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What would be some of the implications of deciding that vacuous material conditionals should be rendered as false? For example, what would happen if we decided that conditionals like "If John squares the circle, then John is the King of France" should be considered false instead of true?

  • Take the contrapositive, which must have the same truth value as the original. So now "If John is not the King of France, then John does not square the circle" is false. What kind of sense does that make? – user4894 Jul 5 '14 at 19:24
  • Sentences of form (⊥ → φ) are not true for all φ because we want to. It's a by-product of the classical interpretation of the material conditional as a special kind of disjunction. (A → B) is simply (¬A ∨ B). So naturally, we're going to have (⊥ → B) be true for all B, because (⊤ ∨ B) is simply ⊤. That being said, there are many non-classical logics that don't define implication in that way and as a result don't have that strange (but expected) feature. – Hunan Rostomyan Jul 6 '14 at 5:11
  • @user4894 Your paradox is rooted in a subtle equivocation. The material conditional is indeed equivalent to it's contrapositive, but if you are using some other mode of conditional then all bets are off (or if not all bets are off then you are not using some other mode of conditional, as the case may be ;) ). The OP has effectively defined a new type of conditional operator that doesn't satisfy the contrapositive equivalence law; rather, it satisfies a "converse equivalence" law. Keeping this in mind, your paradox is avoided. – David H Jul 6 '14 at 5:25
  • Surely if the title of the question is "Making Vacuous Material Conditionals False" I am entitled to assume we're talking about material conditionals. Did I miss something? I agree that I'm often bad at understanding what people mean as opposed to what they actually say. But then in the BODY of the question the OP also used the phrase material conditionals so really, I am not following your point here. – user4894 Jul 6 '14 at 8:49
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In classical Boolean logic at least, logical connectives can be completely understood semantically through their actions as truth-functions of Boolean values. In other words, the truth table associated with a particular logical operator provides a complete description of that operator. The standard truth table for the material conditional → is:

  • p | q || pq

    T | T || T

    T | F || F

    F | T || T

    F | F || T

In your question you proposed a modified conditional (which we'll call ⇒ to distinguish it from the regular material conditional) where vacuously truths are replaced with vacuous falsehoods. The modified truth table that goes with this new operator is:

  • p | q || pq

    T | T || T

    T | F || F

    F | T || F

    F | F || F

At this point you probably notice something peculiar, namely that this truth table is awfully familiar. It turns out your modified conditional would just be completely equivalent to standard logical conjunction. Another oddity to note about this modified "condition": just like conjunction, it's symmetric with respect to p and q! That means, it doesn't matter which one of p and q you call the antecedent and which one you call the consequent. It's this property which it makes virtually useless as an operator whose intended purpose is to capture our intuitive notion of a conclusion predicated on a conclusion.

As example to illustrate how important order is in changing the meaning of a statement, consider the Zeroth Law of Biology. The Zeroth Law states that "If you have had kids, then your parents have had kids." The converse statement, however, is obviously empirically false, so order matters here. Hence, we'd naturally disregard any alternative kind of conditional operator that fails to distinguish order.

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The most important would probably be that we would lose the equivalence between:

  • If P, Q,
  • Not P or Q
  • Not both P and not Q

This would mean that you would no longer be able to infer from:

  • Either Ludwig's not here, or he's in the cellar.

That:

  • If Ludwig is here he's in the cellar.

This is because, of course, this last sentence will be false if Ludwig is not here.

And you wouldn't be able to make this inference the other way round either. Seeing as these types of inference are very useful to us, as well as being very reliable, we'd be in a bit of a poke!

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