| propositional logic |
Only a single unit $p$, with no access to $x$ or to $x^2 \geq 0$
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In propositional logic, $\forall x \, (x^2 \geq 0)$ can be represented only as a single propositional variable $p$, because propositional logic has no quantifiers and no variables for objects.
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| first order logic |
A sentence of first-order logic
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$\forall x \, (x^2 \geq 0)$ is a sentence of first-order logic, because the quantifier $\forall x$ ranges over the objects of the domain of discourse, here the real numbers.
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| sentence |
✓ |
$\forall x \, (x^2 \geq 0)$ is a sentence, because the quantifier binds its only variable $x$.
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| proposition |
✓ |
With $x$ ranging over $\reals$, $\forall x \, (x^2 \geq 0)$ is a proposition, because it is true.
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| formula logic |
✓ |
$\forall x \, (x^2 \geq 0)$ is a formula, because $\forall x \, (x^2 \geq 0)$ applies the quantifier $\forall x$ to the formula $x^2 \geq 0$.
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| expression |
✓ |
$\forall x \, (x^2 \geq 0)$ is an expression in the logic sense, because $\forall x \, (x^2 \geq 0)$ is a well-formed string of symbols; the expression is a formula, not a term.
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| predicate |
✗ |
$\forall x \, (x^2 \geq 0)$ is not a predicate, because it has no free variables.
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| propositional formula |
✗ |
$\forall x \, (x^2 \geq 0)$ is not a propositional formula, because the formula uses the quantifier $\forall$, which propositional logic does not have.
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| term |
✗ |
$\forall x \, (x^2 \geq 0)$ is not a term, because $\forall x \, (x^2 \geq 0)$ makes a claim, which is true, instead of naming an object.
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