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For the interpretation of formulas, consider these structures: the positive real numbers, the real numbers, and complex numbers. The following example in first-order logic (=) is a sentence. This sentence means that for every y, there is an x such that =.
For example, primitive signs must permit expression of the concepts to be modeled; sentential formulas are chosen so that their counterparts in the intended interpretation are meaningful declarative sentences; primitive sentences need to come out as true sentences in the interpretation; rules of inference must be such that, if the sentence is ...
A propositional formula may also be called a propositional expression, a sentence, [1] or a sentential formula. A propositional formula is constructed from simple propositions, such as "five is greater than three" or propositional variables such as p and q, using connectives or logical operators such as NOT, AND, OR, or IMPLIES; for example:
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning.
A well-formed formula is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar. The language L {\displaystyle {\mathcal {L}}} , then, is defined either as being identical to its set of well-formed formulas, [ 48 ] or as containing that set (together with ...
A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable. If A is a formula of a first-order language in which the variables v 1, …, v n have free occurrences, then A preceded by ∀v 1 ⋯ ∀v n is a universal closure of A.
Let ψ be a sentence in first-order logic.The spectrum of ψ is the set of natural numbers n such that there is a finite model for ψ with n elements.. If the vocabulary for ψ consists only of relational symbols, then ψ can be regarded as a sentence in existential second-order logic (ESOL) quantified over the relations, over the empty vocabulary.
A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are φ {\displaystyle \varphi } (or φ ( x ) {\displaystyle \varphi (x)} to indicate x {\displaystyle x} is the unbound variable in φ {\displaystyle \varphi } ) and ψ {\displaystyle \psi } (or ψ ( x ...