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The language of Propositional Calculus comprises of the logical connectives and sentential symbols $A,B,C$ etc. The sentential letters can have arbitrary semantics and truth values.

Two wff $\phi$ and $\psi$ can be 'independent' if there are truth assignments to the sentential symbols which make all four combinations TT,TF,FT,FF possible (and there is no wff $\theta$, such that $\theta(A,B) \iff (A \land B)$ are independent).

So sentential symbols $A,B$ etc are independent of each other.

If $\phi \rightarrow A$ is a tautology, then $\phi$ must be equivalent to $(A \land \psi)$ for some wff $\psi$. So no wff $\phi$ other than a contradiction can tautologically imply an infinite number of independent wffs, for than it would be an infinite conjunction.

However in first order logic, with an infinite number of constant symbols $a, b, c, ...$ in the language, we take as axioms, and hence as always true, the infinite number of wffs $\forall xP(x) \rightarrow P(a)$.

We could interpret $\forall xP(x)$, $P(a)$, etc as sentence symbols of propositional logic. So, how do we reconcile the above dichotomy?

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  • $\begingroup$ Why do you say "phi must be equivalent to (A and psi) for some wff psi", instead of just saying "phi must be equivalent to (A and phi)? $\endgroup$
    – bof
    Oct 30 '13 at 6:50
  • $\begingroup$ Why do you say that no wff phi other than a contradiction can tautologically imply an infinite number of independent wffs? If phi is any wff other than a tautology, then phi tautologically implies the independent wffs (phi or A), (phi or B), (phi or C), etc.; an infinite number of them, if your language has an infinite number of sentence symbols. What am I missing? $\endgroup$
    – bof
    Oct 30 '13 at 6:56
  • $\begingroup$ If we use the conjunctive normal forms for phi,we would see that it has the form [A and (B and C OR D and E etc]. On the second comment,I agree.the definition of independent needs to change.But I would think that phi and (phi or A ) are not independent in the sense of phi being True automatically ensures (phi or A) being True.Perhaps,I should say, no wff other than a contradiction can tautologically imply an infinite number of sentence symbols. $\endgroup$
    – Sudhir
    Oct 30 '13 at 13:01

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