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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.

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    $\begingroup$ Well $Q$ is a semantic consequence of $P\land \neg P$ because in every possibility in which $P\land \neg P$ holds (there is no such possibility !) , $Q$ holds as well. Semantics is about possible worlds, so this is why that's called semantic implication $\endgroup$ – Max Jul 25 '18 at 9:38
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    $\begingroup$ As far as I can tell, your difficulty comes from the part of your question that says "we can see a clear logical connection". That's far too vague to be mathematics. If, by "clear logical connection", you mean something that includes truth tables, then you're probably OK, because truth tables do allow you to infer $Q$ from $P\land\neg P$. But if you mean something more vague than truth tables, then you need to revise your understanding of $\models$. $\endgroup$ – Andreas Blass Jul 25 '18 at 13:16
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    $\begingroup$ @DougSpoonwood it should be clear to anyone reading my comment that "same as" here is an abuse of language meaning "equivalent to", please don't bicker about details like that. $\endgroup$ – Max Jul 25 '18 at 19:48
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    $\begingroup$ The basic issue is that we may have logical languages without the conditional connective ($\to, \Rightarrow$): we may use $\land, \lnot$ or $\lor, \lnot$ and they are enough to formalize arguments as well as the concept of tautology. But the concept of logical consequence: $\vDash$ does not change. $\endgroup$ – Mauro ALLEGRANZA Jul 29 '18 at 13:40
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    $\begingroup$ "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?" YES... but in propositional logic the only "meaning" of a sentence is its TRUTH-VALUE. $\endgroup$ – Mauro ALLEGRANZA Jul 30 '18 at 9:46
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"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."

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    $\begingroup$ Bickering about parentheses really isn't relevant here. Moreover, no one is claiming that $\models$ is the same as $\implies$ here. The OP asks what $\models$ is, why it's called semantic consequence, how it differs from material implication, how a sense of "logical connection" is attained: this has nothing to do with your answer $\endgroup$ – Max Jul 25 '18 at 19:51
  • $\begingroup$ @Max It's not a bicker. You can't have a valid rule of uniform substitution without parentheses. That rule works out as essential for axiomatic systems which only have axioms and rules of inference. |= is certainly a meta-language symbol, while ⟹ an object language symbol. $\endgroup$ – Doug Spoonwood Jul 25 '18 at 22:15
  • $\begingroup$ $(p \land \neg p) \implies q$ is a tautology. See the truth table at wolframalpha.com/input/?i=truth+table+(p+and+~p)+%3D%3E+q $\endgroup$ – Dan Christensen Jul 27 '18 at 4:01
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    $\begingroup$ @DougSpoonwood Actually most mathematical textbook about model theory or proof theory establishes at some point $P \mathbin{\mathcal B} Q$ as syntactic sugar for $(P \mathbin{\mathcal B} Q)$ when it is not harmful (with $\mathcal B$ a binary connective). Moreover, replace everywhere in OP's question the "wrong" statement by your well-formed variation and still your answer does not address the question. At best, you could have just made it a comment. $\endgroup$ – Pece Jul 29 '18 at 9:50
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    $\begingroup$ @DougSpoonwood I think I understand what you were trying to say in you answer, namely that $\models$ is a relation between finite sequences of wffs on the left and wff on the right, while $\implies$ is a binary connective in the language that is used to construct formulas. If so, first I will remove my -1 because your answer is not about nitpicking, but this is not at all clear that this is the message you want to convey, but secondly and more importantly I'm not even sure this is the problem OP is struggling on... $\endgroup$ – Pece Jul 29 '18 at 13:17

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