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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 ...
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.
propositional logic, Boolean algebra: The statement is true if and only if A is false. A slash placed through another operator is the same as placed in front. The prime symbol is placed after the negated thing, e.g. ′ [2]
The following example in first-order logic (=) is a sentence. This sentence means that for every y, there is an x such that =. This sentence is true for positive real numbers, false for real numbers, and true for complex numbers. However, the formula
In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Propositional logic is typically studied with a formal language, [c] in which propositions are represented by letters, which are called propositional variables. These are then used, together with symbols for connectives, to make propositional formula.
Where ψ and φ represent formulas of propositional logic, ψ is a substitution instance of φ if and only if ψ may be obtained from φ by substituting formulas for propositional variables in φ, replacing each occurrence of the same variable by an occurrence of the same formula.
The only non-logical symbols in a formal language for propositional logic are the propositional symbols, which are often denoted by capital letters. To make the formal language precise, a specific set of propositional symbols must be fixed.
In propositional logic, a propositional formula is a type of syntactic formula which is well formed. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, [1] or a sentential formula.