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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:
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 ...
These requirements make the ω-rule sound in every ω-model. As a corollary to the omitting types theorem, the converse also holds: the theory T has an ω-model if and only if it is consistent in ω-logic. There is a close connection of ω-logic to ω-consistency. A theory consistent in ω-logic is also ω-consistent (and arithmetically sound).
A logical principle that states that a conditional statement is logically equivalent to its contrapositive, transforming "If P, then Q" into "If not Q, then not P". contrapositive The statement resulting from swapping the antecedent and consequent of a conditional statement and negating both, maintaining logical equivalence. contrary
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.
The simplest constituents are atomic sentences. A contemporary semantic definition of truth would define truth for the atomic sentences as follows: An atomic sentence F(x 1,...,x n) is true (relative to an assignment of values to the variables x 1, ..., x n)) if the corresponding values of variables bear the relation expressed by the predicate F.
Hence logically equivalent sentences are often identified. A state description for a finite set of constants is a conjunction of atomic sentences (predicates or their negations) instantiated exclusively by these constants, such that for any eligible atomic sentence either it or its negation (but not both) appears in the conjunction.