Logical implication I had asked this earlier but perhaps I could not put it precisely enough. 
Consider the atomic formulae $\forall xPx$ and $Pa$, and the logical axiom
$\forall xPx \rightarrow Pa$.
Can we define a function $T$, per an interpretation, from the set of sentences to $\{0,1\}$, depending on whether the sentence is true in that interpretation?
Further, if we let $T(\forall xPx)=u$ and $T(Pa) = v$, $u$ and $v$ being Boolean variables, taking values 0 or 1, depending on the interpretation, then do they always satisfy the equation,
$$
T(\forall xPx \rightarrow Pa)=[1-{u\cdot (1-v)}]=1?
$$
@Mauro:with this analysis, if our language has constants $a_1, a_2, \dots$ and we let $T(Pa_1)=v_1, \space T(Pa_2)=v_2, \dots$ then we should have, $u=c_1.v_1, \space u=c_2.v_2, \dots$ for all independent variables $v_1,v_2, \dots$ ($u$ being a Boolean function of $v1,v2, \dots$).
But the only Boolean function that can satisfy all these equations is the function that always takes only the value $0$.
How does one resolve this?
 A: The boolean interpretation of a formula with conditional-sign is :
$|P \rightarrow Q| = 1 - |P| + |P| |Q|$.
So, if $|P| = u$ and $|Q| = v$, we have that: $[1-u+uv] = [1-u(1-v)]$.
But this is NOT always 1.
A: I think again that you are missing the different use of  tautological consequence (and tautology) and of logical consequence (and validity) [please, see again Peter Smith's yesterday answer].
In sentential (or propositional) logic we can use boolean valuations and truth-tables; in first-order logic you need the more complicated notion of satisfaction, based on assignment of value to free variables.
Is a basic result of MathLog that a tautology is logically valid , but not vice versa.
If you "downsize" a first-order formula into sentential logic, you treat all the atoms (like $\forall xP(x)$) as single sentential letters. Because $\forall xP(x)$ is different from $P(a)$, when you "downsize" them you must use different sentential letters; say $A$ and $B$. 
In sentential logic, your formula becomes : $A \rightarrow B$, and this is not a tautology.
Tha first-order formula $\forall xP(x) \rightarrow P(a)$ is an axiom of first-order logic (and not of sentential logic) because, according to the standard semantic of FOL (that take in account the "more sophisticated" logical form of the formula, and this is not possible in the sentential version) it is logically valid.
