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}}

The ''form'' or '''logical form''' of an argument is the representation of its sentences using the formal grammar and symbolism of a logical system to display its similarity with all other arguments of the same type.

{{About|the term as used in logic|the linguistics term|Logical form (linguistics)|the term used in knowledge representation|Logic form}}

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

In [[logic]], the '''logical form''' of a [[Statement (logic)|statement]] is a precisely specified [[Semantics|semantic]] version of that statement in a [[formal system]]. Informally, the logical form attempts to [[Logic translation#Natural language formalization|formalize]] a possibly [[Syntactic 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>

In consists in stripping out all spurious grammatical features elements from the sentence (such as gender, and passive forms), and replacing all the expressions specific to ''the subject matter'' of the argument by schematic variables. 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.

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

== History ==

== History ==

ThatThe importance of the concept of form is fundamental 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 (in the [[Prior analytics]]).{{Citation needed|date=December (For2024}} which reasonTherefore, [[Jan Łukasiewicz | Łukasiewicz]] saysclaims that the introduction of variables was '"one of Aristotle's greatest inventions')."{{Citation needed|date=December 2024}}

According to the followers of Aristotle (such aslike [[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'', and so &cforth 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>

== more to follow ==

== Example of argument form ==

An open question in the philosophy of logic concerns the "ontological status" of logical form: in this, two questions arise: whether logical form corresponds to entities and whether any one logical form should be privileged as being in best correspondence to the world.

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

Wittgenstein's Tractatus was the most rigorous exposition of a case for the standing of logical form as grounded in ontology, because in the Tractatus the substance of the world corresponds to logical form. That is, because we can recognize in schematics such as "all a is b" the necessity for a myriad of atomic facts, because generalizations can be made from them if a logical form is followed, the world contains not one thing but many facts.

:All humans are mortal.

:Socrates is human.

:Therefore, Socrates is mortal.

;Argument form

Wittgenstein, however, was almost immediately misinterpreted as having knowledge-that these facts corresponded to atomic sense data reports when a close reading of the Tractatus gives grounds only for a terribly abstract belief in some sort of multiplicity of substance, and not for its realization as sense data reports.

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

:''S'' is ''H''.

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

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

More recently, work by Quine et al. privileges certain kinds of logic, mostly two-valued, as having a higher ontological status than newer forms of logic such as "fuzzy" or "three-valued" logic. The overall critique of such "deviant" logics shows that they are notationally reducible to special-purpose tools for talking about an underlying two-valued metaphysic.

==Importance of argument form==

For example, the "three-valued" logic of true, false, or "indeterminate" is reduced in this critique to a two-valued way of talking about knowledge and as such not as ontologically significant as two-valued logic.

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 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>{{Cite book |last=Bassham, Gregory |title=Critical thinking : a student's introduction |date=2012 |publisher=McGraw-Hill |isbn=978-0-07-803831-0 |edition=5th}}</ref> Two invalid argument forms are [[affirming the consequent]] and [[denying the antecedent]].

;Affirming the consequent

The conservative view finds significance, in some cases, even in the fact that parentheses must be used and it reports, in some cases, that their elimination in Polish logic doesn't eliminate the need to evaluate operators "in order", a fact or factoid which in the conservative view corresponds to an ontological necessity.

:All dogs are animals.

:Coco is an animal.

:Therefore, Coco is a dog.

;Denying the antecedent

An empirical fact which lends support to Quine's view is that all or most digital computers happen to have two states in their memory units and that DIGITAL computers constructed of n-value elements are rarities. In response, the deviant logician would remark that actual people deal in 8 bit bytes and that older "analog" machines were in a modern reading, fuzzy or infinite valued machines.

:All cats are animals.

:Missy is not a cat.

: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">{{Cite book |last=J. C. Beall |url=https://books.google.com/books?id=FLnTavNvIqYC&pg=PA18 |title=Logic: 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 schemata or inferential structure of the argument.<ref name="Tomassi1999">{{Cite book |last=Paul Tomassi |url=https://books.google.com/books?id=TUVQr6InyNYC&pg=PA386 |title=Logic |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">{{Cite book |last=Robert C. Pinto |url=https://books.google.com/books?id=eK0a5CgyV7kC&pg=PA84 |title=Argument, inference and dialectic: collected papers on informal logic |publisher=Springer |year=2001 |isbn=978-0-7923-7005-5 |page=84}}</ref>

Dangerously, the two-valued conservative view becomes ethnocentric, because it privileges the greater proximity, say, of Russell's notation of Principia Mathematica, to "reality" for no good reason.

InIt consists inof stripping out all spurious grammatical features elements from the sentence (such as gender, and passive forms), and replacing all the expressions specific to ''the subject matter'' of the argument by [[schematic variablesvariable]]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.

However, the fact remains that a speaker of an n-valued language all the way down (in which the concepts of truth and falsity become, let us say, lime, coconut, lime in de coconut, and drink and fold up in a four valued logic) can after some pain communicate with a well-meaning two-valued missionary.

==Logical form in modern logic==

This is what is meant by "logical form", a ground of communication. The theory of translation and conflicts of law both demand a clearer answer in an era of globalization from philosophical logic as to whether, for example, a language of guilty versus not guilty, isomorphic to true and false by a simple transformation, is applicable when either translating texts or in international law.

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.

Quine was clearly concerned that removing Aristotelean logic, and its direct successor Western modern logic, from centrality, would open a Pandora's box of mutual incomprehensibility (as in the case of the Scots law verdicts guilty, not guilty and unproven). But he may have been just confused by the fact that as soon as a common minimal logical form is agreed-upon, whether consciously by philosophers or unconsciously by lawyers and such riff raff, it in turn is spoken-of immediately in the philosopher's native tongue, which appears to return privilege to a particular way of speaking.

* 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.

Only a Kantian purity of heart can remind itself that "all a is b", (x)(a(x) -> b(x)), or lime in the coconut are each just syntactical paths to a reality which, if it exists, is Taoist and in this way prior and unspeakable. Wittgenstein had this purity of heart ("wovon Mann nicht sprechen kann, daruber must Mann schweigen), but Quine's genuine purity became to some readers ill-temper.

==Logical forms in natural language processing==

We can, sort of, teach logic. But part of its difficulty is the fact that the is-ness of logical form is invisible to lawyers and others who are in fact able to construct valid arguments and criticise invalid arguments, who have received B in Logic and who consider it useless. For most people, once drained of color and life, a and b, a or b, a implies b, become husks of meaning. Their only possible interest lies in deep translation and mutual understanding.

In [[semantic parsing]], statements in natural languages are converted into logical forms that represent their meanings.<ref name="Ovchinnikova2012">{{Cite book |last=Ekaterina Ovchinnikova |url=https://books.google.com/books?id=jfJUHOncFzkC&q=%22logical+form%22 |title=Integration 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}}

* {{Annotated link|Informal fallacy}}

* {{Annotated link|Categorial grammar}}

* {{Annotated link|Sense and reference}}

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

* [[Semantic argument]]

* [[List of valid argument forms]]

==References==

{{reflist}}

== Further reading ==

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

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

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

== External links ==

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

* {{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.)]

{{Philosophy of language}}

{{Logic}}

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