Logical form: Difference between revisions

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The fundamental difference between modern formal logic and traditional, or Aristotelian logic, lies in their differing analysis of the logical form of the sentences they treat:

*On the traditional view, the form of the sentence consists of (1) a subject (e.g., "man") plus a sign of quantity ("all" or "some" or "no"); (2) the [[Copula (linguistics)|copula]], which is of the form "is" or "is not"; (3) a predicate (e.g., "mortal"). Thus: 'all men are mortal'. The logical constants such as "all", "no" and so on, plus sentential connectives such as "and" and "or" were called [[syncategorematic]] terms (from the Greek ''kategorei'' – to predicate, and ''syn'' – together with). This is a fixed scheme, where each judgment has a specific quantity and copula, determining the logical form of the sentence.

*The modern view is more complex, since a single judgement of Aristotle's system involves two or more logical connectives. For example, the sentence "All men are mortal" involves, in term logic, two non-logical terms "is a man" (here ''M'') and "is mortal" (here ''D''): the sentence is given by the judgement ''A(M,D)''. In [[predicate logic]], the sentence involves the same two non-logical concepts, here analyzed as <math>m(x)</math> and <math>d(x)</math>, and the sentence is given by <math>\forall x. (m(x) \rightarrow d(x))</math>, involving the logical connectives for [[universal quantification]] and [[material conditional|implication]].

The more complex modern view comes with more power. On the modern view, the fundamental form of a simple sentence is given by a recursive schema, like natural language and involving [[logical connective]]s, which are joined by juxtaposition to other sentences, which in turn may have logical structure. Medieval logicians recognized the [[problem of multiple generality]], where Aristotelian logic is unable to satisfactorily render such sentences as "Some guys have all the luck", because both quantities "all" and "some" may be relevant in an inference, but the fixed scheme that Aristotle used allows only one to govern the inference. Just as linguists recognize recursive structure in natural languages, it appears that logic needs recursive structure.