The standard view of logical form is that logical forms are synthetic structures which are the forms of sentences and of other linguistic entities. This is often associated with a more general linguistic view of logic which is articulated in different ways by various authors. This paper contains a critical discussion of such linguistic approaches to logical form, with special emphasis on Quine’s formulation of a logical grammar in Philosophy of Logic. An account of logical forms as higher-order properties, which (...) essentially builds on Frege’s analysis of quantification as higher-order predication, is suggested at the end. (shrink)

Ontology played a very large role in Quine’s philosophy and was one of his major preoccupations from the early 30’s to the end of his life. His work on ontology provided a basic framework for most of the discussions of ontology in analytic philosophy in the second half of the Twentieth Century. There are three main themes (and several sub-themes) that Quine developed in his work. The first is ontological commitment: What are the existential commitments of a theory? The second (...) is ontological reduction: How can an ontology be reduced to (or substituted by) another? And what is the most economical ontology that can be obtained for certain given purposes? The third is criteria of identity: When are entities of some kind (sets, properties, material objects, propositions, meanings, etc.) the same or different? In this paper I discuss Quine’s development of these three themes and some of the problems that were aised in connection with his work. (shrink)

Otávio Bueno gives a positive and accurate summary of my defense of Platonism, with special emphasis on the epistemological issues. He criticizes “skeptical nominalism”, and proposes instead an “agnostic nominalism”, which treats mathematical objects as “objects of thought”, and neither rejects nor accepts abstract entities. In my response I argue that the main problem for nominalism is to account for abstract properties and relations, and that treating mathematical objects as objects of thought does not provide a satisfactory solution to that (...) end.Otávio Bueno apresenta um resumo positivo e correto de minha defesa do Platonismo, com ênfase especial nos aspectos epistemológicos. Ele critica o “nominalismo cético” e propõe um “nominalismo agnóstico” que trata os objetos matemáticos como “objetos de pensamento”, sem rejeitar ou aceitar entes abstratos. Em minha réplica argumento que o problema principal para o nominalismo é dar conta das propriedades e relações abstratas, e que tratar os objetos matemáticos como objetos de pensamento não é uma solução satisfatória para esse fim. (shrink)

In §1 I examine Boole’s “principle of wholistic reference” in relation to Frege’s postulation of truth-values as referents for sentences. I also consider in this connection Frege’s interpretation of quantification and his view that functions and concepts must be defined for all objects. I then present my own contrasting views on the reference of sentences. In §2 I discuss Boole’s introduction of the notion of universe of discourse and consider whether one of the issues implicit in John’s paper is a (...) confrontation between absolute interpretations of logic and relativistic interpretations of logic. I conclude with a brief examination of Tarski’s views on this issue. (shrink)

Classical logic is often characterized through certain laws such as bi-valence and sharpness of concepts, among others. My view is that its most fundamental feature is a commitment to an objective conception of truth, which goes together with a realistic metaphysical view. Truth is objective in that it derives from the nature of reality, and is not dependent on beliefs, theories, practices, and the like. Classical logic is a theory of logical properties, logical truths, and logical states of affairs.

After some preliminary remarks in §1, I argue in §2 that Claudio’s considerations about my treatment of Quine’s Bizet-Verdi counterfactuals do not constitute a difficulty for the structural analysis of such counterfactuals. I discuss some of his other examples and argue that counterfactuals are ambiguous both structurally and contextually. I conclude with an examination of the principle of transitivity for counterfactuals.

Abel Lasalle Casanave’s comments on the rhetorical and dialectical aspects of the discussion of proof in chapters 19 to 21. I center most of my response on two issues raised in his concluding questions; the first being the hermeneutic aspect of the development of mathematics and of mathematical knowledge, and the second “the symbolic conception of mathematics” and the “algebraic mode of thought”.As considerações de Abel Lasalle Casanave dizem respeito aos aspectos retóricos e dialéticos da discussão da noção de prova (...) nos capítulos 19 à 21, e minha réplica está centrada em duas questões levantadas em suas observações finais. A primeira refere-se ao aspecto hermenêutico do desenvolvimento da matemática e do conhecimento matemático; e a segunda à “concepção simbólica da matemática” e ao “modo algébrico de pensar”. (shrink)

My disagreement with the deflationist treatment of truth affects my attitude to Paul Horwich’s approach to meaning and intentionality. In my response I summarize objections to the deflationist account of truth developed in some detail in chapters 2, 7, and 12, and argue that the notion of intentionality should be treated naturalistically in a broader context than the context of the referential import of the locution “means that”.Minha discordância com a visão deflacionista da verdade afeta minha atitude em relação às (...) propostas deflacionistas de Paul Horwich para a intencio-nalidade. Em minha réplica faço um resumo das objeções ao deflacionismo em relação à verdade, detalhadas nos capítulos 2, 7 e 12, e argumento que a noção de intencionalidade deveria ser tratada naturalisticamente em um contexto mais amplo que o contexto do alcance referencial da locução “significa que”. (shrink)

In §§1-2 I argue that Marco misidentifies my main objections to the Church and Gödel slingshot arguments and that his defense of these arguments does not overcome those objections. In §3 I discuss his criticisms of my theory of descriptions in relation to Church’s argument.

Guido Imaguire proposes an epistemological formulation of Ockham’s razor in terms of the notion of explanatory reduction. Although in my response I express reservations about some aspects of the specific formulations, I agree with the general epistemological idea.Guido Imaguire propõe uma formulação epistemológica da navalha de Occam em termos da noção de redução explicativa. Apesar de em minha réplica expressar reservas com respeito à alguns aspectos das formulações, concordo com a idéia geral epistemológica.

In §1 I explain that my rejection of possible states of affairs as a basis for an account of falsity is not part of a general rejection of modal notions but is a rejection of possible and impossible entities of any sort. I then show that my account of senses and of propositions is indeed a modal account. In §2 I examine some of Wittgenstein’s ideas about falsity, as presented by Luiz Carlos, in relation to my account of falsity and (...) negation. In §3 I discuss the modal aspects of identification for propositions. (shrink)

Goodman’s paper “The Problem of Counterfactual Conditionals” played a central role in the debate concerning the proper analysis of counterfactual conditionals. In what follows I examine Goodman’s paper in detail and discuss objections and suggestions by Parry in his “Reexamination of the Problem of Counterfactual Conditionals”. I restrict my discussion to what Goodman termed “the problem of relevant conditions”, which is the main subject of Parry’s criticism, and which I also consider to be the central issue for Goodman’s approach.

The question that I discuss in this paper is whether Frege has a criterion of identity for the objects the True and the False that he introduces as denotation of sentences. My answer is that he does not, either in general or within the system of Basic Laws.

Walter Carnielli argues that my position about knowledge and justification emphasizes individual knowledge, is too demanding, and is anti-psychologistic. In my response I argue that this reflects a misunderstanding of the view of knowledge and justification sketched in my book, and that our views on these issues are much closer than he imagines.Walter Carnielli argumenta que minha posição com respeito ao conhecimento e à justificação enfatiza o conhecimento individual, é demasiadamente exigente e é anti-psicologista. Em minha réplica argumento que isto (...) reflete um mal entendido em relação ao ponto de vista sobre conhecimento e justificação esboçado em meu livro, e que nossas visões dessas questões são muito mais próximas do que ele imagina. (shrink)

In §1 I examine the connections between my account of logical properties and Tarski’s account of logical notions. In §2 I briefly present some of my views on modality and the basis for my claim that there are intensional as well as extensional relations between properties. In §3 I compare my views on the nature of logic and of mathematics with Gödel’s views.

In his paper John Corcoran examines in detail many issues relating to logical form, and raises some questions about my formulations. In my response I emphasize two main distinctions that may clear up some of the issues. One is the distinction between logical forms, in the sense of logical properties of an abstract character, and logical form, in the sense in which we speak of the logical form of a sentence, or of a proposition. Another is the distinction, emphasized by (...) Boole, between primary propositions , and secondary propositions —which I illustrate through the distinction between predicate negation and sentential negation.Em seu artigo John Corcoran examina em detalhe muitas questões sobre forma lógica e levanta alguns problemas relativos à minhas formulações. Na réplica enfatizo duas distinções principais, que podem esclarecer algumas questões. A primeira é a distinção entre formas lógicas, no sentido de propriedades lógicas de caráter abstrato, e forma lógica, no sentido em que falamos da forma lógica de uma sentença ou de uma proposição. A segunda é a distinção, enfatizada por Boole, entre proposições primárias e proposições secundárias , exemplificada com a distinção entre negação predicativa e negação sentencial. (shrink)

Oscar Esquisabel gives an overview of Chapter 13, tracing connections with several philosophers and traditions in philosophy, especially with the hermeneutic tradition. In my response I address his concluding questions about hermeneutics, and about the relation between senses, meanings, and concepts.Oscar Esquisabel apresenta uma visão geral do capítulo 13, traçando conexões com vários filósofos e tradições filosóficas, especialmente com a tradição hermenêutica. Dirijo minha réplica à suas questões finais sobre hermenêutica e sobre a relação entre sentidos, significados e conceitos.

Danielle Macbeth disagrees with the view that there are logical truths in an ontological sense, and argues that we have no adequate epistemological account of our access to such features of reality. In my response I recall some main aspects of my ontological and epistemological formulation of logic as a science, and argue that neither Quine’s considerations against meaning, nor Benacerraf’s considerations against Gödel’s realism, show the untenability of an approach to logical truth in terms of logical propositions that denote (...) logical states of affairs.Danielle MacBeth discorda da tese que há verdades lógicas em um sentido ontológico e argumenta que não há um tratamento adequado de nosso acesso epistêmico à tais aspectos da realidade. Em minha réplica relembro algumas características principais de minha formulação ontológica e epistemológica da lógica como ciência, e argumento que nem as considerações de Quine contra a noção de significado, nem as considerações de Benacerraf contra o realismo de Gödel, mostram a invalidade de uma concepção de verdade lógica em termos de proposições que denotam estados de coisa lógicos. (shrink)

In my response to Guillermo Rosado-Haddock I discuss the two main issues raised in his paper. The first is that by allowing Henkin’s general models as a legitimate model-theoretic interpretation of second-order logic, I undermine my defense of second-order logic against Quine’s views concerning the primacy of first-order logic. The second is that my treatment of logical truth and logical properties does not take into account various systems of logic and properties of systems of logic such as the Löwenheim-Skolem property.Em (...) minha réplica à Guillermo Rosado-Haddock discuto as duas questões centrais levantadas em seu artigo. A primeira é que ao permitir modelos gerais de Henkin como uma interpretação legítima da lógica de segunda ordem, desvirtuo minha defesa da lógica de segunda ordem contra a visão de Quine respeito à primazia da lógica de primeira ordem. A segunda é que meu tratamento da verdade lógica e das propriedades lógicas não leva em consideração diversos sistemas de lógica e propriedades de sistemas de lógica tais como a propriedade de Löwenheim-Skolem. (shrink)

Mark Wilson argues that in order to make physical first-order properties suitable for inclusion in the bottom levels of a logical hierarchy of properties, their proper treatment must take into account the methods of applied mathematics. I agree that the methods of applied mathematics are essential for studying physical properties, and in my response focus on the nature of the logical hierarchy and on the requirements of classical logic.Mark Wilson argumenta que um tratamento adequado para tornar as propriedades físicas de (...) primeira ordem apropriadas para inclusão nos níveis inferiores de uma hierarquia de propriedades lógicas deve levar em consideração os métodos da matemática aplicada. Concordo que os métodos da matemática aplicada são essenciais para estudar as propriedades físicas, e em minha réplica enfoco a natureza da hierarquia lógica e os requisitos da lógica clássica. (shrink)

In my response to Dirk Greimann I maintain that whereas one can recognize some specific appeals to “parsimony” or “simplicity” in the sciences and in philosophy as correct and legitimate, there is no precise adequate formulation of Ockham’s razor as a general methodological principle, and argue that the formulations he examines in his paper exemplify this imprecision.Em minha réplica à Dirk Greimann mantenho que mesmo sendo possível reconhecer certos apelos à “parcimônia” e à “simplicidade” nas ciências e na filosofia como (...) corretos e legítimos, não há uma formulação precisa e adequada da navalha de Occam como princípio metodológico geral, e argumento que as formulações examinadas em seu artigo exemplificam esta imprecisão. (shrink)

I argue in §1 that there is a clear distinction between predicate negation and sentential negation and that sentential negation is a special case of predicate negation operating on the predicate ‘is true’. In §2 I reply to Richard’s objections to negative properties on the basis of the conception of properties as identity conditions presented in Chapter 12 of Logical Forms.

José Seoane centers his commentary on my critique of the standard formal analysis of proof as an elucidation of the informal notion of proof, and I basically agree with his considerations throughout the paper. In my response I argue that the notion of formal proof is fundamentally an analysis of the notion of logical consequence, rather than an elucidation of the informal notion of proof. O comentário de José Seoane está centrado na crítica da análise formal de prova como elucidação (...) da noção informal de prova, e estou basicamente de acordo com suas considerações. Em minha réplica argumento que a noção de prova formal é fundamentalmente uma análise da noção de conseqüência lógica ao invés de uma elucidação da noção informal de prova. (shrink)

The main issue André Porto raises in his paper concerns the use of dot notation to indicate an infinite set of hypotheses. Whereas I agree that one cannot extract a unique infinite expansion from a finite initial segment, in my response I argue that this holds for finite expansions as well. I further explain how my remarks on infinite proof structures are neither motivated by the impact of Gödel’s incompleteness theorems on Hilbert’s program, nor by a negative view of strict (...) finitism.O problema central que André Porto discute em seu artigo diz respeito ao uso da notação de pontos para indicar um conjunto infinito de hipóteses. Mesmo estando de acordo não ser possível extrair uma expansão infinita a partir de um segmento inicial finito, em minha réplica argumento que isto vale igualmente para expansões finitas. Explico também que minhas observações sobre estruturas de prova infinitas não são motivadas pelo impacto dos teoremas de incompletude de Gödel no programa de Hilbert, e tampouco por uma visão negativa do finitismo estrito. (shrink)

Norma Goethe addresses my criticisms of the notion of formal proof as a representation of the practice of proving, and in the process revisits large portions of my discussion of proof. I agree with many of her comments, and direct my response to two specific issues. The first concerns the essential features of proof, and the second the distinction between actual proofs and idealized proofs.Norma Goethe tece seus comentários entorno de minhas críticas à noção de prova formal como representação da (...) prática de provar e retoma diversos aspectos da discussão de prova em meu livro. Concordo com muitas de suas considerações e dirijo minha réplica a duas questões específicas. A primeira é sobre as características essenciais das provas, e a segunda sobre a distinção entre provas atuais e provas idealizadas. (shrink)

I agree with Richard Vallée that predicates like ‘big’, ‘tall’, etc., are comparatives, and that there are no properties of being big, or tall, etc., simpliciter. I also agree that sentence-types are not the primary carriers of truth, and, with one qualification, I too reject the three main assumptions - critically examined by him.Concordo com Richard Vallée que predicados como ‘grande’, ‘alto’, etc., são comparativos, e que não há propriedades de ser simplesmente grande, ou alto, etc. Concordo também que sentenças-tipo (...) não são os portadores de verdade primários e, com uma qualificação, também rejeito as três suposições principais - examinadas criticamente por ele. (shrink)

The paper by Jairo José da Silva is mainly concerned with the character of mathematical proof and with the nature of mathematics and its ontology. Although there is a fair amount of agreement in our views, I focus my response on three issues on which we disagree. The first is his view of mathematical proof as generally unconstrained by language and by a previous proof apparatus. The second is his discussion of Brouwer’s views on proof and formalization. The third is (...) his nominalistic account of structuralism.O artigo de Jairo José da Silva explora principalmente o caráter das provas matemáticas e a natureza ontológica da matemática. Apesar de haver bastante concordância em nossos pontos de vista, o foco de minha réplica são três questões em que discordamos. A primeira é sua visão da prova matemática como completamente livre de restrições impostas pela linguagem e por um aparato prévio de prova. A segunda é sua discussão de Brouwer em relação à prova e à formalização. A terceira é sua formulação nominalista do estruturalismo. (shrink)

In this paper I examine arguments by Benacerraf and by Chihara against Gödel’s platonistic philosophy of mathematics.Neste artigo examino argumentos de Benacerraf e de Chihara contra a filosofia platônica da matemática de Gödel.

In this paper I examine arguments by Benacerraf and by Chihara against Gödel’s platonistic philosophy of mathematics.

In this paper I examine arguments by Benacerraf and by Chihara against Gödel’s platonistic philosophy of mathematics.

Ken López-Escobar questions the timeless status of various entities—propositions, numbers, etc.—as well as my characterization of pure propositional logic as an ontological theory. In my response I argue that my characterization of propositional logic does not depend on timeless propositions, or on other abstract truth bearers, but is a characterization in terms of truth relations between any truth bearers. I also discuss his views on numbers as cultural constructs, as well as his use of quantification in propositional logic.Ken López-Escobar questiona (...) o estatuto atemporal de vários entes—pro-posições, números, etc.— assim como minha caracterização da lógica proposicional pura como teoria ontológica. Na réplica argumento que minha caracterização não depende de proposições atemporais, ou de outros portadores de verdade abstratos, mas é uma caracterização em termos de relações de verdade entre quaisquer portadores de verdade. Examino também suas considerações sobre números como construtos culturais, assim como seu uso de quantificação na lógica proposicional. (shrink)

In §1 I suggest that although my theory of truth as identification is not a correspondence theory of truth in the sense of these terms used by the logical positivists, it can nevertheless be naturally interpreted as a correspondence theory of truth. In §2 I argue that whereas a correspondence theory of truth need not be a realist theory of truth, any sufficiently elaborate realist theory of truth should be interpretable as a correspondence theory of truth. I illustrate this with (...) Frege’s denotational theory of truth. (shrink)

Javier Legris examines my views on symbolism and logical form in relation to two important distinctions emphasized by Jean van Heijenoort—the distinction between logic as calculus and logic as universal language, and the distinction between absolutism and relativism in logic. I generally agree with his considerations and focus my response on some relevant aspects of classical logic.Javier Legris examina minhas considerações sobre simbolismo e forma lógica em relação à duas distinções enfatizadas por Jean van Heijenoort: a distinção entre lógica como (...) cálculo e lógica como linguagem universal, e a distinção entre absolutismo e relativismo na lógica. Estou basicamente de acordo com suas observações e em minha réplica enfoco alguns aspectos relevantes da lógica clássica. (shrink)

Marco Ruffino compares the notion of sense developed in my book with Frege’s notion of sense, and argues that whereas there are ontological similarities, my notion faces epistemological and semantic problems. In my response I discuss the various issues he raises, arguing that my notion of sense can confront them at least as well as Frege’s notion.Marco Ruffino compara a noção de sentido desenvolvida em meu livro com a noção de sentido de Frege, argumentando que mesmo havendo semelhanças ontológicas, minha (...) noção está sujeita a problemas epistemológicos e semânticos. Em minha réplica considero os problemas levantados por Marco, argumentando que minha noção de sentido pode confrontá-los pelo menos tão adequadamente quanto a noção de Frege. (shrink)

In §§1-2 I consider some issues that Guillermo raises in connection with Husserl, especially the distinction between the notion of state of affairs and the more general notion of situation of affairs conceived as a common substratum for different states of affairs. After a few remarks about Church’s slingshot argument in §3, I discuss several objections that Guillermo raises to my interpretation of Frege , to Kripke’s notion of rigid designator and to my objections to Tarski’s semantic conception of truth.

In §1 I discuss Dedekind and Frege on the logical and structural analysis of natural numbers and present my view that the logical analysis of the notion of number involves a combination of their analyses. In §2 I answer some of the specific questions that Abel raises in connection with Chapter 9 of Logical Forms.

Frank Sautter’s questions are directed at the precise senses of the words ‘invention’ and ‘creation’ used in my remarks on the origin of language, and at the connection between Jørgensen’s and my views on the development of language. In my response I clarify my use of the words ‘invention’ and ‘creation’ vis-à-vis Frank’s suggested interpretations, and examine Jørgensen’s distinction of stages in the development of language in relation to imperatives and the “directive use of language”.As perguntas de Frank Sautter estão (...) direcionadas ao sentido mais preciso das palavras ‘invenção’ e ‘criação’ utilizadas em minhas observações sobre a origem da linguagem, e à relação de meu ponto de vista sobre o desenvolvimento da linguagem ao de Jørgensen. Em minha réplica esclareço meu uso das palavras ‘invenção’ e ‘criação’ comparativamente às diversas interpretações sugeridas por Frank, e examino a distinção de estágios no desenvolvimento da linguagem proposta por Jørgensen em relação aos imperativos e ao “uso direcionado da linguagem”. (shrink)

In §1 I reply to Jairo’s objections to my account of truth and falsity showing that my account of falsity does not imply that false sentences refer to something. In §2 I argue that Jairo’s main objection to my account of propositions as abstract properties is based on a misunderstanding concerning the purpose of this account. In §3 I examine Jairo’s suggestion that contradictory sentences can be said to describe possible states of affairs.

Paul Gochet raises several interesting issues about my Chapter 13 discussion of productivity, compositionality, the context principle, meaning, and formalization. In my response I concentrate on the question of units of meaning in relation to the context principle, and on the question of infinity and formalization.Paul Gochet levanta várias questões interessantes sobre minhas considerações, no capítulo 13, sobre produtividade, composicionalidade, o princípio do contexto e formalização. Concentro minha réplica na relação das unidades de significação em relação ao principio do contexto (...) e na questão de infinitude e de formalização. (shrink)

In §1 I discuss the pragmatic and semantic objections that Dirk raises against the claim that sentences refer to states of affairs. In §2 I explain in which sense I maintain that true sentences identify states of affairs.