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. – Frank Hubeny 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 "→". – Mauro ALLEGRANZA 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. – Mauro ALLEGRANZA Aug 29 '18 at 6:44

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.

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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
  • Proof by contradiction
  • 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.

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