Logic Definitions

xokihele's version from 2017-06-15 15:27

Section 1

Question Answer
Reflexivityiff for all d in S the pair ⟨d, d⟩ is an element of R.
Functioniff for all d, e, f: if ⟨d, e⟩ ∈ R and ⟨d, f ⟩ ∈ R then e = f .
Equivalence Relationiff R is reflexive on S, symmetric on S and transitive on S.
Transitivityiff for all d, e, f, if ⟨d, e⟩ ∈ R and ⟨e, f ⟩ ∈ R, then also ⟨d, f ⟩ ∈ R
Seta collection of objects, the elements of the set.
Binary Relationiff it contains only ordered pairs.
Symmetryiff for all d, e of S: if ⟨d, e⟩ ∈ R then ⟨e, d⟩ ∈ R.
Asymmetryiff for no elements d, e of S, if ⟨d, e⟩ ∈ R then ⟨e, d⟩ ∈ R.
Antisymmetryiff for no two distinct elements d, e of S: if ⟨d, e⟩ ∈ R then ⟨e, d⟩ ∈ R.

Section 2

Question Answer
Logical Validityiff there is no interpretation under which the premisses are all true and the conclusion is false.
Logical Entailment (Double Turnstile)if the set of sentences on the left are true, the sentence on the right must be true
Truth Functionaliff the truth-value of the compound sentence cannot be changed by replacing a direct subsentence with another having the same truth-value.
Semantic Consistencyiff there is an L1 structure under which all sentences of Γ are true.
Tautology (Logical Truth)iff: φ is true under all L1 structures, true under any interpretation
Contradictioniff: φ is not true under any L1 structure, false under any interpretation.
Logical Equivalenceiff: φ and ψ are true in exactly the same L1 structures.