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tautologies in predicate logic∗
CWoo†
2013-03-22 3:55:32
Let FO(Σ) be a first order language over signature Σ. Recall that the axioms for FO(Σ) are (universal) generalizations of wff’s belonging to one of the
following six schemas:
1. A → (B → A)
2. (A → (B → C)) → ((A → B) → (A → C))
3. ¬¬A → A
4. ∀x(A → B) → (∀xA → ∀xB), where x ∈ V
5. A → ∀xA, where x ∈ V is not free in A
6. ∀xA → A[a/x], where x ∈ V , a ∈ V (Σ), and a is free for x in A
where V is the set of variables and V (Σ) is the set of variables and constants,
with modus ponens as its rule of inference: from A and A → B we may infer B.
The first three axiom schemas and the modus ponens tell us that predicate
logic is an extension of the propositional logic. On the other hand, we can also
view predicate logic as a part of propositional logic if we treat all quantified
formulas as atoms. This can be done precisely as follows:
Call a wff of FO(Σ) quasi-atom if it is either atomic, or of the form ∀xA,
where A is a wff of FO(Σ). Let Γ be the set of all quasi-atoms of FO(Σ).
Proposition 1. Every wff of FO(Σ) can be uniquely built up from Γ using only
logical connectives → and ¬.
Proof. Induction on the complexity of wff. For any wff A of FO(Σ), it has one
of the following forms: B → C, ¬B, or ∀xB, where B, C are wff’s. If A were
B → C or ¬B, by induction, since B and C were in Γ, A is in Γ as a result. If
A were ∀xB, then A is quasi-atomic, and therefore in Γ by the definition of Γ.
Unique readability follows from the unique readability of wff’s of propositional
logic.
∗ hTautologiesInPredicateLogici
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Since no quantifiers are involved in the built-up process, the language based
on Γ as the set of atoms can be considered as the language of propositional logic.
Call this logic PL(Γ). So the atoms of PL(Γ) are precisely the quasi-atoms of
FO(Σ).
Definition. We call a wff of FO(Σ) a tautology if it is a tautology of PL(Γ)
(via truth tables).
Proposition 2. In FO(Σ), every tautology is a theorem, but not conversely.
Proof. Suppose wff A is a tautology in FO(Σ). Then it is a tautology in PL(Γ)
by definition, and therefore a theorem in PL(Γ) since propositional logic is
complete with respect to truth-value semantics. If A1 , , An is a deduction of A
(with An = A), then each Ai is either an axiom, or is obtained by an application
of modus ponens. If Ai is an axiom (of PL(Γ)), it is an instance of one of the
first three axiom schemas of FO(Σ) above, which means that Ai is an axiom
of FO(Σ). Furthermore, since modus ponens is a rule of inference for FO(Σ),
A1 , . . . , An is a deduction of A in FO(Σ), which means that A is a theorem of
FO(Σ).
On the other hand, any wff that is an instance of one of the last two axiom
schemas of FO(Σ) is a theorem that is not a tautology. For example, ∀x(x =
y) → (z = y) is an instance of the fourth axiom schema, which takes the form
C → D, where C is a quasi-atom, and D an atom, both of which are atoms of
PL(Γ). Any truth-valuation that takes C to 1 and D to 0, takes C → D to 0.
Therefore, ∀x(x = y) → (z = y) is not a tautology.
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