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Double Negation
In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.[1] Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic,[2] but it is disallowed by intuitionistic logic.[3] The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as: ∗ 4 ⋅ 13 .     ⊢ .   p   ≡   ∼ ( ∼ p ) displaystyle mathbf *4cdot 13 . vdash
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Law Of The Excluded Middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for any proposition, either that proposition is true or its negation is true. It is the third of the three classic laws of thought. The law is also known as the law (or principle) of the excluded third, in Latin
Latin
principium tertii exclusi. Another Latin
Latin
designation for this law is tertium non datur: "no third [possibility] is given". The earliest known formulation is in Aristotle's discussion of the principle of non-contradiction, first proposed in On Interpretation,[1] where he says that of two contradictory propositions (i.e
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Sequent
In mathematical logic, a sequent is a very general kind of conditional assertion. A 1 , … , A m ⊢ B 1 , … , B n . displaystyle A_ 1 ,,dots ,A_ m ,vdash ,B_ 1 ,,dots ,B_ n . A sequent may have any number m of condition formulas Ai (called "antecedents") and any number n of asserted formulas Bj (called "succedents" or "consequents"). A sequent is understood to mean that if all of the antecedent conditions are true, then at least one of the consequent formulas is true
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Symbol (formal)
A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern.[citation needed] Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in the formal languages studied in mathematics and logic, the term "symbol" refers to the idea, and the marks are considered to be a token instance of the symbol.[dubious – discuss] In logic, symbols build literal utility to illustrate ideas.Contents1 Overview 2 Can words be modeled as formal symbols? 3 References 4 See alsoOverview[edit] Symbols of a formal language need not be symbols of anything. For instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses)
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Metalogic
Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.[1] Logic
Logic
concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.[2] The basic objects of metalogical study are formal languages, formal systems, and their interpretations
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Formal Proof
A formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of natural deduction is a generalization of the concept of proof.[1] The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof
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Converse (logic)
In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. For the implication P → Q, the converse is Q → P. For the categorical proposition All S are P, the converse is All P are S. In neither case does the converse necessarily follow from the original statement.[1]Contents1 Implicational converse1.1 Converse of a theorem2 Categorical converse 3 See also 4 References 5 Further readingImplicational converse[edit] Let S be a statement of the form P implies Q (P → Q). Then the converse of S is the statement Q implies P (Q → P)
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Inference
Inferences are steps in reasoning, moving from premises to conclusions. Charles Sanders Peirce
Charles Sanders Peirce
divided inference into three kinds: deduction, induction, and abduction. Deduction is inference deriving logical conclusions from premises known or assumed to be true,[1] with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. Abduction is inference to the best explanation. Human inference (i.e. how humans draw conclusions) is traditionally studied within the field of cognitive psychology; artificial intelligence researchers develop automated inference systems to emulate human inference. Statistical inference uses mathematics to draw conclusions in the presence of uncertainty. This generalizes deterministic reasoning, with the absence of uncertainty as a special case
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Validity
In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.[1] It is not required that a valid argument have premises that are actually true,[2] but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.Contents1 Arguments 2 Valid formula 3 Statements 4 Soundness 5 Satisfiability 6 Preservation 7 See also 8 References 9 Further readingArguments[edit] An argument is valid if and only if the truth of its premises entails the truth of its conclusion and each step, sub-argument, or logical operation in the argument is valid. Under such conditions it would be self-contradictory to affirm the premises and deny the conclusion
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Principia Mathematica
Hardy, G. H. (2004) [1940]. A Mathematician's Apology. Cambridge: University Press. p. 83. ISBN 978-0-521-42706-7. He [Russell] said once, after some contact with the Chinese language, that he was horrified to find that the language of Principia Mathematica was an Indo-European oneLittlewood, J. E. (1985). A Mathematician's Miscellany. Cambridge: University Press
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Alfred North Whitehead
Alfred North Whitehead
Alfred North Whitehead
OM FRS FBA (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy,[19] which today has found application to a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology, among other areas. In his early career Whitehead wrote primarily on mathematics, logic, and physics. His most notable work in these fields is the three-volume Principia Mathematica
Principia Mathematica
(1910–13), which he wrote with former student Bertrand Russell
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Predicate Logic
First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic
First-order logic
uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists X such that X is Socrates and X is a man" and there exists is a quantifier while X is a variable.[1] This distinguishes it from propositional logic, which does not use quantifiers or relations.[2] A theory about a topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold for those things
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Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally 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 often 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, which is experimental.[2] Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises
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Law Of Thought
The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally they are taken as laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc. However such classical ideas are often questioned or rejected in more recent developments, such as intuitionistic logic, dialetheism and fuzzy logic. According to the 1999 Cambridge Dictionary of Philosophy,[1] laws of thought are laws by which or in accordance with which valid thought proceeds, or that justify valid inference, or to which all valid deduction is reducible. Laws of thought are rules that apply without exception to any subject matter of thought, etc.; sometimes they are said to be the object of logic
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Double Negative
A double negative is a grammatical construction occurring when two forms of negation are used in the same sentence. Multiple negation is the more general term referring to the occurrence of more than one negative in a clause. In some languages, double negatives cancel one another and produce an affirmative; in other languages, doubled negatives intensify the negation. Languages where multiple negatives affirm each other are said to have negative concord or emphatic negation.[1] Portuguese, Persian, Russian, Spanish, Neapolitan, Italian, Bulgarian, Czech, Polish, Hebrew, and some dialects of English are examples of negative-concord languages,[citation needed] while Latin and German do not have negative concord
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Classical Logic
Classical logic (or standard logic[1][2]) is an intensively studied and widely used class of formal logics
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