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When I read the definitions of material and logical implications, they seem to me pretty much equivalent. Could someone give me an example illustrating the difference?

(BTW, I have no problem with the equivalence between $\lnot p \vee q$ and $p \to q$, aka "if $p$ then $q$". My confusion is with the idea that there are two different forms of implication, material and logical.)

Thanks!

kjo
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1 Answers1

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There is one level at which they can be distinguished. The following definitions are relatively common.

  • Material implication is a binary connective that can be used to create new sentences; so $\phi \to \psi$ is a compound sentence using the material implication symbol $\to$. Alternatively, in some contexts, material implication is the truth function of this connective.

  • Logical implication is a relation between two sentences $\phi$ and $\psi$, which says that any model that makes $\phi$ true also makes $\psi$ true. This can be written as $\phi \models \psi$, or sometimes, confusingly, as $\phi \Rightarrow \psi$, although some people use $\Rightarrow$ for material implication.

In this distinction, material implication is a symbol at the object level, while logical implication is a relation at the meta level. In other words, material implication is a function of the truth value of two sentences in one fixed model, but logical implication is not directly about the truth values of sentences in a particular model, it is about the relation between the truth values of the sentences when all models are considered.

There is a close relationship between the two notions in first-order logic. It is somewhat immediate from the definitions that if $\phi \to \psi$ holds in every model then $\phi \models \psi$, and conversely if $\phi \models \psi$ then $\phi \to \psi$ is true in every model. This relationship becomes more fuzzy when we begin to look at other logics, and in particular it can be quite fuzzy when philosophers talk about material conditionals and logical implication independent of any formal system.

Carl Mummert
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  • @AsafKaragila: it's not clear to me how the statement after "in particular" follows from the theorem. What is $T$ in this particular case? – kjo Oct 01 '11 at 12:30
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    @Asaf: That complicates things, because then you have to talk about provability. Also, not every logical system satisfies the deduction theorem. (Also, you stated the converse of the actual deduction theorem, which says that if $\alpha \vdash \beta$ then $\vdash \alpha \to \beta$; the converse you stated is essentially modus ponens.) I thought about it and decided against it. – Carl Mummert Oct 01 '11 at 12:31
  • @Carl: I see. Thanks for the correction anyway. – Asaf Karagila Oct 01 '11 at 13:47
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    Isn't there another form of logical "implication", since ϕ⊨ψ means we have ψ as a semantic consequence of ϕ, so ϕ implies ψ in a semantic sense, while ϕ|-ψ means we have ψ as a syntactic consequence of ϕ, so ϕ implies ψ in a syntactic sense? If not, why is "ϕ|-ψ" not also an implication? – Doug Spoonwood Oct 23 '11 at 00:16
  • What is "an implication" in general? At least by convention, we don't usually use the term "implication" for the $\vdash$ relation. – Carl Mummert Oct 23 '11 at 21:54
  • I'll add here that A. N. Prior's textbook Formal Logic has parts of it which read like the following: "Rule: Detachment ($\alpha$, D$\alpha$D$\beta$$\gamma$ $\rightarrow$ $\gamma$) and (In all cases the sole rule beside substitution is E-detachment: $\alpha$, E$\alpha$$\beta$ $\rightarrow$ $\beta$. And in my opinion Prior's symbolism comes as clearer here than writing {E$\alpha$$\beta$, $\alpha$} $\vdash$ $\beta$, since the "$\rightarrow$" symbol suggests that one transitions from the left-hand side to the right hand side. – Doug Spoonwood Nov 10 '14 at 19:27
  • @Carl Mummert You said "material implication is a function of the truth value of two sentences", isn't "material implication is a function that returns a sentence of two sentences"? Or more correctly, "$\rightarrow$ is a function that returns a sentence of two sentences"? – Eric Oct 05 '16 at 17:55
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    It is both. We can make a new sentence by joining two existing sentences with $\to$. The truth value of the new sentence is then given by a particular function of the truth values of the existing sentences. So material implication is both the symbol that links the sentences, and the function used to interpret the symbol. Actually, the specific choice of symbol is not as important as the function being used - the function is what makes us call the symbol "material implication". @Eric – Carl Mummert Oct 05 '16 at 20:09
  • @Carl Mummert. In conventional mathematical practice, for example, it's known that Taniyama-Shimura "implies" Fermat's Last Theorem, or $x$ is real "implies" $x^2 \geq 0$. In these contexts, does "implies" refer to logical implication? – Maxis Jaisi Apr 27 '17 at 13:02
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    @Maxis Jaisi: that phrasing usually means that assuming one statement leads to an easy proof of the second statement, assuming some simpler axioms. It's not quite logical implication between the statements because of those additional axioms. But if the necessary axioms are included as part of the hypothesis, then that compound statement will logically imply the conclusion. – Carl Mummert Apr 27 '17 at 18:36
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    @Carl Mummert: Thank you. A final question to drive the point home. Let $P$ be Taniyama-Shimura's Conjecture, and $Q$ Fermat's Last Theorem. When mathematicians say they've "proved" $P \implies Q$, it means we can strike off the $P = \text{True}$ and $Q = \text{False}$ row in the truth table for $P \implies Q$, right? (assuming the necessary axioms are part of the hypothesis) – Maxis Jaisi Apr 28 '17 at 09:36
  • in other words, when mathematicians prove $P \implies Q$, the implicit meaning is they shown that $P \rightarrow Q$ is a tautology. – Maxis Jaisi Apr 28 '17 at 09:39
  • what did you mean by "but logical implication is not directly about the truth values of sentences in a particular model, it is about the relation between the truth values of the sentences when all models are considered."? Is the only difference that logical implication applies with respect to EVERY model and interpretation while material just to one model/interpretation? – Charlie Parker Jan 21 '18 at 16:56
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    can you give some examples on the two implications? – Ooker May 10 '21 at 07:23