Logical form: Difference between revisions

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==History==

The importance of the concept of form to logic was already recognized in ancient times. [[Aristotle]], in the ''[[Prior Analytics]]'', was probably the first to employ variable letters to represent valid inferences. Therefore, [[Jan Łukasiewicz|Łukasiewicz]] claims that the introduction of variables was '"one of Aristotle's greatest inventions'."

According to the followers of Aristotle like [[Ammonius Hermiae|Ammonius]], only the logical principles stated in schematic terms belong to logic, and not those given in concrete terms. The concrete terms ''man'', ''mortal'', etc.,and so forth are analogous to the substitution values of the schematic placeholders ''A'', ''B'', ''C'', which were called the '"matter'" (Greek ''hyle'', Latin ''materia'') of the argument.

The term "logical form" itself was introduced by [[Bertrand Russell]] in 1914, in the context of his program to formalize natural language and reasoning, which he called [[philosophical logic]]. Russell wrote: "Some kind of knowledge of logical forms, though with most people it is not explicit, is involved in all understanding of discourse. It is the business of philosophical logic to extract this knowledge from its concrete integuments, and to render it explicit and pure."<ref>[https://books.google.com/books?id=SsY9g_DDA9MC&pg=PA53 Russell, Bertrand. 1914(1993). Our Knowledge of the External World: as a field for scientific method in philosophy. New York: Routledge. p. 53]</ref><ref name="PreyerPeter2002">{{cite book |editor=Gerhard Preyer |editor2=Georg Peter |title=Logical form and language |url=https://books.google.com/books?id=ioVFUN8yd9QC&pg=PA54 |year=2002 |publisher=Clarendon Press |isbn=978-0-19-924555-0 |page=54 |chapter=What is logical form? |author=Ernie Lepore |author2=Kirk Ludwig}} [http://www.indiana.edu/~socrates/papers/What%20is%20Logical%20Form.pdf preprint]</ref>

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:Therefore, ''S'' is ''M''.

All wethat havehas been done in the ''Argumentargument form'' is to put ''H'' for ''human'' and ''humans'', ''M'' for ''mortal'', and ''S'' for ''Socrates';'. whatWhat results is the ''form'' of the original argument. Moreover, each individual sentence of the ''Argumentargument form'' is the ''sentence form'' of its respective sentence in the original argument.<ref>{{cite book|last=Hurley|first=Patrick J.|title=A concise introduction to logic|publisher=Wadsworth Pub. Co.|___location=Belmont, Calif.|year=1988|isbn=0-534-08928-3|url-access=registration|url=https://archive.org/details/conciseintroduct00hurl_4}}</ref>

==Importance of argument form==

Attention is given to argument and sentence form, because ''form'' is what makes an argument [[Validity (logic)|valid]] or cogent''. All logical form arguments are either inductive or deductive. Inductive logical forms include inductive generalization, statistical arguments, causal argument, and arguments from analogy. Common deductive argument forms are [[hypothetical syllogism]], [[categorical syllogism]], argument by definition, argument based on mathematics, argument from definition. The most reliable forms of logic are [[modus ponens]], [[modus tollens]], and chain arguments because if the premises of the argument are true, then the conclusion necessarily follows.<ref>{{cite book | author1=Bassham, Gregory | title=Critical thinking : a student's introduction | publication-date=2012 | publisher=McGraw-Hill | edition=5th | isbn=978-0-07-803831-0 }}</ref> Two invalid argument forms are [[affirming the consequent]] and [[denying the antecedent]].

;Affirming the consequent

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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 "Somesome 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.

==See also==