# What is the difference between a conditional and material implication?

Can someone help me understand the difference between a conditional and a material implication? Both of them have the "same" (if-then) structure.

For example, "if x=2, then x^2=4". Is this a material implication or a conditional?

As another example, "if John is in school then his bag is not in the house" (since we know that every time John goes to school he takes his bag with him).

And as a last example, "if you get 100/100 on the math exam, then I will buy you a car". Is this a type of implication or a "promise" ?

• I made an edit. You are welcome to roll this back or continue editing. You can see the versions by clicking on the "edited" link above. Regarding your question, do you have a definition of "conditional" and "material implication" that you are working with? If so, that might be worth adding to the question. This would provide more context. Aug 29 '18 at 0:47
• In elementary logic, no difference : The material conditional (also known as material implication, material consequence, or simply implication, implies, or conditional) is a logical connective (or a binary operator) that is often symbolized by a forward arrow "→". Aug 29 '18 at 6:43
• But we may have other use of "conditional" that are not classic, i,.e. not truth-functional; see e.g. Counterfactual conditional. Aug 29 '18 at 6:44
• Hmmm. According to one answer, material implication can be thought of as a “special case of a conditional”. According to another answer, they have “nothing whatsoever” to do with each other. According to a third answer, “there does not seem to be any difference” between the two. Aug 26 '20 at 5:58

Material implication can be thought of as a kind of very simple, special case of a conditional. It is a truth function, which is to say that its truth value depends only on the truth values of its antecedent and consequent, not on any other semantic connection between them. It serves only to express a sufficient condition between the truth of its antecedent and the truth of its consequent. It does not cope with common features of ordinary English conditionals, such as: they might be uncertain, they might hold by default under normal conditions but have exceptions, they might be implicitly quantified, they might serve to imply stronger relationships between the antecedent and conequent such as a causal or evidential relation. Also, we can conditionalise all kinds of speech acts, e.g. questions, commands, offers, threats, bets, promises, etc., which obviously do not have truth values.

In your examples, I would say that the mathematical example does work as a material implication. Indeed, mathematics is a domain where material implications work well because mathematics is not typically concerned with uncertain claims or unstated exceptions. "If John is in school then his bag isn't in the house" could be a material implication at a pinch, but it fails to capture the fact that this might have exceptions: maybe John forgets occasionally, or is taking part in a school event and doesn't need his bag. Also, it skirts over the fact that there is a connection between things: we are given to understand that John's bag isn't in the house because John takes it with him to school. I would say your last example is a conditional promise, not an implication.

What is the difference between a conditional and material implication?

I'm not sure there is a difference. If you accept the following principles of logic, there does not seem to be any difference:

• Direct proof
• Detachment (modus ponens)
• Removal of double negation

In my recent blog on material implication, I derive, among things, the truth table for material implication using only these principles. No proof is longer than 8 lines.

The difference is simple: the material implication has nothing whatsoever to do with a conditional.

It should also be said that we can always use a conditional to express a logical implication, including the logical implications that mathematicians prove that there are from axioms to the theorems that follow from these axioms, and that therefore the material implication has also nothing to do with logical implications in a mathematical context.

The expression "If x = 2 then x² = 4" is a conditional. However, it is also the straightforward interpretation of the logical implication x = 2 → x² = 4. This implication is not formally true, since it depends on a whole set of definitions which are not formally expressed in this expression, but we all understand what those are and we can think of them broadly as the axioms of arithmetic.

It is also apparent in mathematical papers that mathematicians never use the material implication when they prove a theorem. Nearly all mathematical proofs rely on the logical sense of the mathematician, not on any formal calculus based on the material implication.

It should be noted that mathematicians are able to understand each other's proofs simply by reading them, like they have always done since Euclid, essentially in the same way that we can all understand Aristotle's syllogisms, that is, intuitively. Thus, no mathematician would verify a proof by using the definition of the material implication.

Thus, the fact that mathematics usually works well, including in applications in engineering and science, is no evidence that the material implication has any value.