Meaning and Truth

Course Notes

(1) Logic and Language


(A) Meaning and Truth


(i) Analytic philosophy and the philosophy of language


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(ii) The relationship between logic and language

Diana married Charles and became pregnant Diana married Charles in Summer 1981 and Diana became pregnant in Autumn 1981
Diana became pregnant and married Charles Diana became pregnant in Autumn 1981 and Diana married Charles in Summer 1981

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(iii) Meaning and truth


Main theme of the course

the separation out of a logical notion of meaning from the cauldron of linguistic life with the seed of our notion of truth

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(B) Logic and Conceptual Content

(i) New Logic, New Analyses

Frege revolutionized logic, and inaugurated the age of modern logic. His key technical advance was his introduction of quantifier notation, and it was this that allowed him to improve on both Aristotelian logic and Boolean algebra. Aristotelian logicians had had great difficulty formalizing propositions of multiple generality (e.g. ‘Every student loves some philosophy course’) and the central problem with Boole’s system lay in the impossibility of integrating its two parts – the calculus of classes and the calculus of propositions (e.g. showing the relationship between (P) ‘All inhabitants are either Europeans or Asiatics’ and (S) ‘Either all the inhabitants are Europeans, or they are all Asiatics’). Frege made the propositional calculus the more fundamental (and provided a much neater axiomatization of it than Boole), and used both this and function-argument notation to develop the predicate calculus, encompassing both syllogistic theory and Boole’s calculus of classes.


  Aristotle Stoics
syllogistic theory propositional logic
Boole (algebra of logic) logic of categoricals logic of hypotheticals
calculus of classes calculus of propositions
Frege predicate calculus propositional calculus


(ii) The Conception of a Logical Language

Frege was inspired by Leibniz’s conception of a characteristica universalis, or logical language, which was intended to be:

(a) and (b) were envisaged as a ‘universal character’, or ideal notation – a scientifically structured, grammatically simple system of symbols that, at least at a basic level, would bear one-one correlation with the terms of any given natural language. (c) and (d) comprised Leibniz’s calculus ratiocinator, bringing together both synthetic and analytic reasoning, and providing both a ‘logic of discovery’ and a ‘logic of proof’. Frege was sceptical of the possibility of including (a), but did think that a logical language could provide the basis for uniting the sciences, and allow us to dispense with the inadequacies of ordinary language.


(iii) Frege’s Original ‘Sinn’

In his Preface to the Begriffsschrift, in introducing his innovations, Frege wrote: ‘These deviations from what is traditional find their justification in the fact that logic hitherto has always followed ordinary language and grammar too closely. In particular, I believe that the replacement of the concepts subject and predicate by argument and function will prove itself in the long run.’ Within Fregean logic, the need to distinguish subject and predicate position in the formalization of syllogistic propositions disappears; and it was in the justification of this that the notion of ‘conceptual content’ arose.

Consider the following two pairs of syllogistic propositions:

For Aristotle, (Eab) and (Eba) were regarded as ‘identical’, but not (Iab) and (Iba), since the transition from (Iab) to (Iba) was not seen as ‘immediate’. Aristotle’s notion of ‘identity’ was epistemological, not semantic. Within Fregean logic, however, the equivalences are readily demonstrated, as their formalizations show:

In fact, (Iab) and (Iba) may be formalized as either (Iab*) or (Iba*) without further ado, and Frege’s justification for this was that they possess the same ‘conceptual content’ – ‘sense’ as it later became. Two propositions have the same ‘conceptual content’, according to Frege, if they are logically equivalent (have the same possible consequences, as he initially put it), i.e. if whenever one is true, the other is true, and vice versa. Frege’s notion of propositional identity was thus semantic, and it is this conception that entitles him to be regarded as the founder of modern analytic philosophy, in which logic and the philosophy of language are accorded a central role. The issues as to the relationship between meaning and truth, and between ‘semantic intuitions’ and the utilization of a logical system, were thus opened up, and much of Frege’s subsequent thought as well as the work of analytic philosophers who followed him were focused on this.


Mike Beaney

February 1999

Meaning and Truth

Mike Beaney