Logical form: Difference between revisions

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{{short description|Form for logical arguments, obtained by abstracting from the subject matter of its content terms}}

{{HatnoteAbout|This article is about the term as used in logic. For |the linguistics term, see [[|Logical form (linguistics)]].|the term used in knowledge representation|Logic form}}

[[File:Modus ponens logical form.jpg|thumb|This logical argument utilizing [[modus ponens]] makes use of three statements in logical form expressed in simplified English.]]

{{Hatnote|This article is about the term as used in logic. For the linguistics term, see [[Logical form (linguistics)]].}}

{{Redirect|Argument structure|the possible complements of a verb in linguistics|verb argument}}

In [[philosophy]] and [[mathematics]], a '''logical form''' of a [[Syntax (logic)|syntactic]] expression is a precisely-specified [[Semantics|semantic]] version of that expression in a [[formal system]]. Informally, the logical form attempts to formalize a possibly [[Ambiguity (syntax)|ambiguous]] statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal [[formal language]], the meaning of a logical form can be determined unambiguously from [[syntax]] alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one [[string (computer science)|string]] that represents the same logical form in a given language.<ref>The Cambridge Dictionary of Philosophy, CUP 1999, pp. 511–512</ref>

In [[philosophy]] and [[mathematicslogic]], athe '''logical form''' of a [[SyntaxStatement (logic)|syntacticstatement]] expression is a precisely- specified [[Semantics|semantic]] version of that expressionstatement in a [[formal system]]. Informally, the logical form attempts to [[Logic translation#Natural language formalization|formalize]] a possibly [[AmbiguitySyntactic (syntax)ambiguity|ambiguous]] statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal [[formal language]], the meaning of a logical form can be determined unambiguously from [[syntax]] alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one [[string (computer science)|string]] that represents the same logical form in a given language.<ref>The Cambridge Dictionary of Philosophy, CUP 1999, pp. 511–512</ref>

The logical form of an [[argument]] is called the '''argument form''' or ''test form'' of the argument.

The logical form of an [[argument]] is called the '''argument form''' or ''test form'' of the argument.

==History==

The importance of the concept of form to logic was already recognized in ancient times. [[Aristotle]], in the ''[[Prior Analytics]]'', was probablyone of the first people to employ variable letters to represent valid inferences.{{Citation needed|date=December 2024}} Therefore, [[Jan Łukasiewicz|Łukasiewicz]] claims that the introduction of variables was '"one of Aristotle's greatest inventions'."{{Citation needed|date=December 2024}}

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 |last=Ernie Lepore |title=Logical form and language |last2=Kirk Ludwig |publisher=Clarendon Press |year=2002 |isbn=978-0-19-924555-0 |editor-last=Gerhard Preyer |page=54 |chapter=What is logical form? |editor-last2=Georg Peter |chapter-url=https://books.google.com/books?id=ioVFUN8yd9QC&pg=PA54}} [http://www.indiana.edu/~socrates/papers/What%20is%20Logical%20Form.pdf preprint]</ref>

<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>

In artificial intelligence, logical forms have been used in [[semantic parser]]s for [[natural language understanding]].<ref name="Ovchinnikova2012">{{cite book|author=Ekaterina Ovchinnikova|title=Integration of World Knowledge for Natural Language Understanding|url=https://books.google.com/books?id=jfJUHOncFzkC&printsec=frontcover#v=onepage&q=%22logical%20form%22&f=false|date=15 February 2012|publisher=Springer Science & Business Media|isbn=978-94-91216-53-4}}</ref>

== Example of argument form ==

To demonstrate the important notion of the ''form'' of an argument, substitute letters for similar items throughout the sentences in the original argument.

;Original argument

:All humans are mortal.

:Socrates is human.

:Therefore, Socrates is mortal.

;Argument form

:All ''H'' are ''M''.

Line 28 ⟶ 27:

: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>{{citeCite book |last=Hurley |first=Patrick J. |url=https://archive.org/details/conciseintroduct00hurl_4 |title=A concise introduction to logic |publisher=Wadsworth Pub. Co.|___location=Belmont, Calif.|year=1988 |isbn=0-534-08928-3 |___location=Belmont, Calif. |url-access=registration}}</ref>

==Importance of argument form==

Attention is given to argument and sentence form, because ''form'' is what makes an argument [[Validity (logic)|valid]] or [[cogency|cogent]]''. All logical form arguments are either [[inductive reasoning|inductive]] or [[deductive reasoning|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>{{citeCite book | author1last=Bassham, Gregory | title=Critical thinking : a student's introduction | publication-date=2012 | publisher=McGraw-Hill | edition=5th | isbn=978-0-07-803831-0 |edition=5th}}</ref> Two invalid argument forms are [[affirming the consequent]] and [[denying the antecedent]].

;Affirming the consequent

Line 43 ⟶ 42:

:Therefore, Missy is not an animal.

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">{{citeCite book |authorlast=J. C. Beall|title=Logic: the Basics|url=https://books.google.com/books?id=FLnTavNvIqYC&pg=PA18 |yeartitle=2009Logic: the Basics |publisher=Taylor & Francis |year=2009 |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">{{citeCite book |authorlast=Paul Tomassi |title=Logic |url=https://books.google.com/books?id=TUVQr6InyNYC&pg=PA386 |yeartitle=1999Logic |publisher=Routledge |year=1999 |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">{{citeCite book |authorlast=Robert C. Pinto|title=Argument, inference and dialectic: collected papers on informal logic|url=https://books.google.com/books?id=eK0a5CgyV7kC&pg=PA84 |yeartitle=2001Argument, inference and dialectic: collected papers on informal logic |publisher=Springer |year=2001 |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 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:

* 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]].

*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.

==Logical forms in natural language processing==

In artificial[[semantic intelligenceparsing]], logicalstatements formsin havenatural beenlanguages usedare inconverted [[semanticinto parser]]slogical forforms [[naturalthat languagerepresent understanding]]their meanings.<ref name="Ovchinnikova2012">{{citeCite book |authorlast=Ekaterina Ovchinnikova|title=Integration of World Knowledge for Natural Language Understanding|url=https://books.google.com/books?id=jfJUHOncFzkC&printsec=frontcover#v=onepage&q=%22logical%20form+form%22&f |title=falseIntegration of World Knowledge for Natural Language Understanding |date=15 February 2012 |publisher=Springer Science & Business Media |isbn=978-94-91216-53-4}}</ref>

==See also==

*[[ {{Annotated link|Argument map]]}}

* {{Annotated link|Formal fallacy|Logical fallacy}}

*[[Fallacy]]

:*[[Formal fallacy{{Annotated link|LogicalInformal fallacy]]}}

*[[ {{Annotated link|Categorial grammar]]}}

:*[[Informal fallacy]]

*[[ {{Annotated link|Sense and reference]]}}

*[[Categorial grammar]]

*[[ {{Annotated link|Analytic–synthetic distinction]]}}

*[[Sense and reference]]

* [[Semantic argument]]

*[[Analytic–synthetic distinction]]

* [[List of valid argument forms]]

==References==

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== Further reading ==

* {{citeCite book |authorlast=Richard Mark Sainsbury |title=Logical forms: an introduction to philosophical logic|year=2001 |publisher=Wiley-Blackwell |year=2001 |isbn=978-0-631-21679-7}}

* {{citeCite book|editor=Gerhard Preyer, Georg Peter|title=Logical form and language|year=2002 |publisher=Clarendon Press |year=2002 |isbn=978-0-19-924555-0 |editor-last=Gerhard Preyer, Georg Peter}}

* {{citeCite book |authorlast=Gila Sher |title=The bounds of logic: a generalized viewpoint|year=1991 |publisher=MIT Press |year=1991 |isbn=978-0-262-19311-5}}

== External links ==

* {{PhilPapers|category|logical-form}}

* {{cite SEP |url-id=logical-form |title=Logical Form |last=Pietroski |first=Paul}}

* {{InPho|taxonomy|2236}}

* {{cite SEP |url-id=logical-form |title=Logical Form |last=Pietroski |first=Paul}}

* [http://plato.stanford.edu/entries/analysis/#6 Beaney, Michael, "Analysis", The Stanford Encyclopedia of Philosophy (Summer 2009 Edition), Edward N. Zalta (ed.)]

* [http://www.iep.utm.edu/val-snd/ IEP, Validity and Soundness]

{{Philosophy of language}}

{{logicLogic}}

{{DEFAULTSORT:Logical Form}}

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