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Logical connectives can be used to link zero or more statements, so one can speak about n-ary logical connectives. The boolean constants True and False can be thought of as zero-ary operators. Negation is a unary connective, and so on.
The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question. [2] [6] [8] The connective with the largest scope in a formula is called its dominant connective, [9] [10] main connective, [6] [8] [7] main operator, [2] major connective, [4] or principal connective; [4] a connective within the scope of another connective ...
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier ∀ {\displaystyle \forall } in the first order formula ∀ x P ( x ) {\displaystyle \forall xP(x)} expresses that everything in the domain satisfies the property denoted by P ...
In logic, a logical constant or constant symbol of a language is a symbol that has the same semantic value under every interpretation of . Two important types of logical constants are logical connectives and quantifiers .
Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems.
The connectives are usually taken to be logical constants, meaning that the meaning of the connectives is always the same, independent of what interpretations are given to the other symbols in a formula. This is how we define logical connectives in propositional logic: ¬Φ is True iff Φ is False. (Φ ∧ Ψ) is True iff Φ is True and Ψ is True.
A modal connective (or modal operator) is a logical connective for modal logic.It is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-truth-functional in the following sense: The truth-value of composite formulae sometimes depend on factors other than the actual truth-value of their components.
A process in logical deduction where quantifiers are removed from logical expressions while preserving equivalence, often used in the theory of real closed fields. elimination rule A rule in logical inference that allows the derivation of simpler formulas from more complex ones, often by removing logical connectives or quantifiers. empty concept
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