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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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    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.2011-10-01
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    related link: https://www.quora.com/What-is-the-difference-between-material-and-logical-implication2018-01-21
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    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.2018-01-21

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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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    @AsafKaragila: it's not clear to me how the statement after "in particular" follows from the theorem. What is $T$ in this particular case?2011-10-01
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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.2011-10-01
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    @Carl: I see. Thanks for the correction anyway.2011-10-01
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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?2011-10-23
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    What is "an implication" in general? At least by convention, we don't usually use the term "implication" for the $\vdash$ relation.2011-10-23
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    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.2014-11-10
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    @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"?2016-10-05
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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". @Eric2016-10-05
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    @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?2017-04-27
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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.2017-04-27
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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)2017-04-28
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    in other words, when mathematicians prove $P \implies Q$, the implicit meaning is they shown that $P \rightarrow Q$ is a tautology.2017-04-28
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    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?2018-01-21
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In most logic textbooks and classes we hear it stated that in the formula $F \supset G$ (where $F$ and $G$ are syntactic variables), the connective `$\supset$' is a $\textit{"material conditional(or implication)"}$. Kleene in $\textit{Mathematical Logic (published by Dover)}$ p. 69f. offers some illumination. In order to distinguish it from (I take it) the $\textit{formal/logical implication(or conditional)}$, which is symbolized as $F \vdash G$ OR (given soundness and completeness) $ F \vDash G$ OR $\vDash F \supset G$ OR $\vdash F \supset G$, Kleene says something to the effect that formal/logical implication is expressed in the $\textit{metalanguage},$ while the material conditional (or implication) is expressed in the $\textit{object language}$.

So, the reason why we say that $F \supset G$ paraphrases a material conditional is that its truth value (as expressed in the object language) will "depend ordinarily on circumstances outside of logic, e.g. on $\textit{matters}$ of empirical fact." (Kleene, $\textit{Mathematical Logic}$ p. 70) See the connection between 'matters' and 'material'?

Kleene's point, as I understand him, is this: on the one hand, the formal/logical implication, since it is stated in the metalanguage does not concern itself with the interpretation of $F$ and $G$. It says in the $\textit{metalanguage}$ that $\textit{formally}$ or $\textit{logically}$ $F$ and $G$ are connected as $F \supset G$.

On the other hand, the material implication since it is in the object language, its truth value $\textit{depends}$ on the interpretation (or model) we give to $F \supset G$. In any model $F \supset G$ is true if either $F$ is false or $G$ is true; false otherwise i.e. when $F$ is true and $G$ is false. The only thing that the material conditional guarantees is that if $F$ is true, then $G$ is true. If $F$ is false, the material conditional tells us nothing about the truth value of $G$, which might turn out (on the basis of other $\textit{matters}$) to be either true or false.

In short, the differences are two: (1) material conditional is in the object language while the formal conditional is in the metalanguage; (2) the truth value of the material conditional depends on $\textit{matters}$ other than the formal relationship between $F$ and $G$.

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    How is it true that logical implication does not concern about the interpretation of A and B in $A \implies B$? Isn't the definition of a truth table exactly (but implicitly) just the definition of the boolean function $f_{implication}(A,B) = \mathbb{1}[A] \implies \mathbb{1}[B]$ where $\mathbb{1}[\cdot]$ is the indicator (interpretation) function mapping statements to its boolean value True or False?2018-01-21