TheInfoList

In
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

, logical form of a
statement Statement or statements may refer to: Common uses *Statement (computer science), the smallest standalone element of an imperative programming language *Statement (logic), declarative sentence that is either true or false *Statement, a Sentence_(lin ...
is a precisely-specified
semantic Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...
version of that statement in a
formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essen ...
. Informally, the logical form attempts to formalize a possibly
ambiguous Ambiguity is a type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty Uncertainty refers to Epistemology, epistemic sit ...
statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...
, the meaning of a logical form can be determined unambiguously from
syntax In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the ...

alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one
string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
that represents the same logical form in a given language. The logical form of an
argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
is called the argument form of the argument.

# History

The importance of the concept of form to logic was already recognized in ancient times.
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

, in the ''
Prior Analytics The ''Prior Analytics'' ( grc-gre, Ἀναλυτικὰ Πρότερα; la, Analytica Priora) is a work by Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A p ...
'', was probably the first to employ variable letters to represent valid inferences. Therefore,
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. He was born in Lemberg, a city in the Austrian Galicia, Galician Kingdom of Austria-Hungar ...

claims that the introduction of variables was "one of Aristotle's greatest inventions." According to the followers of Aristotle like 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 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 Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
in 1914, in the context of his program to formalize natural language and reasoning, which he called
philosophical logic Understood in a narrow sense, philosophical logic is the area of philosophy that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophi ...
. 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."preprint
/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''. :''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.

# Importance of argument form

Attention is given to argument and sentence form, because ''form'' is what makes an argument 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 In classical logic Classical logic (or standard logic) is the intensively studied and most widely used class of deductive logic Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach ...
,
categorical syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a Logical consequence, conclusion based on two or more propositions that are asser ...
, argument by definition, argument based on mathematics, argument from definition. The most reliable forms of logic are
modus ponens In propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from ...

,
modus tollens In propositional calculus, propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo wiktionary:tollens, tollens'' (Latin language, Latin for "method of removing by taking away") and denying the consequent, is a Deductive r ...
, and chain arguments because if the premises of the argument are true, then the conclusion necessarily follows. Two invalid argument forms are
affirming the consequent Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy In philosophy Philosophy (from , ) is the study of general and fundamental questions, such ...
and
denying the antecedent Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, ex ...
. ;Affirming the consequent :All dogs are animals. :Coco is an animal. :Therefore, Coco is a dog. ;Denying the antecedent :All cats are animals. :Missy is not a cat. :Therefore, Missy is not an animal. A logical
argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
, seen as an
ordered set Image:Hasse diagram of powerset of 3.svg, upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparab ...
of sentences, has a logical form that derives from the form of its constituent sentences; the logical form of an argument is sometimes called argument form. Some authors only define logical form with respect to whole arguments, as the schemata or inferential structure of the argument. In
argumentation theory Two men argue at a political protest in New York City. Argumentation theory, or argumentation, is the interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one ac ...
or
informal logic Informal logic encompasses the principles of logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: g ...
, an argument form is sometimes seen as a broader notion than the logical form. 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 In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label= ...
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, 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 First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses Quantifica ...
, the sentence involves the same two non-logical concepts, here analyzed as $m\left(x\right)$ and $d\left(x\right)$, and the sentence is given by $\forall x \left(m\left(x\right) \rightarrow d\left(x\right)\right)$, involving the logical connectives for
universal quantification In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
and 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 In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
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

In semantic parsing, statements in natural languages are converted into logical forms that represent their meanings.

*
Argument map An argument map or argument diagram is a visual representation of the structure of an argument In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical ...

*
Fallacy A fallacy is the use of invalid or otherwise faulty reason Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of ...
:*
Logical fallacy In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, lang ...
:*
Informal fallacy Informal fallacies are a type of incorrect argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Gre ...
*
Categorial grammarCategorial grammar is a family of formalisms in natural language syntax In linguistics, syntax () is the set of rules, principles, and processes that govern the structure of Sentence (linguistics), sentences (sentence structure) in a given Natur ...
*
Sense and reference In the philosophy of language In analytic philosophy, philosophy of language investigates the nature of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed lan ...
*
Analytic–synthetic distinction The analytic–synthetic distinction is a semantic distinction, used primarily in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowle ...
*
List of valid argument forms Of the many and varied argument forms that can possibly be constructed, only very few are valid argument forms. In order to evaluate these forms, Statement (logic), statements are put into logical form. Logical form replaces any sentences or ideas w ...