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Material Implication (rule Of Inference)
In propositional logic, material implication [1][2] is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated
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Carl Cohen (professor)
Carl Cohen (born April 30, 1931) is Professor of Philosophy
Philosophy
at the Residential College of the University of Michigan, in Ann Arbor, Michigan, USA. He is co-author of "The Animal Rights Debate" (Rowman and Littlefield, 2001),[1] a point-counterpoint volume with Prof. Tom Regan; he is also the author of "Democracy" (Macmillan, 1972); the author of "Four Systems" (Random House, 1982); the editor of "Communism, Fascism, and Democracy" (McGraw Hill, 1997); the co-author (with J. Sterba) of "Affirmative Action and Racial Preference" (Oxford, 2003), co-author (with I. M. Copi) of "Introduction to Logic, 13th edition" (Prentice-Hall, 2008), and author of "A Conflict of Principles: The Battle over Affirmative Action at the University of Michigan" (University Press of Kansas, 2014). He has published many essays in moral and political philosophy in philosophical, medical, and legal journals
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Formal System
A formal system or logical calculus is any well-defined system of abstract thought based on the model of mathematics. A formal system need not be mathematical as such; for example, Spinoza's Ethics imitates the form of Euclid's Elements
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Antecedent (logic)
An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause. In some contexts the antecedent is called the protasis.[1] Examples:If P displaystyle P , then Q displaystyle Q .This is a nonlogical formulation of a hypothetical proposition. In this case, the antecedent is P, and the consequent is Q
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Logical Disjunction
In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is typically written as ∨ or +. "A or B" is true if A is true, or if B is true, or if both A and B are true. In logic, or by itself means the inclusive or, distinguished from an exclusive or, which is false when both of its arguments are true, while an "or" is true in that case. An operand of a disjunction is called a disjunct. Related concepts in other fields are:In natural language, the coordinating conjunction "or". In programming languages, the short-circuit or control structure. In set theory, union. In predicate logic, existential quantification.Contents1 Notation 2 Definition2.1 Truth table3 Properties 4 Symbol 5 Applications in computer science5.1 Bitwise operation 5.2 Logical oper
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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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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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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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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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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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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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Logical Equivalence
In logic, statements p displaystyle p and q displaystyle q are logically equivalent if they have the same logical content. This is a semantic concept; two statements are equivalent if they have the same truth value in every model (Mendelson 1979:56). The logical equivalence of p displaystyle p and q displaystyle q is sometimes expressed as p ≡ q displaystyle pequiv q , E p q displaystyle textsf E pq , or p ⟺ q displaystyle piff q . However, these symbols are also used for material equivalence; the proper interpretation depends on the context
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Logical Consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?[1] All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.[2] Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation.[1] A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e
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Tautology (logic)
In logic, a tautology (from the Greek word ταυτολογία) is a formula or assertion that is true in every possible interpretation. A simple example is "(x equals y) or (x does not equal y)" (or as a less abstract example, "The ball is green or the ball is not green"). Philosopher
Philosopher
Ludwig Wittgenstein
Ludwig Wittgenstein
first applied the term to redundancies of propositional logic in 1921. (It had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternative sense.) A formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent
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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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Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition p to another proposition "not p", written ¬p, which is interpreted intuitively as being true when p is false, and false when p is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa
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