Mathematical Logic
... and the elimination axiom is Eq− : ∀xR(x, x) → Eq(x, y) → R(x, y). It is an easy exercise to show that the usual equality axioms can be derived. All these axioms can be seen as special cases of a general scheme, that of an inductively defined predicate, which is defined by some introduction rules an ...
... and the elimination axiom is Eq− : ∀xR(x, x) → Eq(x, y) → R(x, y). It is an easy exercise to show that the usual equality axioms can be derived. All these axioms can be seen as special cases of a general scheme, that of an inductively defined predicate, which is defined by some introduction rules an ...
The substitutional theory of logical consequence
... of these models. Models have set-sized domains, while the intended interpretation, if it could be conceived as a model, cannot be limited by any cardinality. Similarly, logical truth defined as truth in all models does not imply truth simpliciter. If logical truth is understood as truth under all in ...
... of these models. Models have set-sized domains, while the intended interpretation, if it could be conceived as a model, cannot be limited by any cardinality. Similarly, logical truth defined as truth in all models does not imply truth simpliciter. If logical truth is understood as truth under all in ...
1. Propositional Logic 1.1. Basic Definitions. Definition 1.1. The
... braces: φ, ψ ⇒ γ, δ or Γ ⇒ φ. We will also use juxtaposition to abbreviate union; that is Γ∆ ⇒ ΣΥ abbreviates Γ ∪ ∆ ⇒ Σ ∪ Υ and similarly Γ, φ, ψ ⇒ Σ, γ abbreviates Γ ∪ {φ, ψ} ⇒ Σ ∪ {γ}. When Γ is empty, we simply write ⇒ Σ, or (when it is clear from context that we are discussing a sequent) sometim ...
... braces: φ, ψ ⇒ γ, δ or Γ ⇒ φ. We will also use juxtaposition to abbreviate union; that is Γ∆ ⇒ ΣΥ abbreviates Γ ∪ ∆ ⇒ Σ ∪ Υ and similarly Γ, φ, ψ ⇒ Σ, γ abbreviates Γ ∪ {φ, ψ} ⇒ Σ ∪ {γ}. When Γ is empty, we simply write ⇒ Σ, or (when it is clear from context that we are discussing a sequent) sometim ...
Martin-Löf`s Type Theory
... The notion of constructive proof is closely related to the notion of computer program. To prove a proposition (∀x ∈ A)(∃y ∈ B)P (x, y) constructively means to give a function f which when applied to an element a in A gives an element b in B such that P (a, b) holds. So if the proposition (∀x ∈ A)(∃y ...
... The notion of constructive proof is closely related to the notion of computer program. To prove a proposition (∀x ∈ A)(∃y ∈ B)P (x, y) constructively means to give a function f which when applied to an element a in A gives an element b in B such that P (a, b) holds. So if the proposition (∀x ∈ A)(∃y ...
