Consequence Relations and Natural Deduction

1 Consequence Relations A useful notion that cuts across both semantic (model-oriented) and syntactic (derivation-oriented) issues is the notion of a consequence relation. We will use capital Greek letters like Γ,∆ (Gamma and Delta) to refer to finite sets of formulas, and lower case Greek letters like φ, ψ (phi and psi) to refer to individual formulas. We will save ink by writing Γ,∆ for the set Γ ∪∆, and Γ, φ for the set Γ ∪ {φ}, etc. By φ[t1/x1, . . . tn/xn], we mean the result of plugging in the terms t1, . . . , tn in place of the variables x1, . . . , xn. We assume that all the xi are different variables, and that all of the plugging in happens at once. So, if there are