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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!

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    $\begingroup$ They are indeed identical. The term "material implication" is supposed to distinguish implication, in the logical sense, from the informal notion of implication, which carries some sense of connection. $\endgroup$
    – André Nicolas
    Oct 1, 2011 at 2:38
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    $\begingroup$ related link: quora.com/… $\endgroup$
    – Charlie Parker
    Jan 21, 2018 at 18:16
  • $\begingroup$ one thing I am not 100% clear about is the difference between logical implication and modus ponens. It seems to be a key idea to distinguish material and logical implication. $\endgroup$
    – Charlie Parker
    Jan 21, 2018 at 18:19
  • $\begingroup$ In the context of first-order logic, logical implication (synonymous with entailment $\models$) is a metalogical concept referring to a material conditional $(\to)$ that is a validity, i.e., that is true regardless of interpretation. Related: regular implication $(\Rightarrow)$ versus material conditional $(\to)$. @CharlieParker Modus Ponens (a metalogical concept) is the (valid) argument form $(A \text { and } A\Rightarrow B)\Rightarrow B.$ $\endgroup$
    – ryang
    Sep 15, 2021 at 9:56
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    $\begingroup$ Does this answer your question? Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$) $\endgroup$
    – user1147844
    Feb 24 at 8:01

1 Answer 1

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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.

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    $\begingroup$ @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. $\endgroup$
    – Carl Mummert
    Oct 1, 2011 at 12:31
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    $\begingroup$ 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? $\endgroup$
    – Doug Spoonwood
    Oct 23, 2011 at 0:16
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    $\begingroup$ 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 $\endgroup$
    – Carl Mummert
    Oct 5, 2016 at 20:09
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    $\begingroup$ @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. $\endgroup$
    – Carl Mummert
    Apr 27, 2017 at 18:36
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    $\begingroup$ @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) $\endgroup$
    – Maxis Jaisi
    Apr 28, 2017 at 9:36

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