Let P and Q represent formulas. Would stuff like $(P \wedge \neg P) \models Q$ make sense?

Question

I confess, I am in a state of total confusion right now. And I am still struggling to grasp the underlying distinction between the normal material implication, $\Rightarrow$, and the notion of semantic consequence, $\models$.

I've seen stuff like $(P \wedge \neg P) \models Q$ appear in a book I'm currently reading (A First Course in Logic, by Shawn Hedman), but at the end of the day, it had me wondering what the difference is between material implication and semantic consequence. I mean, really.

We all know that the sentence $(P \wedge \neg P) \Rightarrow Q$ is a tautology. And in a material sense (only thinking in terms of truth values and not in terms of how the two formulas P and Q are connected semantically), this is totally acceptable to me. But to write stuff like $(P \wedge \neg P) \models Q$ blows my entire understanding of the difference between $\Rightarrow$ and $\models$ right out the window. I thought the double turnstile symbol is only to be used in a sense that has more to do with the underlying meaning/interpretation behind the sentences? Like, if we say $P \models Q$ (where P and Q are sentences), then wouldn't it mean that we can see a clear logical connection which allows us to accept that Q follows from P?

If translated into words, the sentence "formula Q is a semantic consequence of $P \wedge \neg P$" is incomprehensible to me. It doesn't have that logical connection that I thought must accompany every use of the double turnstile symbol (for instance, I can easily accept $(P \wedge Q) \models P$, because the logical connection is there, after all if its stated that P and Q are both true, then logically, P must be true). If no logical connection is required for the use of $\models$, then how is it different from the normal material implication?

It feels like I'm missing something crucial, like a bigger picture, or a wider generalization of what the double turnstile symbol represents.

Answer 1

"We all know that the sentence (P∧¬P)⇒Q is a tautology."

I certainly don't know this and gasp when I see statements like this made. Anyone who claims to is also, simply put, wrong. Why things work that way helps to illuminate things.

To say that something is a tautology is to imply that something is an object-level sequence of symbols. If you check your definition of a well-formed formula, or whatever equivalent term gets used, (P∧¬P)⇒Q is not well-formed. Thus, (P∧¬P)⇒Q is not a tautology, because a tautology is by definition a well-formed formula. ((P∧¬P)⇒Q) is a tautology.

On the other hand, (P∧¬P)⊨Q is a meta-language construct. It is not well-formed in the object-language, and since the arity of the predicate |= seems to vary, there might not exist a corresponding well-formed formula (though maybe not also). Also, |= is a predicate, while ((P∧¬P)⇒Q) doesn't have any predicates.

The difference might also get illuminated by looking at other uses of |=. For instance, I think you agree that, "{p, (p⇒q)} |= q" makes sense. Let's suppose that |= is no different from ⇒. Then, "{p, (p⇒q)} |= q" is no different from "{p, (p⇒q)} ⇒ q". There's at least three problems,

1. "{p, (p⇒q)} ⇒ q" is not well-formed, since all well-formed formal implications begin with a '(' and end with a ')'.
2. q is a proposition, but what we have on the left side of the arrow is a set. But, sets of propositions are not propositions. Nor are propositions sets of propositions. So, again, "{p, (p⇒q)} ⇒ q" is not well-formed, and it's not even apparent how to make into a well-formed formula, or even necessarily the case that it can get made into a well-formed formula, since it would even two different types of entities needing to fall under the same roof... so to speak. That sounds like a possible category error.

Pece has also commented that:

"... ⊨ is a relation between finite sequences of wffs on the left and wff on the right, while ⟹ is a binary connective in the language that is used to construct formulas."