In philosophy and mathematics, a logical form of a syntactic expression is a precisely-specified semantic version of that expression in a formal system. Informally, the logical form attempts to formalize a 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; the logical form of an argument is called the argument test form of the argument. The importance of the concept of form to logic was recognized in ancient times. Aristotle, in the Prior Analytics, was the first to employ variable letters to represent valid inferences. Therefore, Łukasiewicz 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, not those given in concrete terms.
The concrete terms man, etc. are analogous to the substitution values of the schematic placeholders'A','B','C', which were called the'matter' 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, to render it explicit and pure." 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 we have done in the Argument form is to put'H' for'human' and'humans','M' for'mortal', and'S' for'Socrates'.
Moreover, each individual sentence of the Argument form is the sentence form of its respective sentence in the original argument. Attention is given to argument and sentence form, because form is what makes an argument valid or cogent. All logical form arguments are either deductive. Inductive logical forms include inductive generalization, statistical arguments, causal argument, 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, chain arguments because if the premises of the argument are true the conclusion follows. Two invalid argument forms are denying the antecedent. 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, seen as an ordered set of sentences, has a logical form that derives from the form of its constituent sentences.
Some authors only define logical form with respect to whole arguments, as the schemata or inferential structure of the argument. In argumentation theory or informal logic, 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, 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, common to the sentences'all men are mortals','all cats are carnivores','all Greeks are philosophers' and so on; 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 a subject plus a sign of quantity. 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.
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" and "is mortal": the sentence is given by the judgement A. In predicate logic, the sentence involves the same two non-logical concepts, here analyzed as m and d, the sentence is given by ∀ x, involving the logical connectives for universal quantification 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 langu
Classical logic is the intensively studied and most used class of logics. Classical logic has had much influence on analytic philosophy, the type of philosophy most found in the English-speaking world; each logical system in this class shares characteristic properties: Law of excluded middle and double negation elimination Law of noncontradiction, the principle of explosion Monotonicity of entailment and idempotency of entailment Commutativity of conjunction De Morgan duality: every logical operator is dual to anotherWhile not entailed by the preceding conditions, contemporary discussions of classical logic only include propositional and first-order logics. In other words, the overwhelming majority of time spent studying classical logic has been spent studying propositional and first-order logic, as opposed to the other forms of classical logic. Most semantics of classical logic are bivalent, meaning all of the possible denotations of propositions can be categorised as either true or false.
Classical logic is a 20th century innovation. The name does not refer to classical antiquity. In fact, classical logic was the reconciliation of Aristotle's logic, which dominated most of the last 2000 years, with the propositional Stoic logic; the two were sometimes seen as irreconcilable. Leibniz's calculus ratiocinator can be seen as foreshadowing classical logic. Bernard Bolzano has the understanding of existential import found in classical logic and not in Aristotle. Though he never questioned Aristotle, George Boole's algebraic reformulation of logic, so called Boolean logic, was a predecessor of modern mathematical logic and classical logic. William Stanley Jevons and John Venn, who had the modern understanding of existential import, expanded Boole's system; the original first-order, classical logic is found in Gottlob Frege's Begriffsschrift. It has a wider application than Aristotle's logic, is capable of expressing Aristotle's logic as a special case, it explains the quantifiers in terms of mathematical functions.
It was the first logic capable of dealing with the problem of multiple generality, for which Aristotle's system was impotent. Frege, considered the founder of analytic philosophy, invented it so as to show all of mathematics was derivable from logic, make arithmetic rigorous as David Hilbert had done for geometry, the doctrine known as logicism in the foundations of mathematics; the notation Frege used never much caught on. Hugh MacColl published a variant of propositional logic two years prior; the writings of Augustus De Morgan and Charles Sanders Peirce pioneered classical logic with the logic of relations. Peirce influenced Ernst Schröder. Classical logic reached fruition in Bertrand Russell and A. N. Whitehead's Principia Mathematica, Ludwig Wittgenstein's Tractatus Logico Philosophicus. Russell and Whitehead were influenced by Peano and Frege, sought to show mathematics was derived from logic. Wittgenstein was influenced by Frege and Russell, considered the Tractatus to have solved all problems of philosophy.
Willard Van Orman Quine insisted on classical, first-order logic as the true logic, saying higher-order logic was "set theory in disguise". Jan Łukasiewicz pioneered non-classical logic; the results of Kurt Goedel and Alfred Tarski undermined the logicist project. With the advent of algebraic logic it became apparent that classical propositional calculus admits other semantics. In Boolean-valued semantics, the truth values are the elements of an arbitrary Boolean algebra. Intermediate elements of the algebra correspond to truth values other than "true" and "false"; the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements. Warren Goldfard, "Deductive Logic", 1st edition, 2003, ISBN 0-87220-660-2
In mathematics, a theorem is a statement, proven on the basis of established statements, such as other theorems, accepted statements, such as axioms. A theorem is a logical consequence of the axioms; the proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, experimental. Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called premises. In light of the interpretation of proof as justification of truth, the conclusion is viewed as a necessary consequence of the hypotheses, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.
Although they can be written in a symbolic form, for example, within the propositional calculus, theorems are expressed in a natural language such as English. The same is true of proofs, which are expressed as logically organized and worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, from which a formal symbolic proof can in principle be constructed; such arguments are easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but explains in some way why it is true. In some cases, a picture alone may be sufficient to prove a theorem; because theorems lie at the core of mathematics, they are central to its aesthetics. Theorems are described as being "trivial", or "difficult", or "deep", or "beautiful"; these subjective judgments vary not only from person to person, but with time: for example, as a proof is simplified or better understood, a theorem, once difficult may become trivial.
On the other hand, a deep theorem may be stated but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a well-known example of such a theorem. Logically, many theorems are of the form of an indicative conditional: if A B; such a theorem does not assert B, only that B is a necessary consequence of A. In this case A is called B the conclusion; the theorem "If n is an natural number n/2 is a natural number" is a typical example in which the hypothesis is "n is an natural number" and the conclusion is "n/2 is a natural number". To be proved, a theorem must be expressible as a formal statement. Theorems are expressed in natural language rather than in a symbolic form, with the intention that the reader can produce a formal statement from the informal one, it is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses are called axioms or postulates.
The field of mathematics known as proof theory studies formal languages and the structure of proofs. Some theorems are "trivial", in the sense that they follow from definitions and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas. Other theorems have a known proof that cannot be written down; the most prominent examples are the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search, verified by a computer program. Many mathematicians did not accept this form of proof, but it has become more accepted.
The mathematician Doron Zeilberger has gone so far as to claim that these are the only nontrivial results that mathematicians have proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities. To establish a mathematical statement as a theorem, a proof is required, that is, a line of reasoning from axioms in the system to the given statement must be demonstrated. However, the proof is considered as separate from the theorem statement. Although more than one proof may be known for a single theorem, only one proof is required to establish the status of a statement as a theorem; the Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs. Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved.