In Geoffrey Hunter's Metalogic ( UC Press, 1971) , §29, is introduced the concept of semantic completeness.

" What do logicians want? The Holy Grail of logic would be a system that caught all truths of pure logic. This nobody has yet found. All truths of pure propositional logic then? But they include non-truth-functional logical truths..."

This assertion ( according to which pure propositional logic contains non-truth-functional truths) is not explained by Hunter.

It sounds strange, for is not propositional logic dealing with propositions that are built using truth functional operators?

Could an example of pure propositional logic be given that is not a truth functional truth?

Is there a part of propositional logic that is not truth-functional. Does non-truth functional propositional logic exist?

  • $\begingroup$ Can you please define "truth functional truth"? $\endgroup$ – Alex Kruckman Dec 8 '19 at 19:50
  • $\begingroup$ The expression " truth-functional truth" cannot be found in the index of the book. The author defines " truth function " ( a function from a set of sequences of truth values to {T,F} and " truth functional connective" as a symbol that can be combined with propositions to form propositions that are totally defined by their truth table. $\endgroup$ – user654868 Dec 8 '19 at 20:03
  • $\begingroup$ Then your first question should probably be "what is the definition of truth-functional truth"? Nevertheless, I will try to give an answer. $\endgroup$ – Alex Kruckman Dec 8 '19 at 20:09
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    $\begingroup$ Maybe modal propositional logic truths. $\endgroup$ – Mauro ALLEGRANZA Dec 8 '19 at 20:12
  • $\begingroup$ I think a " truth functional truth" could be defined as a truth functional proposition that has truth value " true" in all lines of its truth table ( in other words, a truth functional propsition that is true in all interpretation/ truth value assignment). $\endgroup$ – user654868 Dec 8 '19 at 20:13

I am similarly perplexed about the author's intent. Googling "truth-functional logical truth" led me to the book "Simple Formal Logic: With Common-Sense Symbolic Techniques" by Arnold vander Nat. The list of definitions at the back of that book contains the following entries:

truth-functional logical truth (tautology). A sentence is a truth-functional tautology if for every possible assignment of truth-values T and F to all the simple components of the sentence, that sentence always has the resultant value T ... Example, "If both dogs and cats sing, then dogs sing" (if p and q then p).

logical truth (tautology), semantical version. A sentence is a logical truth if for every possible interpretation of all the basic components of that sentence, that sentence always has the resultant value T. ... For example, "All talking animals are animals." [Truth-functional logical truths form a smaller subgroup within the group of logical truths.]

I assume the example "All talking animals are animals" is intended to be an example of a logical truth which is not a truth-functional logical truth, the point being that the fact that the adjective "talking" modifies the noun "animals" implies that the set of talking animals is a subset of the set of animals, regardless of what the words "talking" and "animals" mean.

So presumably a propositional logical truth which is not a propositional truth-functional logical truth could be something like "If all animals sing, then all talking animals sing." If we decide "all animals sing" is true and "all talking animals sing" is false, then the whole sentence is false. But such a truth value assignment wouldn't be coherent, based on the structure of the phrases "all animals sing" and "all talking animals sing". No matter what we decide "animals", "talking animals", and "sing" mean, if "all animals sing" is true, then "all talking animals sing" must also be true.

Now this kind of example clearly goes outside of what I would call "pure propositional logic". That is, if you want to take non-propositional structure like this into account, you have to explicitly include it in your logical system.

In my experience, the kind of distinction that vander Nat is making in these definitions is one that only philosophers (as opposed to mathematical logicians) would make, and it seems to me that it's only meaningful because he's considering natural language sentences (which don't live within any set logical syntax) and using terms like "possible interpretation" and "basic components", which are necessarily vague where natural language is concerned.

One meaningful way to interpret these definitions is this: if you have a logic $L$ which contains propositional logic (such as first-order logic), you could make a distinction between the sentences of $L$ that are merely valid (such as $\forall x\, (x = x)$), and the subclass of sentences of $L$ which are valid propositionally ("truth-functionally") in the sense that they are obtained by substituting into a propositional tautology (such as $(\exists x\, P(x))\lor \lnot (\exists x\, P(x))$). But my point is that it's important to be clear about what logic you're working with: as soon as we move to a larger logic, we're not talking about pure propositional logic anymore. In this reading, there are no purely propositional non-truth-functional truths.

While I'm taking issue with philosophers, let me go back to Geoffrey Hunter's original quote:

"What do logicians want? The Holy Grail of logic would be a system that caught all truths of pure logic. This nobody has yet found."

This is nonsense. Not only do I disagree that this is the Holy Grail of logic, I don't even accept that "all truths of pure logic" is meaningful. There are many logical systems, each with their own syntax and semantics and quirks and reasons for existing, and it's wrongheaded to try to amalgamate them all into one maximalist system. If we fix one logic, we may hope to provide a sound and complete proof system: one such that formal proofs generate all and only the truths of that logic. Needless to say, we have such a system for propositional logic!


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