20th Century Logic

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, (...) contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. One rather conspicuous example is that the process of refining logical terminology has been productive. Future logic students will no longer be burdened by obscure terminology and they will be able to read, think, talk, and write about logic in a more careful and more rewarding manner. Closely related is increased use and study of variable-enhanced natural language as in “Every proposition x that implies some proposition y that is false also implies some proposition z that is true”. Another welcome development is the culmination of the slow demise of logicism. No longer is the teacher blocked from using examples from arithmetic and algebra fearing that the students had been indoctrinated into thinking that every mathematical truth was a tautology and that every mathematical falsehood was a contradiction. A fifth welcome development is the separation of laws of logic from so-called logical truths, i.e., tautologies. Now we can teach the logical independence of the laws of excluded middle and non-contradiction without fear that students had been indoctrinated into thinking that every logical law was a tautology and that every falsehood of logic was a contradiction. This separation permits the logic teacher to apply logic in the clarification of laws of logic. This lecture expands the above points, which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

This paper is a short introduction to Carnap’s writings on semantics with an emphasis on the transition from the syntactic period to the semantic one. I claim that one of Carnap’s main aims was to investigate the possibility of the symmetry between the syntactic and the semantic methods of approaching philosophical problems, both in logic and in the philosophy of science. This ideal of methodological symmetry could be described as an attempt to obtain categorical logical systems, i.e., systems that allow (...) only the intended semantical interpretation. (shrink)

Tarski’s Convention T—presenting his notion of adequate definition of truth (sic)—contains two conditions: alpha and beta. Alpha requires that all instances of a certain T Schema be provable. Beta requires in effect the provability of ‘every truth is a sentence’. Beta formally recognizes the fact, repeatedly emphasized by Tarski, that sentences (devoid of free variable occurrences)—as opposed to pre-sentences (having free occurrences of variables)—exhaust the range of significance of is true. In Tarski’s preferred usage, it is part of the meaning (...) of true that attribution of being true to a given thing presupposes the thing is a sentence. Beta’s importance is further highlighted by the fact that alpha can be satisfied using the recursively definable concept of being satisfied by every infinite sequence, which Tarski explicitly rejects. Moreover, in Definition 23, the famous truth-definition, Tarski supplements “being satisfied by every infinite sequence” by adding the condition “being a sentence”. Even where truth is undefinable and treated by Tarski axiomatically, he adds as an explicit axiom a sentence to the effect that every truth is a sentence. Surprisingly, the sentence just before the presentation of Convention T seems to imply that alpha alone might be sufficient. Even more surprising is the sentence just after Convention T saying beta “is not essential”. Why include a condition if it is not essential? Tarski says nothing about this dissonance. Considering the broader context, the Polish original, the German translation from which the English was derived, and other sources, we attempt to determine what Tarski might have intended by the two troubling sentences which, as they stand, are contrary to the spirit, if not the letter, of several other passages in Tarski’s corpus. (shrink)

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

This chapter clarifies that it was the works Giuseppe Peano and his school that first led Russell to embrace symbolic logic as a tool for understanding the foundations of mathematics, not those of Frege, who undertook a similar project starting earlier on. It also discusses Russell’s reaction to Peano’s logic and its influence on his own. However, the chapter also seeks to clarify how and in what ways Frege was influential on Russell’s views regarding such topics as classes, functions, meaning (...) and denotation, etc., and summarizes the correspondence between Frege and Russell and the light it sheds on the philosophical logic of both. (shrink)

To believe that logic has no history might at first seem peculiar today. But since the early 20th century, this position has been repeatedly conflated with logical monism of Kantian provenance. This logical monism asserts that only one logic is authoritative, thereby rendering all other research in the field marginal and negating the possibility of acknowledging a history of logic. In this paper, I will show how this and many related issues have developed, and that they are founded on only (...) one prominent statement by Kant. I will argue, however, that this statement takes on a very different meaning in a broader context of the history and philosophy of science, and that Kant and his supporters never advocated the logical monism that they are still said to hold today. (shrink)

In the Tractatus, Wittgenstein conceives of tautology as 'saying nothing'. More precisely, he holds -- or so this essay contends -- that it says nothing in virtue of possessing a zero quantity of sense. Insofar as it is the limit of a series of propositions of diminishing quantity of sense, tautology resembles a degenerate conic section. But it also resembles the result of a summing together of equal and opposite linear vector quantities. Both of these models shape Wittgenstein's conception of (...) tautology in the Tractatus, though they are not fully reconciled. Many of the puzzling features of the Tractatus's view of logic arise from this failure to affect the needed reconciliation. These points are argued for by working through the development of Wittgenstein's views on these matters in the Wartime Notebooks and other pre-Tractatus writings. (shrink)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity (...) problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning. (shrink)

This chapter explores the metaphysical views about higher-order logic held by two individuals responsible for introducing it to philosophy: Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970). Frege understood a function at first as the remainder of the content of a proposition when one component was taken out or seen as replaceable by others, and later as a mapping between objects. His logic employed second-order quantifiers ranging over such functions, and he saw a deep division in nature between objects and functions. (...) Russell understood propositional functions as what is obtained when constituents of propositions are replaced by variables, but eventually denied that they were entities in their own right. Both encountered contradictions when supposing there to exist as many objects as functions, and both adopted views about the meaningfulness of higher-order discourse that were difficult to state from within their own strictures. (shrink)

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

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

The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

En The paradox of Addition and its dissolution (1969), Mario Bunge presenta algunos argumentos para mostrar que la Regla de Adición puede ocasionar paradojas o problemas semánticos. Posteriormente, Margáin (1972) y Robles (1976) mostraron que las afirmaciones de Bunge son insostenibles, al menos desde el punto de vista de la lógica clásica. Aunque estamos de acuerdo con las críticas de Margáin y Robles, no estamos de acuerdo en el diagnóstico del origen del problema y tampoco con la manera en la (...) que se proponen solucionarlo. En esta medida, en este texto mostraremos cómo la Regla de Adición sí trae problemas semánticos que pueden ser planteados desde diferentes perspectivas contemporáneas, como la extensionalidad del condicional, los principios proscriptivos y la validez conexiva. (shrink)

Logicism is the view that mathematical truths are logical truths. But a logical truth is commonly thought to be one with a universally valid form. The form of “7 > 5” would appear to be the same as “4 > 6”. Yet one is a mathematical truth, and the other not a truth at all. To preserve logicism, we must maintain that the two either are different subforms of the same generic form, or that their forms are not at all (...) what they appear. The historical record shows that Russell pursued both these options, but that the struggle with the logical paradoxes pushed him away from the first kind of response and toward the second. An object cannot itself have a kind of inner logical complexity that makes a proposition have a different logical form merely in virtue of being about it, nor can their representatives in logical forms be single things different for different forms, at least not without postulating too many such objects and thereby creating Cantorian diagonal paradoxes. There are only apparent objects which are actually fragments of logical forms, different in different cases. (shrink)

Ayda Ignez Arruda was a key figure in the development of the Brazilian school of Paraconsistent logic and the first person to write a historical survey of the field. Despite her importa...

Carnap and Quine first met in the 1932-33 academic year, when the latter, fresh out of graduate school, visited the key centers of mathematical logic in Europe. In the months that Carnap was finishing his Logische Syntax der Sprache, Quine spent five weeks in Prague, where they discussed the manuscript “as it issued from Ina Carnap’s typewriter”. The philosophical friendship that emerged in these weeks would have a tremendous impact on the course of analytic philosophy. Not only did the meetings (...) effectively turn Quine into Carnap's disciple, they also paved the way for their seminal debates about meaning, language, and ontology---the very discussions that would change the course of analytic philosophy in the decades after the Second World War. -/- Yet surprisingly little is known about these first meetings. Although Quine has often acknowledged the impact of his Prague visit, there appears to be little information about these first encounters, except for the fact that the Quines “were overwhelmed by the kindness of the Carnaps” and that it was Quine’s “most notable experience of being intellectually fired by a living teacher”. Neither their correspondence nor their autobiographies offer a detailed account of these meetings. In this paper, I shed new light on Carnap’s and Quine’s first encounters by examining a set of previously unexplored material from their personal and academic archives. Why did Quine decide to visit Carnap? What did they discuss? And in what ways did the meetings affect Quine’s philosophical development? In what follows, I address these questions by means of a detailed reconstruction of Quine’s year in Europe based on a range of letters, notes, and reports from the early 1930s. (shrink)

Susanne K. Langer is best known as a philosopher of culture and student of Ernst Cassirer. In this chapter, however, I argue that this standard picture ignores her contributions to the development of analytic philosophy in the 1920s and 1930s. I reconstruct the reception of Langer’s first book *The Practice of Philosophy*—arguably the first sustained defense of analytic philosophy by an American philosopher—and describe how prominent European philosophers of science such as Moritz Schlick, Rudolf Carnap, and Herbert Feigl viewed her (...) as one of the most important allies in the United States. In the second half of this chapter, I turn to Langer’s best-selling *Philosophy in a New Key* and reconstruct her attempts to broaden the scope of the, by then, rapidly growing U.S. analytic movement. I argue that her book anticipated various developments in analytic philosophy but was largely ignored by her former colleagues. I end the chapter by offering some clues as to why New Key did not incite the same laudatory responses from analytic philosophers as her earlier work. (shrink)

Arthur Norman Prior was born on 4 December 1914 in Masterton, New Zealand. He studied philosophy in the 1930s and was a significant, and often provocative, voice in theological debates until well into the 1950s. He became a lecturer in philosophy at Canterbury University College in Christchurch in 1946 succeeding Karl Popper. He became a full professor in 1952. He left New Zealand permanently for England in 1959, first taking a chair in philosophy at Manchester University, and then becoming a (...) fellow of Balliol College, Oxford, in 1966. Prior died on 6 October 1969 in Trondheim, Norway. After Prior’s death, many logicians and philosophers have analysed and discussed his approach to formal and philosophical logic. In particular, his contributions to modal logic, tense-logic and deontic logic have been studied. -/- In 1957, A.N. Prior proposed the three-valued modal logic Q as a ‘correct’ modal logic from his philosophical motivations, see Prior (1957). Prior developed Q in order to offer a logic for contingent beings, in which one could intelligibly and rationally state that some beings are contingent and some are necessary, see Akama & Nagata (2005). According to Akama & Nagata (2005), Q has a natural semantics. In other words, from the philosophical point of view, Q can be regarded as an ‘actualist’ modal logic. -/- This review article is a developed description of, and discussion on, ‘The System Q’ that is the fifth chapter of Prior (1957). In addition, in his logical analysis of ‘Time & Existence’ (that is the eights chapter of Prior (1967)), Prior has worked on system Q. Thus, Prior (1967) has also been very useful for this article. This article analyses the logical structure of system Q in order to provide a more understandable description as well as logical analysis for today’s logicians, philosophers, and information-computer scientists. In the paper, the Polish notations are translated into modern notations in order to be more comprehensible and to support the developed formal descriptions as well as semantic analysis. (shrink)

In his 1918 logical atomism lectures, Russell argued that there are no molecular facts. But he posed a problem for anyone wanting to avoid molecular facts: we need truth-makers for generalizations of molecular formulas, but such truth-makers seem to be both unavoidable and to have an abominably molecular character. Call this the problem of generalized molecular formulas. I clarify the problem here by distinguishing two kinds of generalized molecular formula: incompletely generalized molecular formulas and completely generalized molecular formulas. I next (...) argue that, if empty worlds are logically possible, then the model-theoretic and truth-functional considerations that are usually given address the problem posed by the first kind of formula, but not the problem posed by the second kind. I then show that Russell’s commitments in 1918 provide an answer to the problem of completely generalized molecular formulas: some truth-makers will be non-atomic facts that have no constituents. This shows that the neo-logical atomist goal of defending the principle of atomicity—the principle that only atomic facts are truth-makers—is not realizable. (shrink)

John Maynard Keynes’s A Treatise on Probability is the seminal text for the logical interpretation of probability. According to his analysis, probabilities are evidential relations between a hypothesis and some evidence, just like the relations of deductive logic. While some philosophers had suggested similar ideas prior to Keynes, it was not until his Treatise that the logical interpretation of probability was advocated in a clear, systematic and rigorous way. I trace Keynes’s influence in the philosophy of probability through a heterogeneous (...) sample of thinkers who adopted his interpretation. This sample consists of Frederick C. Benenson, Roy Harrod, Donald C. Williams, Henry E. Kyburg and David Stove. The ideas of Keynes prove to be adaptable to their diverse theories of probability. My discussion indicates both the robustness of Keynes’s probability theory and the importance of its influence on the philosophers whom I describe. I also discuss the Problem of the Priors. I argue that none of those I discuss have obviously improved on Keynes’s theory with respect to this issue. (shrink)

Frege's conception of the logical categories has vexed commentators for decades. In this dissertation, I argue that it revolves around two forms of internality. The first is the internality of its use in the expression of judgment to the sign. A proper understanding of that internality reveals how Frege's philosophical logic cannot be fit into the framework given by the contemporary syntax/semantics distinction. The second is the internality that obtains between the way in which Begriffsschrift signs stratify into different categories, (...) on the one hand, and the way in which the realm of Bedeutungen stratifies into logical categories, on the other hand. A proper understanding of this internality reveals how Frege's conception of the relation between language and the world is neither a form of realism nor a form of linguistic idealism. To come to understand Frege's conception of the logical categories, is to come to understand both these forms of internality by coming to understand how they hang together. (shrink)

According to the Bad Company objection, the fact that Frege’s infamous Basic Law V instantiates the general definitional pattern of higher-order abstraction principles is a good reason to doubt the soundness of this sort of definitions. In this paper I argue against this objection by showing that the definitional pattern of abstraction principles – as extrapolated from §64 of Frege’s Grundlagen– includes an additional requirement (which I call the specificity condition) that is not satisfied by the Basic Law V while (...) is satisfied by other higher-order abstractions such as Hume’s Principle. I also show that the failure of this additional requirement in the case of Basic Law V is engendered by an essential feature of Frege’s conception of logic and thus that Frege himself should not have regarded the Basic Law V as a definition by abstraction. (shrink)

Val Plumwood’s 1993 paper, “The politics of reason: towards a feminist logic” (hence- forth POR) attempted to set the stage for what she hoped would begin serious feminist exploration into formal logic – not merely its historical abuses, but, more importantly, its potential uses. This work offers us: (1) a case for there being feminist logic; and (2) a sketch of what it should resemble. The former goal of Plumwood’s paper encourages feminist theorists to reject anti-logic feminist views. The paper’s (...) latter aim is even more challenging. Plumwood’s critique of classical negation (and classical logic) as a logic of domination asks us to recognize that particular logical systems are weapons of oppression. Against anti-logic feminist theorists, Plumwood argues that there are other logics besides classical logic, such as relevant logics, which are suited for feminist theorizing. Some logics may oppress while others may liberate. We provide details about the sources and context for her rejection of classical logic and motivation for promoting relevant logics as feminist. (shrink)

The Aristotelian origins of formal systems are outlined, together with Aristotle's use of causal terms in describing syllogisms. The precision and exactness of a formalism, based on the projection of logical forms into perceptive signs, is contrasted with foundational, abstract concepts, independent of any formalism, which are presupposed for the understanding of a formal language. The definition of a formal system by means of a Turing machine is put in the context of Wittgenstein's general considerations of a machine understood as (...) a sign. A modification of Łukasiewicz's logic Ł3 with the inclusion of justification terms is proposed in order to formally analyse some features of formalistic reasoning as a mechanical, causal affair within a wider context of "indeterminacy". (shrink)

Peirce is best known as the founder of pragmatism, but his dissatisfaction with how others understood and appropriated it prompted him to rename his own doctrine “pragmaticism” and to compose several variants of his original maxim defining it, as well as numerous restatements and elaborations. This paper presents an extensive selection of such formulations, followed by analysis and commentary demonstrating that for Peirce the ultimate meaning of an intellectual concept is properly expressed as a conditional proposition about the deliberate, self-controlled (...) conduct of its interpreters, not the law-governed behavior of its object. (shrink)

This paper aims to answer the question of whether or not Frege's solution limited to value-ranges and truth-values proposed to resolve the "problem of indeterminacy of reference" in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It (...) is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book. (shrink)

Vann McGee has recently argued that Belnap’s criteria constrain the formal rules of classical natural deduction to uniquely determine the semantic values of the propositional logical connectives and quantifiers if the rules are taken to be open-ended, i.e., if they are truth-preserving within any mathematically possible extension of the original language. The main assumption of his argument is that for any class of models there is a mathematically possible language in which there is a sentence true in just those models. (...) I show that this assumption does not hold for the class of models of classical propositional logic. In particular, I show that the existence of non-normal models for negation undermines McGee’s argument. (shrink)

There is a long tradition of logic, from Aristotle to Gödel, of understanding a proof from the concepts of necessity and causality. Gödel's attempts to define provability in terms of necessity led him to the distinction of formal and absolute (abstract) provability. Turing's definition of mechanical procedure by means of a Turing machine (TM) and Gödel's definition of a formal system as a mechanical procedure for producing formulas prompt us to understand formal provability as a mechanical causality. We propose a (...) formalism which makes explicit the mechanical causal nature of a TM's work. We claim that Gödel's axiomatised ontotheology and his ontological proof give a clue for the understanding of the concept of absolute provability and the pattern of the corresponding absolute completeness proof, respectively. (shrink)

The historical consensus is that logical evidence is special. Whereas empirical evidence is used to support theories within both the natural and social sciences, logic answers solely to a priori evidence. Further, unlike other areas of research that rely upon a priori evidence, such as mathematics, logical evidence is basic. While we can assume the validity of certain inferences in order to establish truths within mathematics and test scientifi c theories, logicians cannot use results from mathematics or the empirical sciences (...) without seemingly begging the question. Appeals to rational intuition and analyticity in order to account for logical knowledge are symptomatic of these commitments to the apriority and basicness of logical evidence. This chapter argues that these historically prevalent accounts of logical evidence are mistaken, and that if we take logical practice seriously we fi nd that logical evidence is rather unexceptional, sharing many similarities to the types of evidence appealed to within other research areas. (shrink)

As analytic philosophy is becoming increasingly aware of and interested in its own history, the study of that field is broadening to include, not just its earliest beginnings, but also the mid-twentieth century. One of the towering figures of this epoch is W.V. Quine (1908-2000), champion of naturalism in philosophy of science, pioneer of mathematical logic, trying to unite an austerely physicalist theory of the world with the truths of mathematics, psychology, and linguistics. Quine's posthumous papers, notes, and drafts revealing (...) the development of his views in the forties have recently begun to be published, as well as careful philosophical studies of, for instance, the evolution of his key doctrine that mathematical and logical truth are continuous with, not divorced from, the truths of natural science. But one central text has remained unexplored: Quine's Portuguese-language book on logic, his 'farewell for now' to the discipline as he embarked on an assignment in the Navy in WWII. Anglophone philosophers have neglected this book because they could not read it. Jointly with colleagues, I have completed the first full English translation of this book. In this accompanying paper I draw out the main philosophical contributions Quine made in the book, placing them in their historical context and relating them to Quine's overall philosophical development during the period. Besides significant developments in the evolution of Quine's views on meaning and analyticity, I argue, this book is also driven by Quine's indebtedness to Russell and Whitehead, Tarski, and Frege, and contains crucial developments in his thinking on philosophy of logic and ontology. This includes early versions of some arguments from 'On What There Is', four-dimensionalism, and virtual set theory. (shrink)

This essay discusses Wittgenstein's conception of logic, early and late, and some of the types of logical system that he constructed. The essay shows that the common view according to which Wittgenstein had stopped engaging in logic as a philosophical discipline by the time of writing Philosophical Investigations is mistaken. It is argued that, on the contrary, logic continued to figure at the very heart of later Wittgenstein's philosophy; and that Wittgenstein's mature philosophy of logic contains many interesting thoughts that (...) have gone widely unnoticed. (shrink)

The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School – a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its serious approach towards philosophical studies (...) and teaching of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines.The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives of the School (among others, Ajdukiewicz, Lesniewski, Łukasiewicz,Tarski), accompanying the names with short descriptions of their achievements. The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name – Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period – the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of followers are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic or computer science in Poland. (shrink)

Alfred Tarski was one of the greatest logicians of the twentieth century. His influence comes not merely through his own work but from the legion of students who pursued his projects, both in Poland and Berkeley. This chapter focuses on three key areas of Tarski's research, beginning with his groundbreaking studies of the concept of truth. Tarski's work led to the creation of the area of mathematical logic known as model theory and prefigured semantic approaches in the philosophy of language (...) and philosophical logic, such as Kripke's possible worlds semantics for modal logic. We also examine the paradoxical decomposition of the sphere known as the Banach–Tarski paradox. Finally we examine Tarski's work on decidable and undecidable theories, which he carried out in collaboration with students such as Mostowski, Presburger, Robinson and others. (shrink)

Tarski's analysis of the concept of truth gives rise to a hierarchy of languages. Does this fragment the concept all the way to philosophical unacceptability? I argue it doesn't, drawing on a modification of Kaplan's theory of indexicals.

It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, the standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as the first-order completeness is (...) concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete. (shrink)

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. -/- There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably (...) infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. -/- Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects. (shrink)

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, (...) contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. This lecture expands points which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

Pour des raisons essentiellement liées à la vocation des textes où la notion de présupposition a fait son apparition, c’est la présupposition d’existence qui s’est imposée la première à l’attention des philosophes du langage. Elle a également déterminé l’orientation des débats en les focalisant sur quelques problèmes traditionnels, au premier chef desquels le problème de l’absence de référence de certaines expressions et celui des imperfections du langage naturel. Contrairement aux noms propres et aux descriptions définies, les termes qui signifient des (...) universaux ont joué un rôle très marginal dans la discussion de ce qui est présupposé par les énoncés que nous échangeons. Ce sont pourtant les noms d’espèces et de genres qui ont été associés les tout premiers à des phénomènes qui relèvent de la présupposition. Afin de montrer que cette association n’a rien perdu de son intérêt, nous avons étudié le traitement qu’Aristote réserve à la famille de paralogismes qui exploitent la similarité morphologique entre noms propres et noms communs pour rattacher la même présupposition d’existence aux uns et aux autres, alors même que cette présupposition est légitime dans un cas (celui des expressions qui désignent des choses particulières), abusive dans l’autre cas (celui des expressions qui signifient des universaux). (shrink)

We argue that Wittgenstein’s philosophical perspective on Gödel’s most famous theorem is even more radical than has commonly been assumed. Wittgenstein shows in detail that there is no way that the Gödelian construct of a string of signs could be assigned a useful function within (ordinary) mathematics. — The focus is on Appendix III to Part I of Remarks on the Foundations of Mathematics. The present reading highlights the exceptional importance of this particular set of remarks and, more specifically, emphasises (...) its refined composition and rigorous internal structure. (shrink)

In contemporary natural languages semantics one will often see the use of special brackets to enclose a linguistic expression, e.g. ⟦carrot⟧. These brackets---so-called denotation brackets or semantic evaluation brackets---stand for a function that maps a linguistic expression to its "denotation" or semantic value (perhaps relative to a model or other parameters). Even though this notation has been used in one form or another since the early development of natural language semantics in the 1960s and 1970s, Montague himself didn't make use (...) of this notation in his series of groundbreaking papers on semantics. That raises the question: When was the ⟦.⟧-notation introduced to semantics? This note answers that question. (shrink)

This paper describes Tarski’s project of rehabilitating the notion of truth, previously considered dubious by many philosophers. The project was realized by providing a formal truth definition, which does not employ any problematic concept.

PARC is an "appended numeral" system of natural deduction that I learned as an undergraduate and have taught for many years. Despite its considerable pedagogical strengths, PARC appears to have never been published. The system features explicit "tracking" of premises and assumptions throughout a derivation, the collapsing of indirect proofs into conditional proofs, and a very simple set of quantificational rules without the long list of exceptions that bedevil students learning existential instantiation and universal generalization. The system can be used (...) with any Copi-style set of inference rules, so it is quite adaptable to many mainstream symbolic logic textbooks. Consequently, PARC may be especially attractive to logic teachers who find Jaskowski/Gentzen-style introduction/elimination rules to be far less "natural" than Copi-style rules. The PARC system is also keyboard-friendly in comparison to the widely adopted Jaskowski-style graphical subproof system of natural deduction, viz., Fitch diagrams and Copi "bent arrow" diagrams. (shrink)

Logical pluralism is the claim that there is more than one adequate logic. Many authors consider Carnap as one of the forerunners of logical pluralism. More than that, they claim that Carnap’s Principle of Tolerance consists in one of the first explicit formulations a logical pluralism. Nonetheless, there is little detailed investigation to evaluate if the Principle of Tolerance necessarily implies a logical pluralism, and if so, of which kind. The aim of this paper is to analyze the Principle of (...) Tolerance, as well as its context, and to investigate the relation between such principle and logical pluralism. (shrink)

In this paper I sketch some arguments that underlie Hegel's chapter on judgment, and I attempt to place them within a broad tradition in the history of logic. Focusing on his analysis of simple predicative assertions or ‘positive judgments’, I first argue that Hegel supplies an instructive alternative to the classical technique of existential quantification. The main advantage of his theory lies in his treatment of the ontological implications of judgments, implications that are inadequately captured by quantification. The second concern (...) of this paper is the manner in which Hegel makes logic not only dependent on ontology generally, but also variant in regard to domains of objects. In other words, he offers a domain-specific logical theory, according to which the form of judgment or inference is specific to the subject of judgment. My third concern lies with the metaphilosophical consequences of this theory, and this includes some more familiar Hegelian themes. It is well known that Hegel frequently questioned the adequacy of the sentential form for expressing higher order truths. My reading of his theory of predication explains and contextualizes this tendency by demystifying notions like the so-called speculative proposition. (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)