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Rule Of Replacement
In logic, a rule of replacementMoore and Parker is a transformation rule that may be applied to only a particular segment of an logical expression, expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for Well-formed formula, logical expressions in the system. Whereas a rule of inference is always applied to a whole logical expression, a rule of replacement may be applied to only a particular segment. Within the context of a logical proof, logically equivalent expressions may replace each other. Rules of replacement are used in propositional logic to manipulate propositions. Common rules of replacement include de Morgan's laws, Commutative property, commutation, Associative property, association, Distribution (logic), distribution, double negation, Transposition (logic), transposition, Material implication (rule of inference), material implication, logical equivalence, Exportation (logic), exportation, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Property
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or Quantifier (logic), quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (logical conjunction, conjunction), "or" (lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Salva Veritate
The literal translation of the Latin "''salva veritate''" is "with (or by) unharmed truth", using ablative of manner: "''salva''" meaning "rescue," "salvation," or "welfare," and "''veritate''" meaning "reality" or "truth". Thus, ''Salva veritate'' (or intersubstitutivity) is the logical condition by which two expressions may be interchanged without altering the truth-value of statements in which the expressions occur. Substitution ''salva veritate'' of co-extensional terms can fail in opaque contexts. Leibniz The phrase occurs in two fragments from Gottfried Leibniz's ''General Science. Characteristics'': * In Chapter 19, Definition 1, Leibniz writes: "Two terms are the same (''eadem'') if one can be substituted for the other ''without altering the truth of any statement'' (''salva veritate'')." * In Chapter 20, Definition 1, Leibniz writes: "Terms which can be substituted for one another wherever we please ''without altering the truth of any statement'' (''salva veritate''), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (rule Of Inference)
In propositional logic, tautology is either of two commonly used rules of replacement. The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs. They are: The principle of idempotency of disjunction: : P \lor P \Leftrightarrow P and the principle of idempotency of conjunction: : P \land P \Leftrightarrow P Where "\Leftrightarrow" is a metalogical symbol representing "can be replaced in a logical proof with." Formal notation Theorems are those logical formulas \phi where \vdash \phi is the conclusion of a valid proof,Logic in Computer Science, ''p. 13'' while the equivalent semantic consequence \models \phi indicates a tautology. The ''tautology'' rule may be expressed as a sequent: : P \lor P \vdash P and : P \land P \vdash P where \vdash is a metalogical symbol meaning that P is a syntactic consequence of P \lor P, in the one case, P \land P in the other, in some logical system; or as a rule of inference: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exportation (logic)
Exportation is a Validity (logic), valid rule of replacement in propositional logic. The rule allows material conditional, conditional statements having Logical conjunction, conjunctive antecedent (logic), antecedents to be replaced by statements having conditional consequents and vice versa in formal proof, logical proofs. It is the rule that: ((P \land Q) \to R) \Leftrightarrow (P \to (Q \to R)) Where "\Leftrightarrow" is a metalogical Symbol (formal), symbol representing "can be replaced in a proof with." In strict terminology, ((P \land Q) \to R) \Rightarrow (P \to (Q \to R)) is the law of exportation, for it "exports" a proposition from the antecedent of (P \land Q) \to R to its consequent. Its converse, the law of importation, (P \to (Q \to R))\Rightarrow ((P \land Q) \to R) , "imports" a proposition from the consequent of P \to (Q \to R) to its antecedent. Formal notation The ''exportation'' rule may be written in sequent notation: :((P \land Q) \to R) \dashv\vdash (P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Equivalence
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related. Logical equivalences In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these. General logical equivalences Logical equivalences involving conditional statements :#p \implies q \equiv \neg p \vee q :#p \implies q \equiv \neg q \implies \neg p :#p \vee q \equiv \neg p \implies q :#p \wedge q \equiv \neg (p \implies \neg q) :#\neg (p \implies q) \equiv p \wedge \neg q :#(p \implies q) \wedge (p \implie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Implication (rule Of Inference)
In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that ''P implies Q'' is logically equivalent to ''not-P or Q'' and that either form can replace the other in logical proofs. In other words, if P is true, then Q must also be true, while if Q is true, then P cannot be true either; additionally, when P is not true, Q may be either true or false. P \to Q \Leftrightarrow \neg P \lor Q Where "\Leftrightarrow" is a metalogical symbol representing "can be replaced in a proof with," and P and Q are any given logical statements. To illustrate this, consider the following statements: * P: Sam ate an orange for lunch * Q: Sam ate a fruit for lunch Then, to say, "Sam ate an orange for lunch" "Sam ate a fruit for lunch" (P \to Q). Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transposition (logic)
In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "''A'' implies ''B''" to the truth of "Not-''B'' implies not-''A''", and conversely. It is very closely related to the rule of inference modus tollens In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. ''Modus tollens' .... It is the rule that (P \to Q) \Leftrightarrow (\neg Q \to \neg P) where "\Leftrightarrow" is a metalogical Symbol (formal), symbol representing "can be replaced in a proof with". Formal notation The ''transposition'' rule may be expressed as a sequent: :(P \to Q) \vdash (\neg Q \to \neg P) where \vdash is a metalogical symbol mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, but it is disallowed by intuitionistic logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in ''Principia Mathematica'' as: :: \mathbf. \ \ \vdash.\ p \ \equiv \ \thicksim(\thicksim p)PM 1952 reprint of 2nd edition 1927 pp. 101–02, 117. ::"This is the principle of double negation, ''i.e.'' a proposition is equivalent of the falsehood of its negation." Elimination and introduction Double negation elimination and double negation introduction are two valid rules of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (logic)
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, one has 2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3). One says that multiplication ''distributes'' over addition. This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted \,\land\,) and the logical or (denoted \,\lor\,) distributes over the other. Definition Given a set S and two binary operators \,*\, and \,+\, on S, *the operation \,*\, is over (or with respect to) \,+\, if, given any elements x, y, \text z of S, x * (y + z) = (x * y) + (x * z); *the operation \,*\, is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
