# What's the difference between material implication and logical implication?

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!

• 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. – André Nicolas Oct 1 '11 at 2:38
• related link: quora.com/… – Pinocchio Jan 21 '18 at 18:16
• 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. – Pinocchio Jan 21 '18 at 18:19

## 2 Answers

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.

• @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 1 '11 at 12:30
• @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 1 '11 at 12:31
• @Carl: I see. Thanks for the correction anyway. – Asaf Karagila Oct 1 '11 at 13:47
• 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 5 '16 at 20:09
• @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

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 $G$.

• 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? – Pinocchio Jan 21 '18 at 18:14