Logical form: Difference between revisions

Revision as of 22:19, 23 May 2023

Revision as of 20:22, 8 September 2023

Line 43:

A logical [[argument]], seen as an [[ordered set]] of sentences, has a logical form that [[compositionality|derives]] from the form of its constituent sentences; the logical form of an argument is sometimes called argument form.<ref name="BeallBeall2009">{{cite book|author=J. C. Beall|title=Logic: the Basics|url=https://books.google.com/books?id=FLnTavNvIqYC&pg=PA18|year=2009|publisher=Taylor & Francis|isbn=978-0-415-77498-7|page=18}}</ref> Some authors only define logical form with respect to whole arguments, as the [[schema (logic)|schemata]] or inferential structure of the argument.<ref name="Tomassi1999">{{cite book|author=Paul Tomassi |title=Logic |url=https://books.google.com/books?id=TUVQr6InyNYC&pg=PA386|year=1999|publisher=Routledge|isbn=978-0-415-16696-6|pages=386}}</ref> In [[argumentation theory]] or [[informal logic]], an argument form is sometimes seen as a broader notion than the logical form.<ref name="Pinto2001">{{cite book|author=Robert C. Pinto|title=Argument, inference and dialectic: collected papers on informal logic|url=https://books.google.com/books?id=eK0a5CgyV7kC&pg=PA84|year=2001|publisher=Springer|isbn=978-0-7923-7005-5|page=84}}</ref>

It consists of stripping out all spurious grammatical features from the sentence (such as gender, and passive forms), and replacing all the expressions specific to ''the subject matter'' of the argument by [[schematic variable]]s. Thus, for example, the expression "all A's are B's" shows the logical form which is common to the sentences "all men are mortals,", "all cats are carnivores,", "all Greeks are philosophers,", and so on.

==Logical form in modern logic==

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.

==Logical forms in natural language processing==