes-no questions. It yields a straightforward formalization of the view of acts of asking questions as requests for information. (shrink)

This article is part of a larger project in which I attempt to show that Western formal logic, from its inception in Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. In contrast to this trend, the present article concerns the major philosopher whose contribution to logic has been perhaps the most derided and marginalized, and yet whose character and politics are, from a contemporary perspective, drastically superior—John Stuart Mill. (...) My approach to my core concern will be one of narrowing concentric circles. I will begin with Mill’s occasional political writings that bear on the issue of racism, including “The Negro Question.” From there, the core of the article will explore the political dimensions of Mill’s A System of Logic. (shrink)

One of the foremost objections to theological voluntarism is the contingency objection. If God’s will fixes moral facts, then what if God willed that agents engage in cruelty? I argue that even unrestricted theological voluntarists should accept some logical constraints on possible moral systems—hence, some limits on ways that God could have willed morality to be—and these logical constraints are sufficient to blunt the force of the contingency objec­tion. One constraint I defend is a very weak accessibility requirement, related to (...) (but less problematic than) existence internalism in metaethics. The theo­logical voluntarist can maintain: Godcouldn’t have loved cruelty, and even though he could have willed behaviors we find abhorrent, he could only have done so in a world of deeply alien moral agents. We cannot confidently declare such a world unacceptable. (shrink)

Accounts of Hobbes’s ‘system’ of sciences oscillate between two extremes. On one extreme, the system is portrayed as wholly axiomtic-deductive, with statecraft being deduced in an unbroken chain from the principles of logic and first philosophy. On the other, it is portrayed as rife with conceptual cracks and fissures, with Hobbes’s statements about its deductive structure amounting to mere window-dressing. This paper argues that a middle way is found by conceiving of Hobbes’s _Elements of Philosophy_ on the model of (...) a mixed-mathematical science, not the model provided by Euclid’s _Elements of Geometry_. I suggest that Hobbes is a test case for understanding early-modern system-construction more generally, as inspired by the structure of the applied mathematical sciences. This approach has the additional virtue of bolstering, in a novel way, the thesis that the transformation of philosophy in the long seventeenth century was heavily indebted to mathematics, a thesis that has increasingly come under attack in recent years. (shrink)

In Winter 2017, the first author piloted a course in formal logic in which we aimed to (a) improve student engagement and mastery of the content, and (b) reduce maths anxiety and its negative effects on student outcomes, by adopting student oriented teaching including peer instruction and classroom flipping techniques. The course implemented a partially flipped approach, and incorporated group-work and peer learning elements, while retaining some of the traditional lecture format. By doing this, a wide variety of student (...) learning preferences could be provided for. (shrink)

In the final book of Logical Investigations from 1901, Husserl develops a theory of knowledge based on the intentional structure of consciousness. While there is some textual evidence that Husserl considered this to entail a critique of Kantian philosophy, he did not elaborate substantially on this. This paper reconstructs the covert critique of Kant’s theory of knowledge which LI contains. With respect to Kant, I discuss three core aspects of his theory of knowledge which, as Husserl’s reflections on Kant indicate, (...) Husserl was familiar with. These are the cooperation of two faculties for the justification of beliefs; the concept of a priori structures of knowledge Kant operated with; and the delivered transcendental proof of these structures. Regarding Logical Investigations, I first briefly outline the intentional structure of consciousness as presented in the fifth book and then turn to the theory of knowledge in the sixth book. I then clarify, partially on the basis of manuscripts and lecture notes, the covert critique of the three core aspects of Kant’s theory which the sixth book contains. (shrink)

I provide a theory of the metaphysical foundations of identity: an account what grounds facts of the form a=b. In particular, I defend the claim that indiscernibility grounds identity. This is typically rejected because it is viciously circular; plausible assumptions about the logic of ground entail that the fact that a=b partially grounds itself. The theory I defend is immune to this circularity.

Truthmaker semantics is a recent development in formal and philosophical semantics, with similar motivation and scope to possible worlds semantics. The technical background is rather different, however, and results in a more fine-grained hyperintensional notion of content, allowing us to distinguish between classically equivalent propositions. After briefly introducing the main ideas, this entry will describe the technical apparatus of state spaces and the central notions of content and partial content. It will then outline applications of truthmaker semantics in language, (...) logic, and other areas of philosophy. (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)

Many results in logic and mathematics rely on techniques that allow for concrete, often visual, representations of abstract concepts. A primary example of this phenomenon in logic is the distinction between syntax and semantics, itself an example of the more general duality in mathematics between algebra and geometry. Such representations, however, often rely on the existence of certain maximal objects having particular properties such as points, possible worlds or Tarskian first-order structures. -/- This dissertation explores an alternative to (...) such representations known as possibility semantics. Its core idea is to replace maximal objects with ordered systems of partial approximations. Although it originates in the semantics of modal logic and the representation of abstract ordered structures, I argue that it has far-reaching mathematical, foundational and philosophical significance, especially in the context of semiconstructive mathematics, a foundational framework that does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. -/- The dissertation is divided in two main parts. The first part explores various applications of the mathematical framework underlying possibility semantics to lattice theory and non-classical propositional logics. A major theme is the development of constructive dualities for various categories of lattices, which are related to standard non-constructive dualities via Vietoris constructions. -/- The second part of the dissertation explores the alternative foundational setting of semi-constructive mathematics, focusing on three applications of possibility semantics for classical first-order logic to the philosophy of the mathematical infinite. In particular, we introduce generic powers, a semi-constructive analogue of ultrapowers in classical model theory, and we explore the merits of these structures from a foundational, conceptual and historical viewpoint. (shrink)

This paper examines the evolution of Edmund Husserl’s theory of perceptual occlusion. This task is accomplished in two stages. First, I elucidate Husserl’s conclusion, from his 1901 Logical Investigations, that the occluded parts of perceptual objects are intended by partial signitive acts. I focus on two doctrines of that account. I examine Husserl’s insight that signitive intentions are composed of Gehalt and I discuss his conclusion that signitive intentions sit on the continuum of fullness. Second, the paper discloses how (...) Husserl transforms his 1901 philosophy in his 1913 revisions to the Sixth Logical Investigation, affirming that the occluded parts of perceptual objects are intended by empty contiguity acts. I demonstrate how he overturns the two core doctrines of his theory from the Investigations in these revisions, claiming that empty intentions are not composed of Gehalt and asserting that those acts break with the continuum of fullness. Husserl implements these changes to solve problems that arise from his recognition of two new kinds of intentions; darker and completely dark acts. Finally, in the conclusion, I cash out this analysis, by indicating that, in 1913, Husserl transforms his theory of fulfillment on the basis of his new insights about empty acts. (shrink)

In order to decide whether a discursive product of human reason corresponds or not to the logical order, one must analyze it in terms of syntactic correctness, consistency, and validity. The first step in logical analysis is formalization, that is, the process by which logical forms of thoughts are represented in different formal languages or logical systems. Although each thought can be properly formalized in different ways, the formalization variants are not equally adequate. The adequacy of formalization seems to depend (...) on several essential features: parsimony, accuracy, transparency, fertility and reliability. Because there is a partial antinomy between these traits, it is impossible to find a perfectly adequate variant of formalization. However, it is possible and preferable to reach a reasonable compromise by choosing the variant of formalization which satisfies all of these fundamental characteristics. (shrink)

Abstract. The aim of this paper is to show that the topological interpretation of knowledge as an interior kernel operator K of a topological space (X, OX) comes along with a partially ordered family of belief modalities B that fit K in the sense that the pairs (K, B) satisfy all axioms of Stalnaker’s KB logic of knowledge and belief with the exception of the contentious axiom of negative introspection (NI). The new belief modalities B introduced in this paper (...) are defined with the help of the (dense) nuclei of the Heyting algebra OX of open subsets on the topological space (X, OX). In this way, the natural context for the belief operators B related to topological knowledge operator K is shown to be the Heyting algebra NUC(OX) of the nuclei of the Heyting algebra OX.1 More precisely, the dense nuclei of NUC(OX) can be used to define a variety of bimodal logics of knowledge operators K and belief operators B. The operators K and B are compatible with each other in the sense that the pairs (K, B) satisfy all axioms of Stalnaker’s KB system with the exception of the axiom (NI). Therefore, for (X, OX), one obtains a bounded, partially ordered family of belief operators B defined by the elements of NUC(OX). (shrink)

Recent work in formal philosophy has concentrated over-whelmingly on the logical problems pertaining to epistemic shortfall - which is to say on the various ways in which partial and sometimes incorrect information may be stored and processed. A directly depicting language, in contrast, would reflect a condition of epistemic perfection. It would enable us to construct representations not of our knowledge but of the structures of reality itself, in much the way that chemical diagrams allow the representation (at a (...) certain level of abstractness) of the structures of molecules of different sorts. A diagram of such a language would be true if that which it sets out to depict exists in reality, i.e. if the structural relations between the names (and other bits and pieces in the diagram) map structural relations among the corresponding objects in the world. Otherwise it would be false. All of this should, of course, be perfectly familiar. (See, for example, Aristotle, Metaphysics, 1027 b 22, 1051 b 32ff.) The present paper seeks to go further than its predecessors, however, in offering a detailed account of the syntax of a working universal characteristic and of the ways in which it might be used. (shrink)

This paper offers an expressivist account of logical form, arguing that in order to fully understand it one must examine what valid arguments make us do (or: what Achilles does and the Tortoise doesn’t, in Carroll’s famed fable). It introduces Charles Peirce’s distinction between symbols, indices and icons as three different kinds of signification whereby the sign picks out its object by learned convention, by unmediated indication, and by resemblance respectively. It is then argued that logical form is represented by (...) the third, iconic, kind of sign. It is noted that icons uniquely enjoy partial identity between sign and object, and argued that this holds the key to Carroll’s puzzle. Finally, from this examination of sign-types metaphysical morals are drawn: that the traditional foes metaphysical realism and conventionalism constitute a false dichotomy, and that reality contains intriguingly inference-binding structures. (shrink)