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Truth-functionally valid: there is no truth value assignment for which the premises (of an argument) are true yet the conclusion is false.

I am told that there are arguments which are logically valid but not truth-functionally valid. I'm having trouble imagining such an argument. Does such an argument even exist?

Edit: To clarify, I am wondering if that argument exists in the context of sentential logic.

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    Is it possible they're just talking about something being "truth-functionally valid" in the context of sentential logic, which is more limited than first-order logic? (this was one of the first results when I googled "truth functionally valid", the phrase is used on p. 3 of the pdf) Or are you certain they talking about "truth functionally valid" vs. "logically valid" within the same system of logic? – Hypnosifl Dec 9 '19 at 23:11
  • I believe they were talking about truth-functional validity in the context of Sentential Logic. Sorry, should have clarified that originally. – iyankv Dec 9 '19 at 23:12
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    But in that case, are you sure they were also talking about logical validity within the context of sentential logic (which is apparently another name for propositional logic)? Sentential/propositional logic lacks the "For all x" and "there exists an x" symbols and their associated rules that are found in first-order logic, so there can be deductions using those symbols which would be logically valid in first-order logic, but couldn't be expressed as truth-functionally valid deductions in sentential logic. – Hypnosifl Dec 9 '19 at 23:16
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    A proposition in first-order logic with quantifiers would't even obey the syntax of sentential logic, so it wouldn't have a truth value in that system--are you asking if there could be some way to "translate" a proposition in first-order logic involving quantifiers into one or more propositions in sentential logic without them? If so I can't think of any way that could be done, even in special cases... – Hypnosifl Dec 9 '19 at 23:29
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    Given the definitions of valid and "truth functionally valid" in the first comment, which correspond to syntactically vs semantically valid in the usual terminology, a syntactically valid but semantically invalid argument would mean that the corresponding proof system is unsound (and hence uninteresting). The converse, on the other hand, can and does happen in interesting systems, sentential or otherwise, some semantically valid inferences are unprovable, Goedel sentences, for example. – Conifold Dec 10 '19 at 0:29
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By " logically valid " I mean here " deductively valid". This is a broad sense of logical validity which requires only one thing : namely, that it is logically impossible for the conclusion to be false in case the premise(s) is/are true.

A classification of logically valid arguments

(1) formally valid ( valid in virtue of its form alone, the meaning of the non-

logical symbols/expressions playing no role in the validity).

 (1) A/ truth functionally valid

 (2) B/ non truth functionally valid 

(2) valid, but not formally ( valid in virtue of the meaning of non logical words).

Examples

Of 1 A)

If I think, therefore I am.

I think.

Therefore I am.

Of 1 B)

All men are mortal

Socrates is a man

Therefore Socrates is mortal.

Of (2)

Peter is a pianist.

Therefore, Peter is a musician.

The form of this reasoning is

(1) P (a) (2) Therefore , M(a)

in the context of predicate logic.

The form is

(1) P (2) Therefore M

in the context of sentential logic.

Consequently, it is easy to see that the reasoning is not valid in virtue of its form.

It is logically valid however since it is logically impossible fo the conclusion to be false if the premise is true.

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