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I know that given a sentence or formula of a formal system, this formula is a logical truth if it is true under all interpretations.

Is it possible to define this same concept of logical truth without the reference to models and interpretations?


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up vote 3 down vote accepted

For first-order logic this is essentially the completeness theorem.

The completeness theorem tells us that if $T$ is a first-order theory, then $\varphi$ is provable from $T$ if and only if $\varphi$ is true in every model of $T$.

If a formula is logically true it means that it is true in every interpretation. Every interpretation is a model for the empty theory, and so by the completeness theorem we can say that something is logically true if and only if it is provable from $\varnothing$.

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What does the quantifier "every" run over? – alancalvitti Feb 15 '13 at 4:28
@alancalvitti: Where? – Asaf Karagila Feb 15 '13 at 4:33
"true in every interpretation. Every interpretation is a model..." - is there a set of all interpretations or similar quantifiable object? – alancalvitti Feb 15 '13 at 15:35
@alancalvitti: No. But there is no set of ordinals when I quantify over "Every ordinal is a transitive set". The class is definable and therefore we can quantify over its elements. – Asaf Karagila Feb 15 '13 at 15:40
When I wrote similar, I meant perhaps class or category if not set. Are interpretations well-ordered and can we do induction over them like ordinals? Do they form a category? – alancalvitti Feb 15 '13 at 15:51

The common distinction is between syntactical truth and semantic truth. Given a deduction system (i.e., some rules telling us what which strings are allowed and how to deduce new the syntactical truth of a sentence given the syntactical truth of others) you get a well-defined notion of syntactic truth as those statements that are derivable in the deduction system from a given theory.

In contact, semantic truth relates to a given model and is means those statements that when interpreted in $M$ are true.

Goedel's completeness theorem states that (for first order logic) in a deduction system, a statement is syntactically true given a theory $T$ if, and only if, it is semantically true in all models of $T$. So, this answers your question.

The fact that syntactic truth implies semantic truth is quite easy to prove. The other direction is involved and requires a slightly weaker axiom than the axiom of choice.

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Six seconds, oh snap. :-) – Asaf Karagila Feb 15 '13 at 2:57
oh my god :D :D – Ittay Weiss Feb 15 '13 at 2:58

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