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In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of p {\displaystyle p} and q {\displaystyle q} is sometimes expressed as p ≡ q {\displaystyle p\equiv q} , p :: q {\displaystyle p::q} , E p q {\displaystyle {\textsf {E}}pq} , or p q ...
For example, the sentence "'Snow is white' is true" becomes materially equivalent with the sentence "snow is white", i.e. 'snow is white' is true if and only if snow is white. Said again, a sentence of the form "A" is true if and only if A is true. The truth of more complex sentences is defined in terms of the components of the sentence:
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
Some examples of convenient definitions drawn from the symbol set { ~, &, (, ) } and variables. Each definition is producing a logically equivalent formula that can be used for substitution or replacement. definition of a new variable: (c & d) = Df s; OR: ~(~a & ~b) = Df (a ∨ b) IMPLICATION: (~a ∨ b) = Df (a → b) XOR: (~a & b) ∨ (a & ~b ...
In some logical calculi (notably, in classical logic), certain essentially different compound statements are logically equivalent. A less trivial example of a redundancy is the classical equivalence between ¬ p ∨ q {\displaystyle \neg p\vee q} and p → q {\displaystyle p\to q} .
logically equivalent Referring to statements that have the same truth value in every possible scenario, indicating that they are interchangeable in logical reasoning. logicism The philosophical belief that mathematics can be reduced to logic and that all mathematical truths can be derived from logical axioms and definitions. Löwenheim–Skolem ...
Venn diagram of (true part in red) In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement "if and only if" (often abbreviated as "iff " [1]), where is known as the antecedent, and the consequent.