Is the expression "valid formulas of a formal system" correct? I've been reading recently some some lecture notes where the expression "valid formulas of a formal system" is used (with reference to predicate calculus). I thought that you can use the expression "valid formula" only when you talk about a formal language, not a formal system; that is, I thought that "valid formula" is a semantic concept not a syntactic one. Could someone help me?
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What is the def of formal system ? – Mauro ALLEGRANZA Apr 03 '17 at 19:55
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Usually "valid" is defined according to a semantics for the language. – Mauro ALLEGRANZA Apr 03 '17 at 19:56
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But someone cam mean with "valid formula" a theorem, i.e. a formula that is derivable from the axioms of the system by way of the inference rules, irerspective of the interpretation. – Mauro ALLEGRANZA Apr 03 '17 at 19:58
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Formal system = a formal language + a set of inference rules and, possibly, a set of axioms. – user405159 Apr 03 '17 at 19:59
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Agreed; thus my third comment (as well as Henning's full answer) applies. – Mauro ALLEGRANZA Apr 03 '17 at 20:00
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Nope, valid formulas are not theorems in the notes I'm reading, because the distinction is made between valid formulas and theorems. This is pretty confusing. – user405159 Apr 03 '17 at 20:01
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See the post are-axioms-assumed-to-be-true-in-a-formal-system for more details. – Mauro ALLEGRANZA Apr 03 '17 at 20:06
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You're right: Usually "valid" when applied to logical formulas will mean "logically valid", that is, it describes formulas that are true for every possible interpretation of the non-logical symbols in them.
However, from time to time writers will slip and use "valid" in its everyday non-technical meaning of "allowed by the rules" - which may well be the case for the instance you have found. If that is so, your "valid formulas" would just mean "well-formed formulas" (in the language of the system).
But we cannot say for sure without knowing more context of your quote.
hmakholm left over Monica
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No, it's not just "well-formed formulas" because in fact the author talks about the "logically valid formulas of predicate calculus"! – user405159 Apr 03 '17 at 20:03
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1@Fishermansfriend: If that's the context, then it's definitely the technical concept of "logically valid". Don't put too much weight on "of predicate calculus" -- that is just used as a reference to the language of pure predicate calculus (presumably with a countably infinite supply of predicate and function symbols of each arity). – hmakholm left over Monica Apr 03 '17 at 20:19