Why the following is not the case: $p(x)\vDash (\forall x)p(x)$?

Question

In my textbook they are arguing that this is not the case: $p(x)\vDash(\forall x) p(x)$ . Till now I have thought that predicate p(x) has a general meaning and therefore it is the same as $(\forall x) p(x)$ . What is the difference between those two than?

Answer 1

Consider e.g. E.Mendelson, Introduction to Mathematical Logic, page 62:

$\mathscr B$ is said to logically imply $\mathscr C$ [$\mathscr B \vDash \mathscr C$] if and only if, in every interpretation, every sequence that satisfies $\mathscr B$ also satisfies $\mathscr C$.

Consider the simple example with $x=0$ as $p(x)$ and interpret it in the domain $\mathbb N$ of natural numbers.

Clearly, we have that $(\forall x)(x=0)$ does not hold in it.

This is formalized by the semantical specifications for the universal quantifier :

A sequence $s$ satisfies $(∀x_i) \mathscr B$ if and only if every sequence that differs from $s$ in at most the $i$th component satisfies $\mathscr B$.

Applying the above definition to $\forall x (x=0)$, we have that e.g. the sequence $s_0$ such that $s(x)=0$ does not satisfy it [consider the sequence $s_0'$ such that $s_0'(x)=1$.]

But $\mathbb N, s \vDash (x=0)$.

Answer 2

I am not sure if that's the terminology you're using but the statement makes sense if $A\vDash B$ means that $A$ is a tautological consequence of $B$.

We fix a first-order-language. Consider an arbitrary function $V$ from the set of all atomic formulas and all formulas of the form $\exists xA$ to the set of truth values. Now inductively expand the domain of $V$ onto the set of all formulas by setting $V(\neg A)=T$ iff $V(A)=F$ and $V(A\vee B) = T$ iff $V(A) = T$ or $V(B) = T$. $\wedge, \to, \leftrightarrow, \forall$ are considered to be standard abbreviations. Obtained $V$ is called a truth-valuation.

$B$ is said to be a tautological consequence of $A$ iff for any truth-valuation $V$ if $V(A)=T$, then $V(B)=T$.

I guess that the motivation behind this definition is to depict the cases when $B$ can be derived from $A$ by "using only logic" (in particular, disengaging from the discussion of objects).

So from the definition $\forall x\ p(x)$ is not a tautological consequence of $p(x)$. It is, however, true that if $\vdash p(x)$, then $\vdash\forall xp(x)$ (The Generalization Rule).