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The concept of logical equivalence between formulas in propositional calculus and provides proofs for certain logical equivalences using de morgan's laws, the adequacy of connectives, and the disjunctive normal form (dnf) theorem. The document also covers the adequacy of specific sets of connectives.
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4.1 Definition Two formulas φ, ψ are logically equivalent if φ |= ψ and ψ |= φ, i.e. if for every valuation v, v˜(φ) = v˜(ψ). Notation: φ |==| ψ
Exercise φ |==| ψ if and only if |= (φ ↔ ψ)
4.2 Lemma (i) For any formulas φ, ψ
(φ ∨ ψ) |==| ¬(¬φ ∧ ¬ψ)
(ii) Hence every formula is logically equivalent to one without ‘∨’.
Proof:
(i) Either use truth tables or observe that, for any valuation v:
v^ ˜(¬(¬φ ∧ ¬ψ)) = F iff v˜((¬φ ∧ ¬ψ)) = T by tt ¬ iff v˜(¬φ) = v˜(¬ψ) = T by tt ∧ iff v˜(φ) = v˜(ψ) = F by tt ¬ iff v˜(φ ∨ ψ) = F by tt ∨
(ii) Induction on the length of the formula φ:
Clear for lenght 1
For the induction step observe that
If ψ |==| ψ′^ then ¬ψ |==| ¬ψ′
and
If φ |==| φ′^ and ψ |==| ψ′^ then (φ⋆ψ) |==| (φ′⋆ψ′),
where ⋆ is any binary connective. (Use (i) if ⋆ = ∨)
2
Let A, B, Ai be formulas. Then
So, inductively,
¬
∨^ n i=
Ai |==|
∧^ n i=
¬Ai
This is called De Morgan’s Laws.
The connectives ¬ (unary) and →, ∧, ∨, ↔ (binary) are the logical part of our language for propositional calculus.
Question:
Answer: yes
5.2 Theorem Our language L is adequate, i.e. for every n and every truth function J : Vn → {T, F } there is some φ ∈ Formn(L) with Jφ = J. (In fact, we shall only use the connectives ¬, ∧, ∨.)
Proof: Let J : Vn → {T, F } be any n-ary truth function.
If J(v) = F for all v ∈ Vn take φ := (p 0 ∧ ¬p 0 ). Then, for all v ∈ Vn: Jφ(v) = v˜(φ) = F = J(v).
Otherwise let U := {v ∈ Vn | J(v) = T } 6 = ∅. For each v ∈ U and each i < n define the for- mula
ψiv :=
{ pi if v(pi) = T ¬pi if v(pi) = F
and let ψv^ := ∧n i=0−^1 ψvi.
Then for any valuation w ∈ Vn one has the following equivalence (⋆):
w^ ˜(ψv) = T iff for all ˜^ i < n^ : w(ψiv ) = T (by tt^ ∧) iff w = v (by def. of ψiv )
Now define φ := ∨ v∈U ψv.
Then for any valuation w ∈ Vn:
w^ ˜(φ) = T iff for some v ∈ U : w˜(ψv) = T (by tt∨) iff for some v ∈ U : w = v (by (⋆)) iff w ∈ U iff J(w) = T
Hence for all w ∈ Vn: Jφ(w) = J(w), i.e. Jφ = J.
5.4 Corollary - ‘The dnf-Theorem’ For any truth function
J : Vn → {T, F }
there is a formula φ ∈ Formn(L) in dnf with Jφ = J.
In particular, every formula is logically equiva- lent to one in dnf.
5.5 Definition Suppose S is a set of (truth-functional) con- nectives – so each s ∈ S is given by some truth table.
(i) Write L[S] for the language with connec- tives S instead of {¬, →, ∧, ∨, ↔} and define Form(L[S]) and Formn(L[S]) accordingly.
(ii) We say that S is adequate (or truth func- tionally complete) if for all n ≥ 1 and for all n-ary truth functions J there is some φ ∈ Formn(L[S]) with Jφ = J.
