Sentence examples for relation on the language from inspiring English sources
The only valuations that respect both these determinants are vT, assigning T to every formula, and vF assigning F to every formula, and the logic determined by {vT,vF} is the smallest trivial consequence relation on the language concerned.
The reference to subsets of Val was ill-conceived, since the point of contrast with, e.g., ⊢∨, taken as the consequence relation (on the language with sole connective ∨) determined by the class of all ∨-boolean valuations is also determined by classes of valuations which are not subsets of this class since we can include also conjunctive combinations of ∨-boolean valuations.
"We are not the first people to have this kind of initiative," said the deputy head of the Duma's committee on culture, Vladimir Bortko in relation to the language bill, Russian agencies reported.
Start with two (generalized) consequence relations each on the same language, that language having a single primitive connective, #, say.
The logician's task is to capture the consequence relation on natural language sentences, but it usually simplifies things to pay attention only to particular expressions in those sentences, just conjunction, negation and disjunction, say, or those expressions plus the universal quantifier and identity.
(The precise dual would be a consequence relation on a language with a single connective, roughly speaking giving that connective all the logical powers possessed by either disjunction or conjunction, and by a result of Wolfgang Rautenberg cited below, according to this consequence relation any formula follows from any other.
If ⊩ is such a relation on a language L then an L-valuation v is said to be consistent with ⊩ when there do not exist Γ, Δ, with v = T for all φ ∈ Γ and v = F for all ψ ∈ Δ.
"Our focus has been on seeking positive outcomes in relation both to the language, and fostering economic development".
Recall that a congruence is an equivalence relation preserving functions and relations occurring in the language.
Section 1 introduces the idea of formal sentential languages as (absolutely free) algebras, their primitive connectives having the status of the fundamental operations of these algebras, and explains the notion of a consequence relation on such a language.
