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The paradoxes of material implication show that the usual interpretation of implication as "if ... then" statements leads to counter-intuitive results. For example, from (P & Q) -> R we can derive (P -> R) v (Q -> R), which doesn't make much sense if we consider them as "if ... then" statements. Quoth wikipedia:

[The statement (P & Q -> R) |- (P -> R) v (Q -> R)] can be read "If both switch A and switch B are closed, then the light is on. Therefore, it is either true that if switch A is closed, the light is on, or if switch B is closed, the light is on." If the two switches are in series, then the premise is true but the conclusion is false. Thus, using classical logic and taking material implication to mean if-then is an unsafe method of reasoning which can give erroneous results

I know that there are methods like relevance logic and connexive logic which try to make logic fit the mold of our intuition. But are there any ways in which we can restate implication as a different intuition which would work better with the laws of classic logic?

  • I suppose you're talking about the paradox of entailment, which says that given two contradictory or inconsistent premises, the conclusion necessarily follows? No, there's no way of resolving that that fits with the laws of classic logic. Our colloquial and intuitive notions of validity have lots of other mediating factors that can't be captured in a formal system of logic. You mentioned relevance logic as at least one of the attempts to capture this in a more formalized system. What else do you want to know? – Cody Gray Jun 16 '11 at 4:47
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    @Cody, I'm pretty sure the OP is not talking about the paradox of entailment. – Ami Jun 16 '11 at 5:14
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Two separate things:

  • The difficulty with the example of switches is simply that the mapping between the real world example (operation of switches) and the logical symbols is faulty. There is a a more complicated relationship in the switch configuration that is not captured by the logical statements.

  • as to an intuition for 'material implication' or the logical 'if-then' P -> Q, consider the truth-valuation and how upset you might be given the values of P and Q.

So suppose I claim P->Q. How upset will you be, how trustworthy do you think I am if: (suppose P is "it rains tonight", and Q is "I'll go to the movies with you")

  • P is true and Q is true: You won't be upset at all, it's what you'd expect. If P occurs you'd expect Q to occur
  • P is true and Q is false: You'd be very upset. If P happens, you expect Q to happen, and when it doesn't, you should think I lied (it's raining and I didn't go to the movies with you means I'm an untrustworthy liar).
  • P is false and Q is true: hm...weird, so we went to the movies without it raining. I didn't say what I would do if it did not rain, so going to the movies is just fine, I haven't lied about it.
  • P is false and Q is false: also weird, but same reasoning, I didn't make any claims about what would happen if it did not rain, so not going, though not great, doesn't make me out to be a liar.

The truth value explanation of material implication may lead to those 'paradoxes' like 'ex falso quodlibet', and it is somewhat stipulative (and, by that, contrary to natural expectations of 'if-then') but the above intuition, when applied to independent statements, makes consistent sense.

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Great question.

There is nothing inherently wrong with interpreting “->” as “if...then.” Paradoxes of material implication arise from an incorrect translation of observation to symbolic logic.

In the example given, we have a system of two closed switches resulting in one illuminated light bulb and we want to describe a truth about this system within symbolic logic.

There is only one thing that we can know based on this observation - namely:

two closed switches -> illumination

In other words: If two switches are closed then the light bulb will be illuminated.

Now, the words “if two switches are closed” are not two propositions, they are one proposition.

The words "if two switches are closed" is a binary proposition that adheres perfectly to the law of the excluded middle. Any system of two switches either satisfies the condition “two switches are closed” or does not.

Therefore, the only logically correct way to represent this true observation about switches and illumination is in the form P |- Q where P stands for the system of two closed switches and Q the illuminated light bulb.

P & Q |- R is wrong because one observed system state can not yield two binary propositions causing a result. From one observed state you may get only one “if...then” in the form of P |- Q.

  • So you're saying the mistake is in assuming, since P' = P & Q, that we can substitute P & Q for P'? I'm even more confused by this :-/ – Xodarap Jun 16 '11 at 15:48
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This blog post contains a neat explanation of the truth table for material conditional.

The basic idea is that if you want a truth-functional connective that behaves something like ordinary language "if then", then you only have 16 choices. That is, there are 16 possible truth tables A->B could have. The only one that even makes any sense as a formal rendering of "if then" is TFTT.

Obviously, that's still fairly disappointing in its consequences. But what this teaches us is that ordinary language "if then" doesn't behave truth functionally.

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The conclusion "it is either true that if switch A is closed, the light is on, or if switch B is closed, the light is on" can be interpreted as equivalent to the premise. Anything follows from falsehood, so if one of the switches is open, one of "if switch A is closed, the light is on" or "if switch B is closed, the light is on" has a falsehood as its premise, and is therefore true. The paradox arises from the ambiguity of language, because we commonly use the same logical words "if", "either", "or", "is", "true" with informal meanings. Consider the statement: "it is either true that closing switch A causes the light to be on, or closing switch B causes the light to be on." It uses almost the same words as the conclusion. If the premise is also restated in terms of "causation", a non-logical term, then the entailment will no longer hold, and we should not expect it to because the meanings are no longer the same.

So one intuition is to try and replace logical connectives with non-logical words and see if it still makes sense. I speculate that this works because in communication it serves no purpose to have two logically equivalent statements of different lengths: we interpret the shorter statement formally to mean what it says, and we interpret the longer formally- and tautologically-equivalent statement more liberally; when persons seemingly use more words than necessary to say what they mean, then we often rightly assume that they mean something else.

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The Wikipedia argument reaches a false conclusion by conflating (I) "if switch X is in the closed position" with (II) "if we close switch X". Let S(X) be the statement "if X is closed, then the light is on" in interpretation I. If S(A) is true, then so is S(A) v S(B). (This holds also under interpretation II.) If S(A) is false, it can only be so in the circumstance that B is open---under which condition, statement S(B), and hence S(A) v S(B), is true since the premise "B is closed" is counterfactual. (Here the choice of interpretation I is critical.) If we change the condition by closing B, then of course S(A), and hence S(A) v S(B), becomes true. (Incidentally, the conflation of interpretations I and II is widespread in the social sciences, which is why governmental action so often doesn't work as expected.)

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