Classical Logic

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined (...) in terms of a \-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints. (shrink)

In this thesis, we will study the embedding of classical first-order logic in first-order S4, which is based on the translation originally introduced in Fitting (1970). The initial main part is dedicated to a detailed model-theoretic proof of the soundness of the embedding. This will follow the proof sketch in Fitting (1970). We will then outline a proof procedure for a proof-theoretic replication of the soundness result. Afterwards, a potential proof of faithfulness of the embedding, read in terms of soundness (...) and completeness, will be discussed. We will particularly highlight the many difficulties coming with it. In the final section, we will relate this discussion to the debate on notational variance in French (2019). We will do this by showing how a weaker version of French's notion of ‘expressive equivalence’ conforms to the model-theoretic soundness result. We will then conclude that the soundness result without completeness might contain rather little overall insight by relating it to the extensibility of classical logic to S4. (shrink)

The paper presents Alfred Tarski’s debate with the semantic antinomies: the basic Liar Paradox, and its more sophisticated versions, which are currently discussed in philosophy: Strengthen Liar Paradox, Cyclical Liar Paradox, Contingent Liar Paradox, Correct Liar Paradox, Card Paradox, Yablo’s Paradox and a few others. Since Tarski, himself did not addressed these paradoxes—neither in his famous work published in 1933, nor in later papers in which he developed the Semantic Theory of Truth—therefore, We try to defend his concept of truth (...) against these antinomies. It is demonstrated that Tarskian theory of truth is resistant to the paradoxes and it is still the best solution to avoid the antinomies and remain within a classical logic, that is, accepting the laws of noncontradiction, excluded middle, and the principle of bivalence. Thus, the goal of the paper is double—firstly, to show that none of the versions of the Liar Paradox’s is a serious threat to Tarski’s concept of truth, and secondly, that Semantic Theory of Truth allows to remain within classical logic, and at the same time, avoid antinomies—which makes it the most attractive among classical theories of truth. (shrink)

This paper and its sequel “look under the hood” of the usual sorts of proof-theoretic systems for certain well-known intuitionistic and classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be the result of prefixing a formula in a propositional modal language with a step-marker, for this paper either 0 or 1. Think of 1 as indicating the taking of “one step away from 0.” Deductions will be constructed using marked formulas. Section 2 presents (...) the model-theoretic concepts, based on those in [7], that guide the rest of this paper. Section 3 presents Natural Deduction systems IK and CK, formalizations of intuitionistic and classical one-step versions of K. In these systems, occurrences of step-markers allow deductions to display deductive structure that is covered over in familiar “no step” proof-theoretic systems for such logics. Box and Diamond are governed by Introduction and Elimination rules; the familiar K rule and Necessitation are derived (i.e. admissible) rules. CK will be the result of adding the 0-version of the Rule of Excluded Middle to the rules which generate IK. Note: IK is the result of merely dropping that rule from those generating CK, without addition of further rules or axioms (as was needed in [7]). These proof-theoretic systems yield intuitionistic and classical consequence relations by the obvious definition. Section 4 provides some examples of what can be deduced in IK. Section 5 defines some proof-theoretic concepts that are used in Section 6 to prove the soundness of the consequence relation for IK (relative to the class of models defined in Section 2.) Section 7 proves its completeness (relative to that class). Section 8 extends these results to the consequence relation for CK. (Looking ahead: Part 2 will investigate one-step proof-theoretic systems formalizing intuitionistic and classical one-step versions of some familiar logics stronger than K.). (shrink)

This book launches a sustained defense of a radical interpretation of the doctrine of the open future. Patrick Todd argues that all claims about undetermined aspects of the future are simply false.

In this paper I try to answer four basic questions: (1) How the concept of God is to be represented? (2) Are there any logical principles governing it? (3) If so, what kind of logic lies behind them? (4) Can there be a logic of the concept of God? I address them by presenting a formal-logical account to the concept of God. I take it as a methodological desideratum that this should be done within the simplest existing logical formalism. I (...) start with first-order logic (FOL) with identity, and then show that its simplest modal extension (SQML, or the simplest quantified modal logic) is enough for us to formalize a minimally satisfactory theory of the concept of God. I focus exclusively on the monotheistic concept of God. (shrink)

In this article we revisit a number of disputes regarding significance logics---i.e., inferential frameworks capable of handling meaningless, although grammatical, sentences---that took place in a series of articles most of which appeared in the Australasian Journal of Philosophy between 1966 and 1978. These debates concern (i) the way in which logical consequence ought to be approached in the context of a significance logic, and (ii) the way in which the logical vocabulary has to be modified (either by restricting some notions, (...) or by adding some vocabulary) to keep as much of Classical Logic as possible. Our aim is to show that the divisions arising from these disputes can be dissolved in the context of a novel and intuitive proposal that we put forward. (shrink)

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width (...) reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. (shrink)

Could there be a single logical system that would allow us to work simultaneously with classical, paraconsistent, and paracomplete negations? These three negations were separately studied in logics whose negations bear their names. Initially we will restrict our analysis to propositional logics by analyzing classical negation, ¬c, as treated by Classical Propositional Logic (LPC); the paraconsistent negation, ¬p, as treated through the hierarchy of Paraconsistent Propositional Calculi Cn (0 ≤ n ≤ ω); and the paracomplete negation, ¬q, as treated by (...) the hierarchy of Paracomplete Propositional Calculi Pn (0 ≤ n ≤ ω). In “Logics that are both paraconsistent and paracomplete” (1989), Newton da Costa proposed a system with approximate characteristics to what we are looking for. In the hierarchy of Non-Alethical Propositional Calculi Nn (0 ≤ n ≤ ω), only one negation is introduced (as primitive), called a “non-alethic” (¬n), whose operation preserves the properties of classical, or paraconsistent or paracomplete negation -- depending on the well or ill behavior of the formula connected to it. However, as we shall see, in the hierarchy Nn we can not reiterate negations with different behaviors in a same formula (e.g., ¬p¬cα or ¬q¬c¬p α), or even analyze a formula like ¬cα → ¬pα. In view of these problems, can we really say that the hierarchy Nn allows us to understand the relationships and interactions of the three types of negations? In order to deal with this, given the initial problem, we will present four axiomatic systems (KG) in which, unlike Nn, the three negations are directly introduced -- offering a semantics and a method of proofs by analytic tableaux. Through the KG Systems we will show how the negations interact, obtaining non-demonstrable theorems in LPC, Cn, Pn, and Nn (0 ≤ n ≤ ω). Finally, we will also offer a first-order extension for the KG Systems. (shrink)

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut (...) involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities. (shrink)

Russell’s role in the controversy about the paradoxes of material implication is usually presented as a tale of how even the greatest minds can fall prey of basic conceptual confusions. Quine accused him of making a silly mistake in Principia Mathematica. He interpreted “if- then” as a version of “implies” and called it material implication. Quine’s accusation is that this decision involved a use-mention fallacy because the antecedent and consequent of “if- then” are used instead of being mentioned as the (...) premise and the conclusion of an implication relation. It was his opinion that the criticisms and alternatives to the material implication presented by C. I. Lewis and others would never be made in the first place if Russell simply called the Philonian construction “material conditional” instead of “material implication”. Quine’s interpretation on the topic became hugely influential, if not universally accepted. This paper will present the following criticisms against this interpretation: (1) the notion of material implication does not involve a use-mention fallacy, since the components of “if-then” are mentioned and not used; (2) Quine’s belief that the components of “if-then” are used was motivated by a conditional-assertion view of conditionals that is widely controversial and faces numerous difficulties; (3) if anything, it was Quine who could be accused of fallacious reasoning: he ignored that in the assertion of a conditional is the whole proposition that is asserted and not its constituents; (4) the Philonian construction remains counter-intuitive even if it is called “material conditional”; (5) the Philonian construction is more plausible when it is interpreted as a material implication. (shrink)

The material account of indicative conditionals faces a legion of counterexamples that are the bread and butter in any entry about the subject. For this reason, the material account is widely unpopular among conditional experts. I will argue that this consensus was not built on solid foundations, since these counterexamples are contextual fallacies. They ignore a basic tenet of semantics according to which when evaluating arguments for validity we need to maintain the context constant, otherwise any argumentative form can be (...) rendered invalid. If we maintain the context fixed, the counterexamples to the material account are disarmed. Throughout the paper I also consider the ramifications of this defence, make suggestions to prevent contextual fallacies, and anticipate some possible misunderstandings and objections. (shrink)

Intensional evidence is any reason to accept a proposition that is not the truth values of the proposition accepted or, if it is a complex proposition, is not the truth values of its propositional contents. Extensional evidence is non-intensional evidence. Someone can accept a complex proposition, but deny its logical consequences when her acceptance is based on intensional evidence, while the logical consequences of the proposition presuppose the acceptance of extensional evidence, e.g., she can refuse the logical consequence of a (...) proposition she accepts because she doesn’t know what are the truth-values of its propositional contents. This tension motivates counterexamples to the negation of conditionals, the propositional analysis of conditionals, hypothetical syllogism, contraposition and or-to-if. It is argued that these counterexamples are non-starters because they rely on a mix of intensionally based premises and extensionally based conclusions. Instead, a genuine counterexample to classical argumentative forms should present circumstances where an intuitively true and extensionally based premise leads to an intuitively false conclusion that is also extensionally based. The other point is that evidentiary concerns about intensionally based beliefs should be constrained by the truth conditions of propositions presented by classical logic, which are nothing more than coherence requirements in distributions of truth value. It is argued that this restriction also dissolves some known puzzles such as conditional stand-offs, Adams pair, the opt-out property, and the burglar’s puzzle. (shrink)

The material account claims that indicative conditionals are material. However, the conventional wisdom even among material account enthusiasts is that the material account cannot be extended to subjunctive conditionals. There are mainly three reasons that motivate this consensus: (1) the belief that if subjunctives were material, most subjunctive conditionals would be vacuously true, which is implausible; (2) its inconsistency with Adams pair, which suggest that indicative and subjunctive conditionals have different truth conditions; and (3) the belief that it is an (...) inferior hypothesis compared to the possible world theories. I will argue against (1) that the counterintuitive aspects of vacuously true conditionals can be explained away in a uniform fashion, regardless of whether they are indicatives or subjunctives. I reinforce this assumption by showing that the positive arguments for the material account of indicatives are also intuitively valid for subjunctives. The point mentioned in (2) is resisted by explaining Adams pair as logically equivalent conditionals that can be appropriate at different times, depending of the speaker’s epistemic situation. Finally, (3) is criticised by making the case that the possible world account faces insurmountable problems and that a full-blown material account of indicatives and subjunctives is overall a more elegant solution. (shrink)

Adam Rieger (2013) has carried out a survey of arguments in favour of the material account of indicative conditionals. These arguments involve simple and direct demonstrations of the material account. I extend the survey with new arguments and clarify the logical connections among them. I also show that the main counter-examples against these arguments are not successful either because their premises are just as counter-intuitive as the conclusions, or because they depend on contextual fallacies. The conclusion is that the unpopularity (...) of the material account is unjustified and that a more systematic approach in the analysis of arguments is long overdue in our attempts to understand the nature of conditionals. (shrink)

This is the 3rd edition. Although a number of new technological applications require classical deductive computation with non-classical logics, many key technologies still do well—or exclusively, for that matter—with classical logic. In this first volume, we elaborate on classical deductive computing with classical logic. The objective of the main text is to provide the reader with a thorough elaboration on both classical computing – a.k.a. formal languages and automata theory – and classical deduction with the classical first-order predicate calculus with (...) a view to computational implementations, namely in automated theorem proving and logic programming. The present third edition improves on the previous ones by providing an altogether more algorithmic approach: There is now a wholly new section on algorithms and there are in total fourteen clearly isolated algorithms designed in pseudo-code. Other improvements are, for instance, an emphasis on functions in Chapter 1 and more exercises with Turing machines. (shrink)

Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational.

Paraconsistent logics are logical systems that reject the classical principle, usually dubbed Explosion, that a contradiction implies everything. However, the received view about paraconsistency focuses only the inferential version of Explosion, which is concerned with formulae, thereby overlooking other possible accounts. In this paper, we propose to focus, additionally, on a meta-inferential version of Explosion, i.e. which is concerned with inferences or sequents. In doing so, we will offer a new characterization of paraconsistency by means of which a logic is (...) paraconsistent if it invalidates either the inferential or the meta-inferential notion of Explosion. We show the non-triviality of this criterion by discussing a number of logics. On the one hand, logics which validate and invalidate both versions of Explosion, such as classical logic and Asenjo–Priest’s 3-valued logic LP. On the other hand, logics which validate one version of Explosion but not the other, such as the substructural logics TS and ST, introduced by Malinowski and Cobreros, Egré, Ripley and van Rooij, which are obtained via Malinowski’s and Frankowski’s q- and p-matrices, respectively. (shrink)

When discussing Logical Pluralism several critics argue that such an open-minded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism. In this paper we show that the arguments usually employed to arrive at this conclusion do not work. The main reason for this is the existence of certain substructural logics which have the same set of valid inferences as Classical Logic—although they are, in a clear (...) sense, non-identical to it. We argue that this phenomenon can be generalized, given the existence of logics which coincide with Classical Logic regarding a number of metainferential levels—although they are, again, clearly different systems. We claim this highlights the need to arrive at a more refined version of the Collapse Argument, which we discuss at the end of the paper. (shrink)

In The Boundary Stones of Thought, Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project.

The truth-functional hypothesis states that indicative conditional sentences and the material implication have the same truth conditions. Haze (2011) has rejected this hypothesis. He claims that a self-referential conditional, coupled with a plausible assumption about its truth-values and the assumption that the truth-functional hypothesis is true, lead to a liar’s paradox. Given that neither the self-referential conditional nor the assumption about its truth-values are problematic, the culprit of the paradox must be the truth-functional hypothesis. Therefore, we should reject it. In (...) this paper I argue that, contrary to what Haze thinks, the truth-functional hypothesis is not to blame. In fact, no liar’s paradox emerges when the truth-functional hypothesis is true; it emerges only if it is false. (shrink)

Brogaard and Salerno (2008) argued that counter-examples to contraposition, strengthening the antecedent, and hypothetical syllogism involving subjunctive conditionals only seem to work because they involve a contextual fallacy where the context assumed in the premise(s) is illicitly shifted in the conclusion. To avoid such counter-examples they have proposed that the context must remain fixed when evaluating an argument for validity. That is the Brogaard-Salerno Stricture. Tristan Haze (2016), however, has recently objected that intuitively valid argumentative forms such as conjunction introduction (...) do not satisfy this constraint. This paper has two goals. First, it argues that the Brogaard-Salerno Stricture is not violated in Haze’s putative counter-example. Second, it argues that since this stricture blocks the usual counter-examples to instances of classical argumentative forms that involve indicative or subjunctive conditionals, it is reasonable to infer that indicative and subjunctive conditionals are material. (shrink)

Iffication, Preiffication, Qualiffication, Reiffication, and Deiffication. -/- Roughly, iffication is the speech-act in which—by appending a suitable if-clause—the speaker qualifies a previous statement. The clause following if is called the qualiffication. In many cases, the intention is to retract part of the previous statement—called the preiffication. I can retract part of “I will buy three” by appending “if I have money”. This initial study focuses on logical relations among propositional contents of speech-acts—not their full conversational implicatures, which will be treated (...) elsewhere. The modified statement—called the iffication—is never stronger than the preiffication. A degenerate iffication is one logically equivalent to its preiffication. There are limiting cases of degenerate iffications. In one, the qualiffication is tautological, as “I will buy three if three is three”. In another, the negation of the qualiffication implies the preiffication, as “I will buy three if I will not buy three”. Reiffication is iffication of an iffication. “I will buy three if I have money” is reifficated by appending “if there are three left”. Deiffication is the speech-act in which—by appending a suitable and-clause—the effect of an iffication is cancelled so that the result implies the preiffication. “I will buy three if I have money” is deifficated by appending “and I have money”. All further examples come from standard (one-sorted, tenseless, non-modal) first-order arithmetic. All theorems are about first-order arithmetic propositions. An easy theorem, hinted above, is that an iffication is degenerate if and only if the negation of the qualiffication implies the preiffication. The iffication of a conjunction using one of the conjuncts as qualiffication need not imply the other conjunct: “Two is an even square if two is square” does not imply “Two is even”. END OF PRINTED ABSTRACT. -/- Some find that the last statement provides a surprise in logic. https://www.academia.edu/s/a5a4386b75?source=link Acknowledgements: Robert Barnes, William Frank, Amanda Hicks, David Hitchcock, Leonard Jacuzzo, Edward Keenan, Mary Mulhern, Frango Nabrasa, and Roberto Torretti. (shrink)

Information-theoretic approaches to formal logic analyse the "common intuitive" concept of propositional implication (or argumental validity) in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; an argument is valid if the conclusion contains no information beyond that of the premise-set. This paper locates information-theoretic approaches historically, philosophically and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves and (...) by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyse validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. (shrink)

This presentation includes a complete bibliography of John Corcoran’s publications relevant on Aristotle’s logic. The Sections I, II, III, and IV list respectively 23 articles, 44 abstracts, 3 books, and 11 reviews. Section I starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article—from Corcoran’s Philadelphia period that antedates his discovery of Aristotle’s natural deduction system—and the Journal of Symbolic Logic article—from his Buffalo period first reporting his original results. It ends with works published in 2015. (...) Some items are annotated as listed or with endnotes connecting them with other work and pointing out passages that, in retrospect, are seen to be misleading and in a few places erroneous. In addition, Section V, “Discussions”, is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 10 items published in the 1970s, 24 in the 1980s, 42 in the 1990s, 60 in the 2000s, and 70 in the current decade. The secondary bibliography is also annotated as listed or with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. Section VI, “Alternatives”, lists recent works on Aristotle’s logic oblivious of Corcoran’s research and, more generally in some cases, even of the Łukasiewicz-initiated tradition. As is evident from Section VII, “Acknowledgements”, Corcoran’s publications benefited from consultation with other scholars, most notably George Boger, Charles Kahn, John Mulhern, Mary Mulhern, Anthony Preus, Timothy Smiley, Michael Scanlan, Roberto Torretti, and Kevin Tracy. All of Corcoran’s Greek translations were done in collaboration with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. (shrink)

JOHN CORCORAN AND WILIAM FRANK. Surprises in logic. Bulletin of Symbolic Logic. 19 253. Some people, not just beginning students, are at first surprised to learn that the proposition “If zero is odd, then zero is not odd” is not self-contradictory. Some people are surprised to find out that there are logically equivalent false universal propositions that have no counterexamples in common, i. e., that no counterexample for one is a counterexample for the other. Some people would be surprised to (...) find out that in normal first-order logic existential import is quite common: some universals “Everything that is S is P” —actually quite a few—imply their corresponding existentials “Something that is S is P”. Anyway, perhaps contrary to its title, this paper is not a cataloging of surprises in logic but rather about the mistakes that did or might have or might still lead people to think that there are no surprises in logic. The paper cataloging of surprises in logic is on our “to-do” list. -/- ► JOHN CORCORAN AND WILIAM FRANK, Surprises in logic. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] There are many surprises in logic. Peirce gave us a few. Russell gave Frege one. Löwenheim gave Zermelo one. Gödel gave some to Hilbert. Tarski gave us several. When we get a surprise, we are often delighted, puzzled, or skeptical. Sometimes we feel or say “Nice!”, “Wow, I didn’t know that!”, “Is that so?”, or the like. Every surprise belongs to someone. There are no disembodied surprises. Saying there are surprises in logic means that logicians experience surprises doing logic—not that among logical propositions some are intrinsically or objectively “surprising”. The expression “That isn’t surprising” often denigrates logical results. Logicians often aim for surprises. In fact, [1] argues that logic’s potential for surprises helps motivate its study and, indeed, helps justify logic’s existence as a discipline. Besides big surprises that change logicians’ perspectives, the logician’s daily life brings little surprises, e.g. that Gödel’s induction axiom alone implies Robinson’s axiom. Sometimes wild guesses succeed. Sometimes promising ideas fail. Perhaps one of the least surprising things about logic is that it is full of surprises. Against the above is Wittgenstein’s surprising conclusion : “Hence there can never be surprises in logic”. This paper unearths basic mistakes in [2] that might help to explain how Wittgenstein arrived at his false conclusion and why he never caught it. The mistakes include: unawareness that surprise is personal, confusing logicians having certainty with propositions having logical necessity, confusing definitions with criteria, and thinking that facts demonstrate truths. People demonstrate truths using their deductive know-how and their knowledge of facts: facts per se are epistemically inert. [1] JOHN CORCORAN, Hidden consequence and hidden independence. This Bulletin, vol.16, p. 443. [2] LUDWIG WITTGENSTEIN, Tractatus Logico-Philosophicus, Kegan Paul, London, 1921. -/-. (shrink)

The word ‘equality’ often requires disambiguation, which is provided by context or by an explicit modifier. For each sort of magnitude, there is at least one sense of ‘equals’ with its correlated senses of ‘is greater than’ and ‘is less than’. Given any two magnitudes of the same sort—two line segments, two plane figures, two solids, two time intervals, two temperature intervals, two amounts of money in a single currency, and the like—the one equals the other or the one is (...) greater than the other or the one is greater than the other [sc. in appropriate correlated senses of ‘equals’, ‘is greater than’ and ‘is less than’]. In case there are two or more appropriate senses of ‘equals’, the one intended is often indicated by an adverb. For example, one plane figure may be said to be equal in area to another and, in certain cases, one plane figure may be said to be equal in length to another. Each sense of ‘equality’ is tied to a specific domain and is therefore non-logical. Notice that in every cases ‘equality’ is definable in terms of ‘is greater than’ and also in terms of ‘is less than’ both of which are routinely considered domain specific, non-logical. The word ‘identity’ in the logical sense does not require disambiguation. Moreover, it is not correlated ‘is greater than’ and ‘is less than’. If it is not the case that a certain designated triangle is [sc. is identical to] an otherwise designated triangle, it is not necessary for the one to be greater than or less than the other. Moreover, if two magnitudes are equal then a unit of measure can be chosen and, no matter what unit is chosen, each magnitude is the same multiple of the unit that the other is. But identity does not require units. In this regard, congruence is like identity and unlike equality. In arithmetic, the logical concept of identity is coextensive with the arithmetic concept of equality. The logical concept of identity admits of an analytically adequate definition in terms of logical concepts: given any number x and any number y, x is y iff x has every property that y has. The arithmetical concept of equality admits of an analytically adequate definition in terms of arithmetical concepts: given any number x and any number y, x equals y iff x is neither less than nor greater than y. As Aristotle told us and as Frege retold us, just because one relation is coextensive with another is no reason to conclude that they are one. (shrink)

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our traditional intuitive logical framework and (...) that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving radical repudiation of a part of our scientific tradition, a tradition that is defended by classical logicians. But it is also viewed as reactionary: as being analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been. (shrink)

Corcoran’s 27 entries in the 1999 second edition of Robert Audi’s Cambridge Dictionary of Philosophy [Cambridge: Cambridge UP]. -/- ancestral, axiomatic method, borderline case, categoricity, Church (Alonzo), conditional, convention T, converse (outer and inner), corresponding conditional, degenerate case, domain, De Morgan, ellipsis, laws of thought, limiting case, logical form, logical subject, material adequacy, mathematical analysis, omega, proof by recursion, recursive function theory, scheme, scope, Tarski (Alfred), tautology, universe of discourse. -/- The entire work is available online free at more than (...) one website. Paste the whole URL. http://archive.org/stream/RobertiAudi_The.Cambridge.Dictionary.of.Philosophy/Robert.Audi_The.Cambrid ge.Dictionary.of.Philosophy -/- The 2015 third edition will be available soon. Before you think of buying it read some reviews on Amazon and read reviews of its competition: For example, my review of the 2008 Oxford Companion to Philosophy, History and Philosophy of Logic,29:3,291-292. URL: http://dx.doi.org/10.1080/01445340701300429 -/- Some of the entries have already been found to be flawed. For example, Tarski’s expression ‘materially adequate’ was misinterpreted in at least one article and it was misused in another where ‘materially correct’ should have been used. The discussion provides an opportunity to bring more flaws to light. -/- Acknowledgements: Each of these entries was presented at meetings of The Buffalo Logic Dictionary Project sponsored by The Buffalo Logic Colloquium. The members of the colloquium read drafts before the meetings and were generous with corrections, objections, and suggestions. Usually one 90-minute meeting was devoted to one entry although in some cases, for example, “axiomatic method”, took more than one meeting. Moreover, about half of the entries are rewrites of similarly named entries in the 1995 first edition. Besides the help received from people in Buffalo, help from elsewhere was received by email. We gratefully acknowledge the following: José Miguel Sagüillo, John Zeis, Stewart Shapiro, Davis Plache, Joseph Ernst, Richard Hull, Concha Martinez, Laura Arcila, James Gasser, Barry Smith, Randall Dipert, Stanley Ziewacz, Gerald Rising, Leonard Jacuzzo, George Boger, William Demopolous, David Hitchcock, John Dawson, Daniel Halpern, William Lawvere, John Kearns, Ky Herreid, Nicolas Goodman, William Parry, Charles Lambros, Harvey Friedman, George Weaver, Hughes Leblanc, James Munz, Herbert Bohnert, Robert Tragesser, David Levin, Sriram Nambiar, and others. -/- . (shrink)

This paper provides both a solution and a problem for the account of compositionality in Christopher Peacocke’s modest inferentialism. The immediate issue facing Peacocke’s account is that it looks as if compositionality can only be understood at the level of semantics, which is difficult to reconcile with inferentialism. Here, following up a brief suggestion by Peacocke, I provide a formal framework wherein compositionality occurs the level of the determining relation between inference and semantics. This, in turn provides a “test” for (...) compositionality, which, problematically, Peacocke’s natural deduction framework for classical logic can not meet. To finish, I briefly outline an alternative, bilateralist, framework for modest inferentialism, for which compositionality holds. (shrink)

McCall's system for contingent syllogisms is modified. A semantics for the resulting system is provided.

Analytical philosophy of the last one hundred years has been heavily influenced by a doctrine to the effect that one can arrive at a correct ontology by paying attention to certain superficial (syntactic) features of first-order predicate logic as conceived by Frege and Russell. More specifically, it is a doctrine to the effect that the key to the ontological structure of reality is captured syntactically in the ‘Fa’ (or, in more sophisticated versions, in the ‘Rab’) of first-order logic, where ‘F’ (...) stands for what is general in reality and ‘a’ for what is individual. Hence “f(a)ntology”. Because predicate logic has exactly two syntactically different kinds of referring expressions—‘F’, ‘G’, ‘R’, etc., and ‘a’, ‘b’, ‘c’, etc.—so reality must consist of exactly two correspondingly different kinds of entity: the general (properties, concepts) and the particular (things, objects), the relation between these two kinds of entity being revealed in the predicate-argument structure of atomic formulas in first-order logic. (shrink)

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

Syllogisms with or without negative terms are studied by using Gergonne's ideas. Soundness, completeness, and decidability results are given.

The paper shows that for any invalid polysyllogism there is a procedure for constructing a model with a domain with exactly three members and an interpretation that assigns non-empty, non-universal subsets of the domain to terms such that the model invalidates the polysyllogism.

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it (...) is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up. (shrink)

The article deals with some current pioneering formal reconstructions and interpretations of the problem well known in antiquity as The Master Argument. This problem is concerning with enrichment of formal logical systems with modal and temporal notions. The opening topic is devoted to reconstruction of Arthur Prior. while the other here included approach to the problem arc mostly reactions. revisions or additions to this one.

This chapter defends a single, fixed, definite answer to the question: Is there a logic that governs the unsayable? The proposed answer is: “Yes, and no. Or yes-but-not-yes. And/or yes-no.” Each component of this answer is examined and used to generate three laws of what I call “synthetic logic”, which correspond directly to the laws of classical (Aristotelian) logic: the law of contradiction (“A=-A”), the law of non-identity (“A≠A”), and the law of the included middle (“-(Av-A)”). We can talk about (...) the unsayable only by assuming that propositions constructed in accordance with these three alternative logical laws can be meaningful. (shrink)

The work of this dissertation, in a broad sense, seeks to rescue what may be in the original project or nucleus of philosophy, from its Socratic arising: the idea of elucidative rationality. This rationality is aimed at expressing our practices in a way that can be confronted with objections and alternatives. The notion of expression is central to this rationality. This centrality is elucidated by the contemporary philosopher Brandom (1994, 2000, 2008a, 2013), from his view of the semantic inferentialism. With (...) this view, this dissertation, in a strict sense, investigates evidences that leads to the distinction of the inferentially expressivist tradition of syllogistic. In this semantic inferentialism, logic is the “organ of semantic self-consciousness”. In this sense, logic does not define the rational, in the most basic sense, but allows us to be aware, through inferential articulation, of the conceptual contents, which govern all our thoughts. Evidences presented in the work of this dissertation seek to show that the syllogism is marked by the logical-elucidative character of this semantic self-consciousness, because of its expressive role as inseparable from the notion of inference. In the first part of this dissertation, then, the tradition of syllogistics is examined, in which the expression is the main notion. This study is based on the tradition of syllogistics, composed of the Aristotelian school of Campinas, organized by Angioni (2014a), the logical economist Keynes (1906), and the “formalist” schools, represented mainly by the logician Łukasiewicz (1957, 1929/1963) and by Corcoran (1972, 1974, 2009, 2015). The main claims of the Campinas school are analyzed: the non-epistemological, but explanatory, exposition of (scientific) knowledge, in the syllogism, the secondary role of the notion of truth, the priority of the predicative structure, and the suggestion of approaching of the syllogistic to the relevant system of logic. In this analysis, we add the discussion about the reasoning of the epagogic type, important to the practical understanding of the “first principles”. The key points of Keynes are also discussed: the semantic priority of propositional judgment and the explanatory role of deductive inference. The second part of this dissertation discusses the relation between expression, inference and the expressive role of logic, based on the semantic inferentialism of Brandom. In order to discuss this relationship, propaedeutic questions are raised, related to the semantic role of sentences and subsentences (terms and predicates) in language, to different conceptions of logical validity, beyond the truth-functional aspect, to the logical demarcation of symbolic rules, to demarcation of logic, the idea of “formal logic”, and the removal of formal semantics from natural language concern. Next, the theoretical framework of Brandom's semantic inferentialism is presented. In this framework, the idea of philosophical semantics, pragmatisms of the semantic and conceptual type, expressivisms of the rationalist, logical and propositional conceptual type, and the semantically primitive notion of incompatibility come into play. Thus, to those who are interested in the correspondence between ancient logic and modern logic, the work of this dissertation offers a useful contribution, especially to projects of formalization of the syllogistics, which need not appeal against or in favor of a strictly formal approach to logic. (shrink)

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz (...) and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic. (shrink)

An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers.

It is a received view of the post-Fregean predicate logic that a universal statement has no existential import and thus does not entail its particular (existential) counterpart. This paper takes issue with the view by discussing the trespasser case, which has widely been employed for supporting the view. The trespasser case in fact involves a shift of context. Properly understood, the case provides no support for the received view but rather suggests that we rethink the ‘quantity view’ of the existential (...) import of quantifiers. (shrink)

The goal of this paper is to present a new reconstruction of Aristotle's assertoric logic as he develops it in Prior Analytics, A1-7. This reconstruction will be much closer to Aristotle's original text than other such reconstructions brought forward up to now. To accomplish this, we will not use classical logic, but a novel system developed by Ben-Yami [2014. ‘The quantified argument calculus’, The Review of Symbolic Logic, 7, 120–46] called ‘QUARC’. This system is apt for a more adequate reconstruction (...) since it does not need first-order variables on which the usual quantifiers act—a feature also not to be found in Aristotle. Further, in the classical reconstruction, there is also need for binary connectives that don't have a counterpart in Aristotle. QUARC, again, does not need them either to represent the Aristotelian sentence types. However, the full QUARC is also not called for so that I develop a subsystem thereof which closely resembles Aristotle's way of developi... (shrink)

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz (...) and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic. (shrink)

Some of Aristotle’s statements about the indemonstrability of the Principle of Non-Contradiction (PNC) in Metaphysics Γ 4 merit more attention. The consensus seems to be that Aristotle provides two arguments against the demonstrability of the PNC, with one located in Γ 3 and the other found in the first paragraph of Γ 4. In this article, I argue that Aristotle also relies upon a third argument for the same conclusion: the argument from truth. Although Aristotle does not explicitly state this (...) argument, it is the best argument that he could use to defend some of his statements in the second paragraph of Γ 4. Since the argument relies on only a few of Aristotle’s core views about truth, I propose that it is faithful to his considered position throughout his corpus, and it may be the strongest argument he could offer for the indemonstrability of the PNC. (shrink)

In this fragment of Opuscula Logica it is displayed an arithmetical treatment of the aristotelic syllogisms upon the previous interpretations of Christine Ladd-Franklin and Jean Piaget. For the first time, the whole deductive corpus for each syllogism is presented in the two innovative modalities first proposed by Hugo Padilla Chacón. A. The Projection method (all the possible expressions that can be deduced through the conditional from a logical expression) and B. The Retrojection method (all the possible valid antecedents or premises (...) conjunction for an expression proposed as a conclusion). The results are numerically expressed, with their equivalents in the propositional language of bivalent logic. (shrink)

Redictio ad absurdum is an important part of Aristotle’s syllogistic. It is connected with direct proof and they are complementary methods. All moods of Aristotle are provable by direct methods and redictio ad absurdum. In this paper, I have studied on the bases and principles of redictio ad absurdum, I showed how to prove by redictio ad absurdum, and how to prove Aristotle by redictio ad absurdum. By redictio ad absurdum, all forms of Aristotle’s method proved in the first figure (...) can be reduced to the perfect forms. (shrink)