The focus of the paper is a sophism based on the proposition ‘This is Socrates’ found in a short treatise on obligational casus attributed to William Heytesbury. First, the background to the puzzle in Walter Burley’s traditional account of obligations (the responsio antiqua), and the objections and revisions made by Richard Kilvington and Roger Swyneshed, are presented. All six types of obligations described by Burley are outlined, including sit verum, the type used in the sophism. Kilvington and Swyneshed disliked the (...) dynamic nature of the responsio antiqua, and Kilvington proposed a revision to the rules for irrelevant propositions. This allowed him to use a form of reasoning, the “disputational meta-argument”, which is incompatible with Burley’s rules. Heytesbury explicitly rejected Kilvington’s revision and the associated meta-argument. Swyneshed also revised Burley’s account of obligations, formulating the so-called responsio nova, characterised by the apparently surprising thesis that a conjunction can be denied both of whose conjuncts are granted. On closer inspection, however, his account is found to be less radical than first appears. (shrink)

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

We consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary evidence-based definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways: (1) in terms of classical algorithmic verifiabilty; and (2) in terms of finitary algorithmic computability. We then show that the two definitions correspond to two distinctly different assignments of satisfaction and truth (...) to the compound formulas of PA over N---I_PA(N; SV ) and I_PA(N; SC). We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both I_PA(N; SV ) and I_PA(N; SC). We then show: (a) that if we assume the satisfaction and truth of the compound formulas of PA are always non-finitarily decidable under I_PA(N; SV ), then this assignment corresponds to the classical non-finitary putative standard interpretation I_PA(N; S) of PA over the domain N; and (b) that the satisfaction and truth of the compound formulas of PA are always finitarily decidable under the assignment I_PA(N; SC), from which we may finitarily conclude that PA is consistent. We further conclude that the appropriate inference to be drawn from Goedel's 1931 paper on undecidable arithmetical propositions is that we can define PA formulas which---under interpretation---are algorithmically verifiable as always true over N, but not algorithmically computable as always true over N. We conclude from this that Lucas' Goedelian argument is validated if the assignment I_PA(N; SV ) can be treated as circumscribing the ambit of human reasoning about `true' arithmetical propositions, and the assignment I_PA(N; SC) as circumscribing the ambit of mechanistic reasoning about `true' arithmetical propositions. (shrink)

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)

Sumário: 1. El caso del método científico, Alberto Oliva; 2. Un capítulo de la prehistoria de las ciencias humanas: la defensa por Vico de la tópica, Jorge Alberto Molina; 3. La figura de lo cognoscible y los mundos, Pablo Vélez León; 4. Lebenswelt de Husserl y las neurociencias, Vanessa Fontana; 5. El uso estético del concepto de mundos posibles, Jairo Dias Carvalho; 6. Realismo normativo no naturalista y mundos morales imposibles, Alcino Eduardo Bonella; 7. En la lógica de pragmatismo, Hércules (...) de Araujo Feitosa, Luiz Henrique da Cruz Silvestrini; 8 La lógica, el tiempo y el lenguaje natural: un sistema formal para tiempos de portugués, Carlos Luciano Manholi. / Temas em filosofia contemporânea II / Jonas Rafael Becker Arenhart, Jaimir Conte, Cezar Augusto Mortari (orgs.) – Florianópolis : NEL/UFSC, 2016. 167 p. : tabs. - (Rumos da epistemologia ; v. 14); ISBN 978-85-87253-27-9 (papel); ISBN 978-85-87253-26-2 (e-book). (shrink)

Mathematics textbooks teach logical reasoning by example, a practice started by Euclid; while logic textbooks treat logic as a subject in its own right without practical application to mathematics. Stuck in the middle are students seeking mathematical proficiency and educators seeking to provide it. To assist them, the article explains in practical detail how to teach logic-based skills such as: making mathematical reasoning fully explicit; moving from step to step in a mathematical proof in logically correct ways; and checking to (...) make sure inferences are logically correct. The methodology can easily be extended beyond the four examples analyzed. (shrink)

Veloso-Pereira-Haeusler_On-what-there-must-be.

In this paper I offer solutions to two problems which our moral practice engenders for expressivism, the meta-ethical doctrine according to which ethical statements aren't propositional, susceptible of truth and falsity, but, rather, express the speaker's non-cognitive attitudes. First, the expressivist must show that arguments which are valid when interpreted propositionally are valid when construed expressivistically, and vice versa. The second difficulty is the Frege-Geach problem. Moral arguments employ atomic sentences, negations, disjunctions, etc., and, by expressivist lights, the meaning of (...) a moral sentence depends on the attitude that it expresses. Since one's attitude varies as one asserts a claim, or negates it, or cites it as a disjunct, etc., the meaning of the relevant phrase changes as well, so the argument equivocates. (Formal proofs are provided in appendices to the paper.). (shrink)

The Research Assessment Exercise was introduced in 1986 by Thatcher, and was continued by Blair. So it has now been running for 21 years. During this time, the rules governing the RAE have changed considerably, and the interval between successive RAEs has also varied. These changes are not of great importance as far as the argument of this paper is concerned. We will concentrate on the main features of the RAE which can be summarised as follows.

This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 (...) and Σ1 formulae. Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions. (shrink)

The purpose of this paper is to present an algebraic generalization of the traditional two-valued logic. This involves introducing a theory of automorphism algebras, which is an algebraic theory of many-valued logic having a complete lattice as the set of truth values. Two generalizations of the two-valued case will be considered, viz., the finite chain and the Boolean lattice. In the case of the Boolean lattice, on choosing a designated lattice value, this algebra has binary retracts that have the usual (...) axiomatic theory of the propositional calculus as suitable theory. This suitability applies to the Boolean algebra of formalized token models [2] where the truth values are, for example, vocabularies. Finally, as the actual motivation for this paper, we indicate how the theory of formalized token models [2] is an example of a many-valued predicate calculus. (shrink)

This article investigates the prospect of giving de dicto- and de re-necessity a uniform treatment. The historical starting point is a puzzle raised by Aristotle's claim, advanced in one of the modal chapters of his Prior Analytics, that universally privative apodeictic premises simply convert. As regards the Prior and the Posterior Analytics, the data suggest a representation of propositions of the type in question by doubly modally qualified formulae of modal predicate logic that display a necessity operator in two distinct (...) positions. Can the N-operator occurring in these positions be given a unified semantical treatment (which would justify dispensing with a notational differentiation)? A positive answer, based on a suitably shaped truth condition for N-formulae, is given, and is supported in the final section with an alternative proof theoretically based conception of a property's essential belonging to an individual. (shrink)

Scientific knowledge develops in an increasingly fragmentary way.A multitude of scientific disciplines branch out. Curiosity for thisdevelopment leads into quests for a unifying understanding. To a certain extent, foundational studies provide such unification. There is a tendency, however, also of a fragmentary growth of foundational studies, like in a multitude of disciplinaryfoundations. We suggest to look at the foundational problem, not primarily as a search for foundations for one discipline in another, as in some reductionist approach, but as a steady (...) revelation ofpresuppositions for individual scientific theories – which are boundto meet, sooner or later, in a common language. A decisive point hereis our holistic conception of language, as a whole of description-interpretation processes which are entangled(complementary) in the language itself. For every language there is alinguistic complementarity. We suggest this unique form ofentanglement as a unifying presupposition, ultimately foundational forall communicable knowledge. Involved is a linguistic realism, in terms ofwhich we critically examine ``language-world'' problems, as exposed byWittgenstein, and Russell, about a foundational interdependence of language andreality (world). Throughout, we attach to the developmentof foundational studies of mathematics, logics, and the naturalsciences. In particular, we study the interpretation problem foraxioms of infinity in some detail. We emphasize that the holistic concept of language contradicts Carnap's semiotic fragmentation thesis (thus, no clean cutbetween syntax, semantics, pragmatics). (shrink)

Bertrand Russell offered an influential paradox of propositions in Appendix B of The Principles of Mathematics, but there is little agreement as to what to conclude from it. We suggest that Russell's paradox is best regarded as a limitative result on propositional granularity. Some propositions are, on pain of contradiction, unable to discriminate between classes with different members: whatever they predicate of one, they predicate of the other. When accepted, this remarkable fact should cast some doubt upon some of the (...) uses to which modern descendente of Russell's paradox of propositions have been put in recent literature. (shrink)

This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositional calculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to extend this constructive (...) method to non-classical fragments equipped with weaker implications (that avoid Curry's objection). This is the case, for example, of Jan Lukasiewicz's n-valued logics and Wilhelm Ackermann's logic of restricted implication. Besides such problems, both Henkin's method and the triviality phenomenon enable us to propose a new positive tableau proof system which uses only positive meta-linguistic resources, and to mo- tivate a new discussion concerning the role of negation in logic proposing the concept of paratriviality. In this way, some relations between positive reasoning and infinity, the possibilities to obtain a ¯first-order positive logic as well as the philosophical connection between truth and meaning are dis- cussed from a conceptual point of view. (shrink)

Arenhart_Krause_Uma-logica-da-indistinguibilidade.

Physical superpositions exist both in classical and in quantum physics. However, what is exactly meant by ‘superposition’ in each case is extremely different. In this paper we discuss some of the multiple interpretations which exist in the literature regarding superpositions in quantum mechanics. We argue that all these interpretations have something in common: they all attempt to avoid ‘contradiction’. We argue in this paper, in favor of the importance of developing a new interpretation of superpositions which takes into account contradiction, (...) as a key element of the formal structure of the theory, “right from the start”. In order to show the feasibility of our interpretational project we present an outline of a paraconsistent approach to quantum superpositions which attempts to account for the contradictory properties present in general within quantum superpositions. This approach must not be understood as a closed formal and conceptual scheme but rather as a first step towards a different type of understanding regarding quantum superpositions. (shrink)

The main aim of this paper is to present a way to improve the e.ciency of an intelligent alarm correlation module by means of an abductive reasoner. The alarm correlation module is integrated in a more complex system that performs monitoring and control tasks over a tract of a highway. On the basis of a specific theory on the domain, explanations of anomalous traffic patterns can be provided taking into account those situations not directly detected by data acquisition technology. The (...) integration of the comprehensive view obtained by means of a correlation phase with additional domain knowledge allows to abduce from the observed anomalous traffic patterns other significant situations. The experience of traffic operators allows to build a theory of abducible explanations that exploits the knowledge of the use context of the system. The correlation module of SAMOT has been modelled within a propositional modal language, named ST-Logic, which will be adopted for the abductive reasoner as well. Finally, some remarks about stratified interpretative cycles made possible within such a formal framework will be addressed. (shrink)

A generator program for a computable function (by definition) generates an infinite sequence of programs all but finitely many of which compute that function. Machine learning of generator programs for computable functions is studied. To motivate these studies partially, it is shown that, in some cases, interesting global properties for computable functions can be proved from suitable generator programs which cannot be proved from any ordinary programs for them. The power (for variants of various learning criteria from the literature) of (...) learning generator programs is compared with the power of learning ordinary programs. The learning power in these cases is also compared to that of learning limiting programs, i.e., programs allowed finitely many mind changes about their correct outputs. (shrink)

The article looks briefly at Fefermans own foundations. Among many different senses of foundations, the one that mathematics needs in practice is a recognized body of truths adequate to organize definitions and proofs. Finding concise principles of this kind has been a huge achievement by mathematicians and logicians. We put ZFC and categorical foundations both into this context.

In much the same way that it is possible to construct a model of hyperbolic geometry in the Euclidean plane, it is possible to model quantum logic within the classical propositional calculus.

Multiple explanatory frameworks may be required to provide an adequate account of human cognition. This paper embeds the classical account within a neural network framework, exploring the encoding of syntacticallystructured objects over the synchronicdiachronic characteristics of networks. Synchronic structure is defined in terms of temporal binding and the superposition of states. To accommodate asymmetric relations, synchronic structure is subject to the type uniqueness constraint. The nature of synchronic structure is shown to underlie Xbar theory that characterizes the phrasal structure of (...) human languages. The derivation differentiates core Xbar properties from languagespecific properties. (shrink)

We develop a series of small infinitary epistemic logics to study deductive inference involving intra-/interpersonal beliefs/knowledge such as common knowledge, common beliefs, and infinite regress of beliefs. Specifically, propositional epistemic logics GL are presented for ordinal α up to a given αo so that GL is finitary KDn with n agents and GL allows conjunctions of certain countably infinite formulae. GL is small in that the language is countable and can be constructive. The set of formulae Lα is increasing up (...) to α = ω but stops at ω We present Kripke-completeness for GL for each α ≤ ω, which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL has a sufficient expressive power to discuss intra-/interpersonal beliefs with infinite lengths. As applications, we discuss the explicit definability of Axioms T, 4, 5, and of common knowledge in GL Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL, α ≤ ω. (shrink)

Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, treated as (...) variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals.1 Thus ‘=’ is a logical constant because no permutation maps two individuals to one or one to two; ‘∈’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables. (shrink)

This article considers how people judge the identity of objects (e.g., how people decide that a description of an object at one time, t₀, belongs to the same object as a description of it at another time, t₁). The authors propose a causal continuer model for these judgments, based on an earlier theory by Nozick (1981). According to this model, the 2 descriptions belong to the same object if (a) the object at t₁ is among those that are causally close (...) enough to be genuine continuers of the original and (b) it is the closest of these close-enough contenders. A quantitative version of the model makes accurate predictions about judgments of which a pair of objects is identical to an original (Experiments 1 and 2). The model makes correct qualitative predictions about identity across radical disassembly (Experiment 1) as well as more ordinary transformations (Experiments 2 and 3). (shrink)

A translation of formulas in a language L1 to formulas in a language L2 is a mapping which preserves the parameters and commutes with the substitution prefix, the propositional connectives and the quantifiers. Every translation generates a corresponding transformation of theories in L1 to theories in L2. We formulate the equiconsistency problem for such transformations and propose a variant of its solution. First, for a transformation F we find the least theory A in L1 such that its inclusion in a (...) theory T guarantees equiconsistency of F and F, then we propose axiomatizations of A for some F's. (shrink)

Modal logic S5 is commonly viewed as an epistemic logic that captures the most basic properties of knowledge. Kripke proved a completeness theorem for the first-order modal logic S5 with respect to a possible worlds semantics. A multiagent version of the propositional S5 as well as a version of the propositional S5 that describes properties of distributed knowledge in multiagent systems has also been previously studied. This article proposes a version of S5-like epistemic logic of distributed knowledge with quantifiers ranging (...) over the set of agents, and proves its soundness and completeness with respect to a Kripke semantics. (shrink)

We motivate and formalize the idea of sameness by default: two objects are considered the same if they cannot be proved to be different. This idea turns out to be useful for a number of widely different applications, including natural language processing, reasoning with incomplete information, and even philosophical paradoxes. We consider two formalizations of this notion, both of which are based on Reiter’s Default Logic. The first formalization is a new relation of indistinguishability that is introduced by default. We (...) prove that the corresponding default theory has a unique extension, in which every two objects are indistinguishable if and only if their non-equality cannot be proved from the known facts. We show that the indistinguishability relation has some desirable properties: it is reflexive, symmetric, and, while not transitive, it has a transitive “flavor.” The second formalization is an extension (modification) of the ordinary language equality by a similar default: two objects are equal if and only if their non-equality cannot be proved from the known facts. It appears to be less elegant from a formal point of view. In particular, it gives rise to multiple extensions. However, this extended equality is better suited for most of the applications discussed in this paper. (shrink)