Do vector spaces always have a Hamel basis without the axiom of choice?

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A well-formed formula (wff) is (syntactically) provable if and only if it is valid. This defines the relation between (Semantic) truth and (Syntactic) provability. I'm not aware of how the term 'hold' is used, by there are wff's in Sentence Logic that are contingent. And, yes, every provable statement in FOL os Sentence Logic is a Tautology; true in every model.
 
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WWGD said:
A well-formed formula (wff) is (syntactically) provable if and only if it is valid. This defines the relation between (Semantic) truth and (Syntactic) provability. I'm not aware of how the term 'hold' is used, by there are wff's in Sentence Logic that are contingent. And, yes, every provable statement in FOL or Sentence Logic is a Tautology; true in every model.
Ok, a well-formed formula (wff) is (syntactically) provable if and only if it is valid since we're assuming a sound and (semantically) complete logic system (like FOL or Sentential (or propositional) logic are).

What does it mean that in Sentential logic there are contingent wffs ?
 
cianfa72 said:
Ok, a well-formed formula (wff) is (syntactically) provable if and only if it is valid since we're assuming a sound and (semantically) complete logic system (like FOL or Sentential (or propositional) logic are).

What does it mean that in Sentential logic there are contingent wffs ?
Statements that aren't tautologies, like ##A \rightarrow B ##, which is not true when A is true and B is false.