Search results
Results from the WOW.Com Content Network
A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values : as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.
Several different graph invariants can be defined from the simplest sentences (with different measures of simplicity) that define a given graph. In particular the logical depth of a graph is defined to be the minimum level of nesting of quantifiers (the quantifier rank ) in a sentence defining the graph. [ 17 ]
Propositional logic, as currently studied in universities, is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered in evaluating the conditions for the truth of a sentence, or whether a sentence logically follows from some other sentence or group of sentences.
¬Q(Y) ∨ R(Y) (The variable in the second clause was renamed to make it clear that variables in different clauses are distinct.) Now, unifying Q(X) in the first clause with ¬Q(Y) in the second clause means that X and Y become the same variable anyway. Substituting this into the remaining clauses and combining them gives the conclusion:
For example, consider the following expression in which both variables are bound by logical quantifiers: ∀ y ∃ x ( x = y ) . {\displaystyle \forall y\,\exists x\,\left(x={\sqrt {y}}\right).} This expression evaluates to false if the domain of x {\displaystyle x} and y {\displaystyle y} is the real numbers, but true if the domain is the ...
A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e., without regard to any personal interpretations of the sentences) the sentence must be true if every sentence in the set is true. [3]
In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form that gives it useful properties for use in logic programming, formal specification, universal algebra and model theory.
A set of sentences is called a (first-order) theory, which takes the sentences in the set as its axioms. A theory is satisfiable if it has a model M ⊨ T {\displaystyle {\mathcal {M}}\models T} , i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set T {\displaystyle T} .