This paper proposes substitutional definitions of logical truth and consequence in terms of relative interpretations that are extensionally equivalent to the model-theoretic definitions for any relational first-order language. Our philosophical motivation to consider substitutional definitions is based on the hope to simplify the meta-theory of logical consequence. We discuss to what extent our definitions can contribute to that.

This paper is devoted to show the development of some of the model-theoretic ideas which are clearly present in the main members of the Peano school (Peano himself, Burali-Forti, Pieri and Padoa) asa result of their conception of nominal definitions. Also, their semantic definition of logical consequence (Pieri, Padoa) is viewed as one of the outcomes of that conception. Some examples of their use of theexpression “nominal definition” are presented first. Second, the main advantages of this kind of definition, as (...) they saw them, are briefly explained, mainly in a philosophical context. Finally, already in the kernel of the paper, some of the details of the model-theoretic view itself are shown, first in Peano, then in Pieri and Padoa, including in both cases some study of their semantic definitions of logicalconsequence. (shrink)

Bilateralists, who accept that there are two primitive speech acts, assertion and denial, can offer an attractive definition of consequence: Y follows from X if and only if it is incoherent to assert all formulas X and to deny all formulas Y. The present paper argues that this definition has consequences many will find problematic, amongst them that truth coincides with assertibility. Philosophers who reject these consequences should therefore reject this definition of consequence.

Resumo Peter van Inwagen no seu An Essay of Free Will e, no muito mais tarde, “The Consequence Argument” formula várias versões daquilo que designou por “o argumento de consequência”. van Inwagen descreveu o “argumento da consequência” como um argumento para a incompatibilidade do determinismo com o livre arbítrio. Contudo, o autor deste artigo argumenta que a mais recente formulação do argumento da consequência não é, tal como está, um argumento para a incompatibilidade do determinismo com o livre arbítrio. Embora (...) possa ser transformado em um, não parece haver nenhuma maneira simples de transformá-lo em um argumento convincente para a incompatibilidade do determinismo com o livre arbítrio. Por isso, sugere-se, para a construção de um argumento convincente para a incompatibilidade do determinismo com o livre-arbítrio, que a melhor aposta passa pela reconstrução de uma definição do determinismo substancialmente diferente da de van Inwagen. Palavras-chave : argumento da consequência, determinismo, livre arbítrio, metafísica, van InwagenIn An Essay on Free Will, and in the much later “The Consequence Argument” Peter van Inwagen formulated various versions of what he calls “the consequence argument”. van Inwagen has described the consequence argument as an argument for the incompatibility of determinism with free will. But, I argue, the latest formulation of the consequence argument is not, as it stands, an argument for the incompatibility of determinism with free will. Although it can be turned into one, there does not seem to be any straightforward way of turning it into a cogent argument for the incompatibility of determinism with free will. I suggest that if we want to construct a cogent argument for the incompatibility of determinism with free will, our best bet is to start with a definition of determinism substantially different from van Inwagen’s. Keywords : consequence argument, determinism, free will, metaphysics, van Inwagen. (shrink)

This paper is devoted to show the development of some of the model-theoretic ideas which are clearly present in the main members of the Peano school (Peano himself, Burali-Forti, Pieri and Padoa) asa result of their conception of nominal definitions. Also, their semantic definition of logical consequence (Pieri, Padoa) is viewed as one of the outcomes of that conception. Some examples of their use of theexpression “nominal definition” are presented first. Second, the main advantages of this kind of definition, as (...) they saw them, are briefly explained, mainly in a philosophical context. Finally, already in the kernel of the paper, some of the details of the model-theoretic view itself are shown, first in Peano, then in Pieri and Padoa, including in both cases some study of their semantic definitions of logicalconsequence. (shrink)

In his classic 1936 paper Tarski sought to motivate his definition of logical consequence by appeal to the inference form: P(0), P(1), . . ., P(n), . . . therefore ∀nP(n). This is prima facie puzzling because these inferences are seemingly first-order and Tarski knew that Gödel had shown first-order proof methods to be complete, and because ∀nP(n) is not a logical consequence of P(0), P(1), . . ., P(n), . . . by Taski's proposed definition. An attempt to resolve (...) the puzzle due to Etchemendy is considered and rejected. A second attempt due to Gómez-Torrente is accepted as far as it goes, but it is argued that it raises a further puzzle of its own: it takes the plausibility of Tarski's claim that his definition captures our common concept of logical consequence to depend upon our common concept being a reductive conception. A further interpretation of what Tarski had in mind when he offered the example is proposed, using materials well known to Tarski at the time. It is argued that this interpretation makes the motivating example independent of reductive definitions which take natural numbers to be higher-order set theoretic entities, and it also explains why he did not regard the distinction between defined and primitive terms as pressing, as was the distinction between logical and nonlogical terms. (shrink)

Among the influential contributions of Alfred Tarski to logic and philosophy, and close in importance to his widely applied and discussed definition of truth, one finds his definition of logical consequence for formal languages. Like his definition of truth, Tarski's definition of logical consequence has been widely and fruitfully applied. Unlike the definition of truth, that of logical consequence has been rarely discussed philosophically. The main aim of this dissertation is to offer a thorough discussion of some philosophical issues arising (...) from the neglected definition. ;The point of departure is an interpretative reading of the relevant writings by Tarski. This reading yields historically interesting data that refute other recent readings. It also provides philosophical illumination, by clarifying how Tarski thought that his definition satisfied some pretheoretic desiderata. ;Next it is discussed whether Tarski's definition satisfies other pretheoretic desiderata, not stated by Tarski and in some cases not endorsed by him, but which are desiderata nonetheless from reasonable standpoints. These include the desiderata that all instances of logical consequence in Tarski's defined sense be instances of analytical implication, and that they be instances of valid implication. The conclusion of this part of the dissertation is that, under reasonable assumptions regarding the field of applicability of the definition and the concepts used in it, there are reasons to believe that these desiderata are met, and none to believe they are not. ;Finally, the problem of logical constants is examined. The concept of a "logical constant" is the only concept not taken from logic and mathematics that appears in Tarski's definition. The problem of logical constants is that of replacing the primitive 'logical constant' with a defined, better understood concept, and in a such a way that the resulting defined concept of logical consequence meets the previously discussed desiderata. The main conclusion of this part is that some attempts to do this replace the obscure with the more obscure, while other attempts that seem to be in the right spirit have shortcomings of their own. (shrink)

The paper is concerned with Quine's substitutional account of logical truth. The critique of Quine's definition tends to focus on miscellaneous odds and ends, such as problems with identity. However, in an appendix to his influential article On Second Order Logic, George Boolos offered an ingenious argument that seems to diminish Quine's account of logical truth on a deeper level. In the article he shows that Quine's substitutional account of logical truth cannot be generalized properly to the general concept of (...) logical consequence. The purpose of this paper is threefold: first, to introduce the reader to the metamathematics of Quine's substitutional definition of logical truth; second, to make Boolos' result accessible to a broader audience by giving a detailed and self-contained presentation of his proof; and, finally, to discuss some of the possible implications and how a defender of the Quinean concepts might react to the challenge posed by Boolos' result. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

The standard semantic definition of consequence with respect to a selected set X of symbols, in terms of truth preservation under replacement (Bolzano) or reinterpretation (Tarski) of symbols outside X, yields a function mapping X to a consequence relation ⇒x. We investigate a function going in the other direction, thus extracting the constants of a given consequence relation, and we show that this function (a) retrieves the usual logical constants from the usual logical consequence relations, and (b) is an inverse (...) to—more precisely, forms a Galois connection with—the Bolzano-Tarski function. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page Tarski consequence-definition paper is based on a monistic fixed-universe framework—like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as ‘the class of all individuals’. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework—like the Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of (...) discourse serving as different ranges of the individual variables in different interpretations—as in post-WWII model theory. In the early 1960s, many logicians—mistakenly, as we show—held the ‘contrary alternative’ that Tarski had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege-Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski’s philosophy underwent many other radical changes. (shrink)

This paper analyses two definitions of contingency. Both definitions have been widely accepted and used as to identify contingent events. One of them is primarily of a philosophical character, whereas the other is more commonly used in mathematics. Evidently, these two definitions do not describe the same set of phenomena, and neither of them determines the completely intuitive notion of contingency.Namely, carefully selected examples testify that the first definition is too narrow and the second too wide. These facts have certain (...) epistemological consequences. They must, therefore, serve as a warning for using the definitions only in a restrictive and cautious way when detecting random beliefs – those we cannot identify as knowledge. (shrink)

In this paper we reconsider the issue of free choice and the role of the wh-morphology employed in it. We show that the property of being an interrogative wh-word alone is not sufficient for free choice, and that semantic and sometimes even morphological definiteness is a pre-requisite for some free choice items (FCIs) in certain languages, e.g. in Greek and Mandarin Chinese. We propose a theory that explains the polarity behaviour of FCIs cross-linguistically, and allows indefinite (Giannakidou 2001) as well (...) as definite-like FCIs. The difference is manifested as a lexical distinction in English between any (indefinite) and wh-ever (definite); in Greek it appears as a choice between a FCI nominal modifier (taking an NP argument), which illustrates the indefinite option, and a FC free relative illustrating the definite one. We provide a compositional analysis of Greek FCIs in both incarnations, and derive in a parallel manner the Chinese FCIs. Here the definite versus indefinite alternation is manifested in the presence or absence of dōu, which we take to express the maximality operator. It is thus shown that what we see in the morphology of FCIs in Greek is reflected in syntax in Chinese. Our analysis has important consequences for the class of so-called wh-indeterminates. In the context of current proposals, free choiceness is taken to come routinely from interrogative semantics, and wh-indeterminates are treated as question words which can freely become FCIs (Kratzer and Shimoyama 2002). Our results from Mandarin and Greek emphasize that wh-indeterminates do not form a uniform class in this respect, and that interrogative semantics alone cannot predict either sensitivity of free choice to definiteness, or the polarity behaviour of FCIs. (shrink)

The official definition of ‘pain’ by the International Association for the Study of Pain (IASP) hasn’t seen much revision since its publication in 1979. There have been various criticisms of the definition in the literature from different quarters: that the definition implies a dubious metaphysical dualism, that it requires a strong form of consciousness as well as linguistic abilities, that it excludes many vulnerable groups that are otherwise perfectly capable of experiencing pain, that it has therefore unacceptable practical as well (...) as ethical consequences, that it is unhelpful to the health-care professional as an operational definition, that it is too narrow, and many others. In my view, most of these criticisms depend on misunderstandings or on uncharitable interpretations. My aim is to go over the definition, clarify some potential ambiguities, and argue that these criticisms don’t cut much ice. At the end, I will present a few slightly reworded versions of the definition that I claim are more felicitous and capture the original intended meaning by the members of the taxonomic committee in a much better and transparent way that avoids all the major extant criticisms in the literature. (Written for interdisciplinary audiences.). (shrink)

Tarski's 1936 paper, “On the concept of logical consequence”, is a rather philosophical, non-technical paper that leaves room for conflicting interpretations. My purpose is to review some important issues that explicitly or implicitly constitute its themes. My discussion contains four sections: terminological and conceptual preliminaries, Tarski's definition of the concept of logical consequence, Tarski's discussion of omega-incomplete theories, and concluding remarks concerning the kind of conception that Tarski's definition was intended to explicate. The third section involves subsidiary issues, such as (...) Tarski's discussion concerning the distinction between material and formal consequence and the important question ofthe criterion for distinguishing between logical and non-logical terms.§1. Preliminaries. In this paper an argument is a two-part system composed of a set of propositions P and a single proposition c. The expression ‘c is a [logical] consequence of P’ is used with the same meaning as the expression ‘c is [logically] implied by P’. The expressions ‘is a logical consequence of’ and the converse ‘implies’ are relational. Often, I shall be talking in the same sense of validity of an argument. Validity is a property of arguments; an argument with premise-set P and conclusion c is valid if and only if P implies c; i.e., c is a logical consequence of P. Notice that this notion of argument is strictly ontic; it does not involve any agent that thinks, determines or establishes that a given proposition is or is not a consequence of a given set of propositions. (shrink)

We generalize the notion ofconsequence relationstandard in abstract treatments of logic to accommodate intuitions ofrelevance. The guiding idea follows theuse criterion, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each beusedin some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure of monotonicity, but also a move to considering consequence relations as obtaining betweenmultisets. We motivate and state basic definitions of relevant consequence relations, both (...) in single conclusion (asymmetric) and multiple conclusion (symmetric) settings, as well as derivations and theories, guided by the use intuitions, and prove a number of results indicating that the definitions capture the desired results (at least in many cases). (shrink)

To understand logic is, first and foremost, to understand logical consequence. This Element provides an in-depth, accessible, up-to-date account of and philosophical insight into the semantic, model-theoretic conception of logical consequence, its Tarskian roots, and its ideas, grounding, and challenges. The topics discussed include: the passage from Tarski's definition of truth to his definition of logical consequence, the need for a non-proof-theoretic definition, the idea of a semantic definition, the adequacy conditions of preservation of truth, formality, and necessity, the nature, (...) structure, and totality of models, the logicality problem that threatens the definition of logical consequence, a general solution to the logicality, formality, and necessity problems/challenges, based on the isomorphism-invariance criterion of logicality, philosophical background and justification of the isomorphism-invariance criterion, and major criticisms of the semantic definition and the isomorphism-invariance criterion. (shrink)

Kant claims that the nominal definition of truth is: “Truth is the agreement of cognition with its object”. In this paper, I analyse the relevant features of Kant's theory of definition in order to explain the meaning of that claim and its consequences for the vexed question of whether Kant endorses or rejects a correspondence theory of truth. I conclude that Kant's claim implies neither that he holds, nor that he rejects, a correspondence theory of truth. Kant's claim is (...) not a generic way of setting aside a correspondence definition of truth, or of considering it uninformative. Being the nominal definition of truth, the formula “truth is the agreement of cognition with its object” illustrates the meaning of the predicate “is true” and people's ordinary conception of truth. True judgements correspond to the objects they are about. However, there could be more to the property of truth than correspondence. (shrink)

This paper aims to establish several interconnected points. First, a particular interpretation of the mathematical definition of information, known as the causal interpretation, is supported largely by misunderstandings of the engineering context from which it was taken. A better interpretation, which makes the definition and quantification of information relative to the function of its user, is outlined. The first half of the paper is given over to introducing communication theory and its competing interpretations. The second half explores three consequences (...) of the main thesis. First, a popular claim that the quantification of information in a signal is irrelevant for the meaning of that signal is exposed as fallacious. Second, a popular distinction between causal and semantic information is shown to be misleading, and I argue it should be replaced with a related distinction between natural and intentional signs. Finally, I argue that recent empirical work from microbiology and cognitive science drawing on resources of mathematical communication theory is best interpreted by the functional account. Overall, a functional approach is shown to be both theoretically and empirically well-supported. (shrink)

Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be defined syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in [4] that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in [3]. Other authors have extended this result to the cases of (...) k-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature. (shrink)

Bolzano’s definition of consequence in effect associates with each set X of symbols (in a given interpreted language) a consequence relation X . We present this in a precise and abstract form, in particular studying minimal sets of symbols generating X . Then we present a method for going in the other direction: extracting from an arbitrary consequence relation its associated set C of constants. We show that this returns the expected logical constants from familiar consequence relations, and that, restricting (...) attention to sets of symbols satisfying a strong minimality condition, there is an isomorphism between the set of strongly minimal sets of symbols and the set of corresponding consequence relations (both ordered under inclusion). (shrink)

Standard definitions of procrastination underscore the irrational nature of this habit, a critical criterion being that the procrastinating individual delays despite expecting to be worse off for the delay. However, an examination of more than 175 items in 18 procrastination scales reveals that they do not address such a forward-looking criterion. Consequently, scales run the risk of not separating maladaptive and irrational delays from other forms of delay. We propose that forward-looking considerations may not be the best way of operationalizing (...) the irrationality involved in procrastination and argue that scales should instead focus on past negative consequences of unnecessary delay. We suggest a new scale to measure such procrastination-related negative consequences and demonstrate that this scale, used separately or combined with established procrastination scales, performs better in predicting negative states and correlates to procrastination than established scales. The new scale seems to be helpful in separating trivial forms of unnecessary delay from maladaptive forms and hence represents a potentially valuable tool in research and clinical/applied efforts. (shrink)

"Composition as identity" is the radical claim that the whole is identical to the parts - radical because it implies that a single object can be identical to many objects. Composition as identity, together with auxiliary assumptions, implies the principle of "collapse": an object is one of some things if and only it is part of the fusion of those things. Collapse has important implications: the comprehension principle of plural logic must be restricted, plural definite descriptions such as "the Cheerios (...) in the bowl" are empty, Composition as identity does not preclude emergent properties (contrary to what McDaniel has argued), and drastic simplifications of mereological and logical ideology are extensionally available. (shrink)

_ Source: _Volume 56, Issue 3-4, pp 340 - 366 The history of thinking about consequences in the Middle Ages divides into three periods. During the first of these, from the eleventh to the middle of the twelfth century, and the second, from then until the beginning of the fourteenth century, the notion of natural consequence played a crucial role in logic, metaphysics, and theology. The first part of this paper traces the development of the theory of natural consequence (...) in Abaelard’s work as the conditional of a connexive logic with an equivalent connexive disjunction and the crisis precipitated by the discovery of inconsistency in this system. The second part considers the accounts of natural consequence given in the thirteenth century as a special case of the standard modal definition of consequence, one for which the principle _ex impossibili quidlibet_ does not hold, in logics in which disjunction is understood extensionally. (shrink)

Zwart and Franssen’s impossibility theorem reveals a conflict between the possible-world-based content-definition and the possible-world-based likeness-definition of verisimilitude. In Sect. 2 we show that the possible-world-based content-definition violates four basic intuitions of Popper’s consequence-based content-account to verisimilitude, and therefore cannot be said to be in the spirit of Popper’s account, although this is the opinion of some prominent authors. In Sect. 3 we argue that in consequence-accounts , content-aspects and likeness-aspects of verisimilitude are not in conflict with each other, but (...) in agreement . We explain this fact by pointing towards the deep difference between possible-world- and the consequence-accounts, which does not lie in the difference between syntactic (object-language) versus semantic (meta-language) formulations, but in the difference between ‘disjunction-of-possible-worlds’ versus ‘conjunction-of-parts’ representations of theories. Drawing on earlier work, we explain in Sect. 4 how the shortcomings of Popper’s original definition can be repaired by what we call the relevant element approach. We propose a quantitative likeness-definition of verisimilitude based on relevant elements which provably agrees with the qualitative relevant content-definition of verisimilitude on all pairs of comparable theories. We conclude the paper with a plea for consequence-accounts and a brief analysis of the problem of language-dependence (Sect. 6). (shrink)

Logic is usually considered to be the study of logical consequence – of the most basic laws governing how a statement’s truth depends on the truth of other statements. Some of the pioneers of modern formal logic, notably Hilbert and Carnap, assumed that the only way to get hold of the relation of consequence was to reconstruct it as a relation of inference within a formal system built upon explicit inferential rules. Even Alfred Tarski in 1930 seemed to foresee no (...) kind of consequence other than one induced by a set of inference rules: "Let A be an arbitrary set of sentences of a particular discipline. With the help of certain operations, the so-called rules of inference , new sentences are derived from the set A , called the consequences of the set A . To establish these rules of inference, and with their help to define exactly the concept of consequence, is again a task of special metadisciplines; in the usual terminology of set theory the schema of such definition can be formulated as follows: The set of all consequences of the set A is the intersection of all sets which contain the set A and are closed under the given rules of inference." (p. 63) Thereby also the concept of truth came to be reconstructed as inferability from the empty set of premises. (More precisely, this holds only for non-empirical, necessary truth; but of course logic never set itself the task of studying empirical truth.) From this viewpoint, logic came to look as the enterprise of explication of consequence in terms of inference. (shrink)

Kerridge et al recently published a paper in the journal about organ transplantation and the diagnosis of death.1 Although I appreciate the authors’ efforts to present their arguments about such a controversial issue, I found some inconsistencies in this article that I would like to discussWhen Kerridge and his collaborators discussed the origins of the concept of brain death , they emphasised that after the report of the medical consultants on the diagnosis of death to the US President’s Commission was (...) published in 1981,2 clinicians equated the concept of BD with brainstem death. In fact, the brainstem criterion was first proposed by Mohandas and Chou in Minnesota, in 1971.3 The Minnesota criteria inspired the UK code, which was mainly adopted in UK commonwealth countries.4 This view was afterwards powerfully defended by Christopher Pallis.5 After the US President’s Commission report,2 a lot of countries, and most US states, accepted the whole brain, and not the brainstem, criterion.6Regarding the “dead donor rule”, with the advent of transplant surgery, interest in definitions and diagnosis of death based on brain formulations really acquired a new urgency. None the less, it is important to point out that the concept of BD as death of the individual, did not appear to benefit organ transplantation, but was a consequence of the development of intensive care. As Pallis emphasised, if organ transplant techniques had never been developed, intensive care procedures would have provided the possibility of supplying life support to those cases with destroyed brains and preserved heart function, and physicians would need to face the clinical syndrome called BD.5 When French neurophysiologists and neurologists described the death of the nervous system and coma dépassé, organ transplant techniques were only in the very early stages of development. …. (shrink)

_ Source: _Volume 56, Issue 3-4, pp 201 - 221 This paper summarizes medieval definitions and divisions of consequences and explains the import of the medieval development of the theory of consequence for logic today. It then introduces the various contributions to this special issue of _Vivarium_ on consequences in medieval logic.

Comment faire aujourd'hui de la théologie trinitaire et de la christologie alors que la notion de personne tend à changer de définition dans la culture actuelle ? L'A., en tenant compte de la Tradition de l'Eglise à ce sujet, tente de définir la personne au sens de relation et de sujet. Après avoir fait le tour rapide de ce que propose d'autres théologies contemporaine à ce propos, il tire les conséquences de sa définition pour la théologie et la christologie.

During the last hundred years medical language has been influenced by scientific and technological progress. As a result uncertainty in medical communication is increasing. This may have serious consequences not only for health care delivery but also for medical science. Disease classification, assessment of the validity of epidemiological investigations and comparison of the results of various investigations are examples of what will become less secure. The purpose of the article is to emphasise a main source of uncertainty — the (...) problem of interpreting definitions. Two issues of interpretation are explored. One concerns the logical status of a statement and the other the function of a statement. It is concluded that the security of medical language can be increased through the elucidation of the logical status and functions of statements making up medical concepts. (shrink)

In the first section, I consider what several logicians say informally about the notion of logical consequence. There is significant variation among these accounts, they are sometimes poorly explained, and some of them are clearly at odds with the usual technical definition. In the second section, I first argue that a certain kind of informal account—one that includes elements of necessity, generality, and apriority—is approximately correct. Next I refine this account and consider several important questions about it, including the appropriate (...) characterization of necessity, the criterion for selecting logical constants, and the exact role of apriority. I argue, among other things, that there is no need to recognize a special logical sense of necessity and that the selection of terms to serve as logical constants is ultimately a pragmatic matter. In the third section, I consider whether the informal account I have presented and defended is adequately represented by the usual technical definition. I show that it is, and provably so, for certain limited ways of selecting logical constants. In the general case, however, there seems to be no way to be sure that the technical and informal accounts coincide. (shrink)

The now standard definition of logical consequence is model-theoretic. Many writers have tried to justify, or to criticise, the model-theoretic definition by arguing that it extensionally captures, or fails to capture, our intuitions about logical consequence, such as its modal character or its being truth-preservation in virtue of form. One popular means of comparing the extension of model-theoretic consequence with some intuitive notion proceeds by adapting Kreisel's squeezing argument. But these attempts get Kreisel wrong, and try to achieve more than (...) he ever intended. This suggests that the model-theoretic definition should be viewed quite differently as an explication of our intuitions about logical consequence. I introduce Kreisel's squeezing argument in Section 1. Then in Section 2, I show how it is adapted by two prominent writers on logical consequence, Etchemendy (1990) and Shapiro (2005). Finally, in Section 3, I argue that these adaptations fail. (shrink)

Query processing in ground definite deductive is known to correspond precisely to a linear programming problem. However, the “groundedness” requirement is a huge drawback to using linear programming techniques for logic program computations because the ground version of a logic program can be very large when compared to the original program. Furthermore, when we move from propositional logic programs to first-order logic programs, this effectively means that functions symbols may not occur in clauses. In this paper, we develop a theory (...) of “instantiate- by-need” that performs instantiations only when needed. We prove that this method is sound and complete when computing answer substitutions for non-ground logic programs including those containing function symbols. More importantly, when taken in conjunction with Colmerauer's result that unification can be viewed as linear programming, this means that resolution with unification can be completely replaced by linear programming as an operational paradigm. Additionally, our tree construction method is not rigidly tied to the linear programming paradigm-we will show that given any method M that can compute the set of atomic logical consequences of a propositional logic program, our method can use M to compute , the set of all atoms that are consequences of a first-order logic program. (shrink)

This paper is dedicated to developing a formalism that takes rejection seriously. Bilateral notation of signed formulas with force indicators is adopted to define signed consequences which can be viewed as the bilateral counterpart of Tarskian consequence relations. Its relation to some other bilateral approaches is discussed. It is shown how David Nelson’s logic N4 can be characterized bilaterally and the corresponding completeness result is proved. Further, bilateral variants of three familiar notions are considered and investigated: that of a (...) fragment, of definitional equivalence, and of a conservative extension. (shrink)

A truth-preservation fallacy is using the concept of truth-preservation where some other concept is needed. For example, in certain contexts saying that consequences can be deduced from premises using truth-preserving deduction rules is a fallacy if it suggests that all truth-preserving rules are consequence-preserving. The arithmetic additive-associativity rule that yields 6 = (3 + (2 + 1)) from 6 = ((3 + 2) + 1) is truth-preserving but not consequence-preserving. As noted in James Gasser’s dissertation, Leibniz has been criticized (...) for using that rule in attempting to show that arithmetic equations are consequences of definitions. -/- A system of deductions is truth-preserving if each of its deductions having true premises has a true conclusion—and consequence-preserving if, for any given set of sentences, each deduction having premises that are consequences of that set has a conclusion that is a consequence of that set. Consequence-preserving amounts to: in each of its deductions the conclusion is a consequence of the premises. The same definitions apply to deduction rules considered as systems of deductions. Every consequence-preserving system is truth-preserving. It is not as well-known that the converse fails: not every truth-preserving system is consequence-preserving. Likewise for rules: not every truth-preserving rule is consequence-preserving. There are many famous examples. In ordinary first-order Peano-Arithmetic, the induction rule yields the conclusion ‘every number x is such that: x is zero or x is a successor’—which is not a consequence of the null set—from two tautological premises, which are consequences of the null set, of course. The arithmetic induction rule is truth-preserving but not consequence-preserving. Truth-preserving rules that are not consequence-preserving are non-logical or extra-logical rules. Such rules are unacceptable to persons espousing traditional truth-and-consequence conceptions of demonstration: a demonstration shows its conclusion is true by showing that its conclusion is a consequence of premises already known to be true. The 1965 Preface in Benson Mates (1972, vii) contains the first occurrence of truth-preservation fallacies in the book. (shrink)

This paper deals with the adequacy of the model-theoretic definition of logical consequence. Logical consequence is commonly described as a necessary relation that can be determined by the form of the sentences involved. In this paper, necessity is assumed to be a metaphysical notion, and formality is viewed as a means to avoid dealing with complex metaphysical questions in logical investigations. Logical terms are an essential part of the form of sentences and thus have a crucial role in determining logical (...) consequence. Gila Sher and Stewart Shapiro each propose a formal criterion for logical terms within a model-theoretic framework, based on the idea of invariance under isomorphism. The two criteria are formally equivalent, and thus we have a common ground for evaluating and comparing Sher and Shapiro philosophical justification of their criteria. It is argued that Shapiro's blended approach, by which models represent possible worlds under interpretations of the language, is preferable to Sher’s formal-structural view, according to which models represent formal structures. The advantages and disadvantages of both views’ reliance on isomorphism are discussed. (shrink)

Besides its intrinsic interest, the definition of ‘game’ is important for three reasons. Firstly, in Wittgenstein's Philosophical Investigations ‘game’ is the paradigm family resemblance concept. If he is wrong in thinking that ‘game’ cannot be defined, then the persuasive force of his argument against definition generally will be considerably weakened. This, in its turn, will have important consequences for our understanding of concepts and philosophical method. Secondly, Wittgenstein's later writings are full of analogies drawn from games—chess alone is mentioned (...) scores of times—and a proper understanding of ‘game’ can lead us to exercise more caution when considering the parallels between games and non-games. Thirdly, games and play are intriguingly and closely related to art and ritual, and an analysis of games can throw considerable light on both of the latter. (shrink)

In the first section, I consider what several logicians say informally about the notion of logical consequence. There is significant variation among these accounts, they are sometimes poorly explained, and some of them are clearly at odds with the usual technical definition. In the second section, I first argue that a certain kind of informal account—one that includes elements of necessity, generality, and apriority—is approximately correct. Next I refine this account and consider several important questions about it, including the appropriate (...) characterization of necessity, the criterion for selecting logical constants, and the exact role of apriority. I argue, among other things, that there is no need to recognize a special logical sense of necessity and that the selection of terms to serve as logical constants is ultimately a pragmatic matter. In the third section, I consider whether the informal account I have presented and defended is adequately represented by the usual technical definition. I show that it is, and provably so, for certain limited ways of selecting logical constants. In the general case, however, there seems to be no way to be sure that the technical and informal accounts coincide. (shrink)

The importance of addressing the existential threat to humanity, climate change, has grown remarkedly in recent years while conflicting views and interests in societies exist. Therefore, climate change agendas have been weaponized to varying degrees, ranging from the international level between countries to the domestic level among political parties. In such contexts, climate change agendas are predominantly driven by political or economic ambitions, sometimes unconnected to concerns for environmental sustainability. Consequently, it can result in an environment that fosters antagonism and (...) disputes over power and position and increases the risk of prolonged confrontations, hindering the collective global efforts to mitigate and adapt to climate change. Through the current discourse, we aim to provide a preliminary definition of the weaponization of climate change and environmental degradation and examine its risks and consequences on international relations, political dynamics, public perception, and the comprehensive integrity of climate action. We also recommend embracing a globally coordinated, scientifically substantiated approach to circumvent climate change by building an eco-surplus cultural value system. (shrink)

This paper examines from a historical perspective Tarski's 1936 essay, "On the concept of logical consequence." I focus on two main aims. The primary aim is to show how Tarski's definition of logical consequence satisfies two desiderata he himself sets forth for it: (1) it must declare logically correct certain formalizations of the -rule and (2) it must allow for variation of the individual domain in the test for logical consequence. My arguments provide a refutation of some interpreters of Tarski, (...) and notably John Etchemendy, who have claimed that his definition does not satisfy those desiderata. A secondary aim of the paper is to offer some basic elements for an understanding of Tarski's definition in the historical logico-philosophical context in which it was proposed. Such historical understanding provides useful insights on Tarski's informal ideas on logical consequence and their internal cohesion. (shrink)