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In the Mendelson's logic book, there are 2 conditionals which Mendelson says they are regarded false even if their antecedent is false. One of them is the following:

If this piece of iron is placed in water at time $t$, then the iron will dissolve.

Why is it considered false even if its antecedent is false - that is, the iron is not placed in water at time $t$?

Zev Chonoles
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Milad Math
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    What?! In classical first-order logic, which is what most mathematicians use, that statement would be true if the iron is not placed in water at time $t$, because that statement is a guarantee that is not broken. – user21820 Jul 27 '15 at 09:50
  • I know, but in the nonlogical interpretation Mendelson says it is regarded false, but why? @user21820 – Milad Math Jul 27 '15 at 09:52
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    I can't possibly guess what he is thinking because even in natural language we certainly agree that the guarantee is not broken. – user21820 Jul 27 '15 at 09:52
  • It's different if we have modal complications in natural language, such as "If I want, I can pass through concrete walls." In this case I agree that there are two interpretations. In the logical interpretation, it is true if I do not want. In the 'usual' interpretation, it is false because it implies that whether I can pass through concrete walls is the same as whether I want to or not. – user21820 Jul 27 '15 at 09:55

2 Answers2

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Mendelson is after the fact that the conditionals we use in everyday language are often not at all like material implication ($\to$) in logic.

The example sentence (intuitively) expresses that iron has a certain disposition (click) rather than being a regular implication.

Example: Let "$x$ is lethally poisonous" be defined as "If someone eats $x$, then he will die". Then, surely, you wouldn't agree that everything that noone ever tried to eat is lethally poisonous. So, despite being of If-then-form, the example definition (intuitively) doesn't express a material implication here. Rather, we take the definition to mean that $x$ has a certain property, a disposition to kill us when eaten.

Another example of commonly used conditionals that are entirely unlike $\to$ are of course counterfactual conditionals like "If you hadn't asked this question on math.SE, someone else would have". Because, well, who knows what would have happened?

You can ignore Mendelson's remark for the rest of the book, just be aware that (as often) the colloquial understandings and the mathematical understanding diverge.

The Stanford Encyclopedia of Philosophy also has something on conditionals and their classification, but it's a long read.

lodrik
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  • Excellent, except for the English. Please change "If you wouldn't have asked this question on SO, someone else would have" to "If you hadn't asked this question on SO, someone else would have". Thanks... – David C. Ullrich Jul 27 '15 at 16:37
  • Thanks, fixed. Non-native speaker at work ;) – lodrik Jul 27 '15 at 16:46
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    Hey thanks! I expected an argument... Trust me, non-native is not the problem - native speakers get exactly that point wrong all the time. Drives me up the wall, like fingernails on blackboard... – David C. Ullrich Jul 27 '15 at 16:48
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If this piece of iron is placed in water at time t , then the iron will dissolve.

Why is it considered false even if its antecedent is false - that is, the iron is not placed in water at time t ?

Because when we express such a statement in natural language, it is often implicitly a modal logic statement; a claim of the necessity of the implication.

We can express the statement as: "In all possible worlds, the iron will dissolve whenever it is placed in water at time t, (for a given definition of "possible")"   So, if there are possible worlds where the iron is placed in water at time t, and in any of those worlds the iron does not dissolve, then the given statement is falsified.

If it is not necessary that the iron would have dissolved if we had placed the iron in the water at that time, then the implication is not necessarily true.

Graham Kemp
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  • "[..] it is often implicitly a modal logic statement; a claim of the necessity of the implication." Yes and No. Natural language conditionals can be analyzed using some form of intensional semantics (like the semantics for modal logic). But an implication in natural language doesn't typically express a necessity i.e. that "it couldn't possibly be different". Mendelson's example, intuitively, just says that a certain piece of iron has a contingent "property" to dissolve, if placed in water at time t. ... – lodrik Jul 28 '15 at 22:45
  • ... It's not a modal claim that this is necessarily so, that e.g. the laws of physics couldn't have possibly been a tiny little bit different... – lodrik Jul 28 '15 at 22:46
  • @lodrik Well, yes, it depends on what possible worlds you consider accessible. Typically its only "just like the actual world but for this one contingent action." – Graham Kemp Jul 28 '15 at 23:05