We consider two formalisations of the notion of a compositionalsemantics for a language, and find some equivalent statements in termsof substitutions. We prove a theorem stating necessary and sufficientconditions for the existence of a canonical compositional semanticsextending a given partial semantics, after discussing what features onewould want such an extension to have. The theorem involves someassumptions about semantical categories in the spirit of Husserl andTarski.
We trace two logical ideas further back than they have previously been traced. One is the idea of using diagrams to prove that certain logical premises do—or don’t—have certain logical consequences. This idea is usually credited to Venn, and before him Euler, and before him Leibniz. We find the idea correctly and vigorously used by Abū al-Barakāt in 12th century Baghdad. The second is the idea that in formal logic, P logically entails Q if and only if every model of (...) P is a model of Q. This idea is usually credited to Tarski, and before him Bolzano. But again we find Abū al-Barakāt already exploiting the idea for logical calculations. Abū al-Barakāt’s work follows on from related but inchoate research of Ibn Sīnā in eleventh century Persia. We briefly trace the notion of model-theoretical consequence back through Paul the Persian and in some form back to Aristotle himself. (shrink)
In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, i.e. separately (...) for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’. (shrink)
From this starting point, and assuming no previous knowledge of logic, Wilfrid Hodges takes the reader through the whole gamut of logical expressions in a ...
We define a logic capable of expressing dependence of a variable on designated variables only. Thus has similar goals to the Henkin quantifiers of [4] and the independence friendly logic of [6] that it much resembles. The logic achieves these goals by realizing the desired dependence declarations of variables on the level of atomic formulas. By [3] and [17], ability to limit dependence relations between variables leads to existential second order expressive power. Our avoids some difficulties arising in the original (...) independence friendly logic from coupling the dependence declarations with existential quantifiers. As is the case with independence friendly logic, truth of is definable inside . We give such a definition for in the spirit of [11] and [2] and [1]. (shrink)
The paper analyses what is said and what is presupposed by thePrinciple of Compositionality for semantics, as it is commonly stated. ThePrinciple of Compositionality is an axiom which some semantics satisfy andsome don’t. It says essentially that if two expressions have the same meaning then they make the same contribution to the meanings of expressionscontaining them. This is a sensible axiom only if one combines it with aconverse, that if two expressions make the same contribution to the meanings of sentences (...) containing them, then they have the same meaning;and some assumption that two expressions which can’t meaningfully besubstituted for each other have different meanings. as a full abstraction principle, and as ‘Husserl’s principle’.) Moreoverthe Principle of Compositionality speaks only about when two expressionshave the same meaning; it adds nothing whatever about what that meaningmight be . Some recent discussions by writers in linguistics and logic are assessed. The paper finishes by reviewing thehistory of the notion of compositionality. (shrink)
[Wilfrid Hodges] During the last forty or so years it has become popular to offer explanations of logical notions in terms of games. There is no doubt that many people find games helpful for understanding various logical phenomena. But we ask whether anything is really 'explained' by these accounts, and we analyse Paul Lorenzen's dialogue foundations for constructive logic as an example. The conclusion is that the value of games lies in their ability to provide helpful metaphors and representations, rather (...) than in any true conceptual analysis. In fact some of the standard explanations of logical notions in terms of competitive games simply don't work. /// [ Erik C. W. Krabbe] In an attempt to redeem the Lorenzen-type dialogues from their detractors, it is perhaps best first to provide a survey of the various benefits these dialogues have been supposed to yield. This will be done in Section I. It will not be possible, within the confines of this paper, to scrutinize them all, but in Section II we shall delve deeper into the capacity of this type of dialogue to yield a model for the immanent criticism of philosophical positions. Section III will extend the concept of a dialogue in such a way as to conform better with our intuitive conceptions of what a rational discussion of a position should contain. This will be followed up by a concept of 'winning a dialogue' that takes position midway between the old conception of 'winning one play' and that of the full-fledged presentation of a winning strategy. Concepts of 'rational discussion' are thus shown to be, plausibly, more fundamental than those of proof. In Section IV, I shall discuss the specific problems about dialogical foundations put forward by Wilfrid Hodges. (shrink)
We consider two formalisations of the notion of a compositionalsemantics for a language, and find some equivalent statements in termsof substitutions. We prove a theorem stating necessary and sufficientconditions for the existence of a canonical compositional semanticsextending a given partial semantics, after discussing what features onewould want such an extension to have. The theorem involves someassumptions about semantical categories in the spirit of Husserl andTarski.
During the last forty or so years it has become popular to offer explanations of logical notions in terms of games. There is no doubt that many people find games helpful for understanding various logical phenomena. But we ask whether anything is really 'explained' by these accounts, and we analyse Paul Lorenzen's dialogue foundations for constructive logic as an example. The conclusion is that the value of games lies in their ability to provide helpful metaphors and representations, rather than in (...) any true conceptual analysis. In fact some of the standard explanations of logical notions in terms of competitive games simply don't work. /// [Erik C. W. Krabbe] In an attempt to redeem the Lorenzen-type dialogues from their detractors, it is perhaps best first to provide a survey of the various benefits these dialogues have been supposed to yield. This will be done in Section I. It will not be possible, within the confines of this paper, to scrutinize them all, but in Section II we shall delve deeper into the capacity of this type of dialogue to yield a model for the immanent criticism of philosophical positions. Section III will extend the concept of a dialogue in such a way as to conform better with our intuitive conceptions of what a rational discussion of a position should contain. This will be followed up by a concept of 'winning a dialogue' that takes position midway between the old conception of 'winning one play' and that of the full-fledged presentation of a winning strategy . Concepts of 'rational discussion' are thus shown to be, plausibly, more fundamental than those of proof. In Section IV, I shall discuss the specific problems about dialogical foundations put forward by Wilfrid Hodges. (shrink)
Tarski's model-theoretic truth definition of the 1950s differs from his 1930s truth definition by allowing the language to have a set of parameters that are interpreted by means of structures. The paper traces how the model-theoretic theorems that Tarski and others were proving in the period between these two truth definitions became increasingly difficult to fit into the framework of the earlier truth definition, making the later one more or less inevitable. The paper also maintains that neither recursiveness nor satisfaction (...) are essential features of the truth definition . The recursive form was a ‘practical’ step towards the explicit definition that was Tarski's target. There are model-theoretic languages for which satisfaction is provably too crude a notion for expressing those properties of formulas that are needed for determining the truth of sentences. (shrink)
[Wilfrid Hodges] During the last forty or so years it has become popular to offer explanations of logical notions in terms of games. There is no doubt that many people find games helpful for understanding various logical phenomena. But we ask whether anything is really 'explained' by these accounts, and we analyse Paul Lorenzen's dialogue foundations for constructive logic as an example. The conclusion is that the value of games lies in their ability to provide helpful metaphors and representations, rather (...) than in any true conceptual analysis. In fact some of the standard explanations of logical notions in terms of competitive games simply don't work. /// [Erik C. W. Krabbe] In an attempt to redeem the Lorenzen-type dialogues from their detractors, it is perhaps best first to provide a survey of the various benefits these dialogues have been supposed to yield. This will be done in Section I. It will not be possible, within the confines of this paper, to scrutinize them all, but in Section II we shall delve deeper into the capacity of this type of dialogue to yield a model for the immanent criticism of philosophical positions. Section III will extend the concept of a dialogue in such a way as to conform better with our intuitive conceptions of what a rational discussion of a position should contain. This will be followed up by a concept of 'winning a dialogue' that takes position midway between the old conception of 'winning one play' and that of the full-fledged presentation of a winning strategy. Concepts of 'rational discussion' are thus shown to be, plausibly, more fundamental than those of proof. In Section IV, I shall discuss the specific problems about dialogical foundations put forward by Wilfrid Hodges. (shrink)
The editors invited us to write a short paper that draws together the main themes of logic in the Western tradition from the Classical Greeks to the modern period. To make it short we had to make it personal. We set out the themes that seemed to us either the deepest, or the most likely to be helpful for an Indian reader.
We record the advances made by the eleventh century Persian logician Ibn Sina—known in the West as Avicenna—away from a purely cognitive view of proofs and towards a more computational view, and the kinds of consideration that led him to these advances. Some of Ibn Sina’s new logics, which stand somewhere between Aristotle’s categorical syllogisms and modern first-order logic, can serve as a kind of laboratory for testing what are the differences between Aristotelian and modern logic, and where these differences (...) come from. (shrink)
We consider two formalisations of the notion of a compositionalsemantics for a language, and find some equivalent statements in termsof substitutions. We prove a theorem stating necessary and sufficientconditions for the existence of a canonical compositional semanticsextending a given partial semantics, after discussing what features onewould want such an extension to have. The theorem involves someassumptions about semantical categories in the spirit of Husserl andTarski.
This chapter surveys set theory, model theory, and computability theory: how they first emerged from the foundations of mathematics, and how they have developed since. There are any amounts of mathematical technicalities in the background, but the chapter highlights those themes that have some philosophical resonance.
