Saturday, April 02, 2011

on the necessity of necessity.

I take it that all true conditional statements express a necessary connection between the antecedent and the consequent. Put slightly different: all true conditional statements are true because they express a necessary connection between the antecedent and the consequent, and all false conditional statements are false because they fail to express a necessary connection between the antecedent and the consequent.

Consider, for instance, the standard conditional form: if p then q (symbolically: p ⊃ q). Quite ordinarily, we express things in the standard conditional form to express sufficient conditions, whereby in saying p ⊃ q we mean p is sufficient for q. How is it that in saying p is sufficient for q we are expressing a necessary connection between p and q? At least in this sense: to say that p is sufficient for q is to say q is a necessary condition for p. For example, consider the following true conditional: if there is life then there is water. I take it that this conditional, if true, is true because water is a necessary condition for life, and hence, this is why the antecedent’s being true (that there is life) is sufficient for the truth of the consequent (that there is water).

Another way to say the same thing is to say that the standard conditional expresses natural dependencies. Again, to say that p is sufficient for q is to say that q is a necessary condition for p, and it seems to me that this second statement (and thereby the first) is another way of saying that p depends upon q. In the same way that someone might say that water is necessary for life, someone might equally say that the possibility of life depends upon the existence of water.

As such, all conditional statements that fail to express necessary conditions or dependency relations between the antecedent and the consequent are false, even if the antecedent and consequent are true. For instance, the following antecedents and consequents for the following conditionals are all true:

If Louis lives in Idaho then I am an uncle

If Soren is a person then Jon is a budding logician

If Max is Max then Max kissed a girl

If Max kissed a girl then Max kissed Brianna.

But alas, all of these conditionals are false, for the truth of each antecedent is neither sufficient nor dependent on the truth of each consequent, and neither is each consequent necessary for the truth of each antecedent.

According to the truth functional interpretation of the conditional, all these conditionals are true. By modus tollens, etc.

.:addendum:.

Here’s my modal interpretation of the truth functional interpretation of the material conditional:

p        q         p ⊃ q

T        T         ◊T
T        F        ~◊T
F        T         ◊T
F        F         ◊T

Since, on my view, p ⊃ q is true if and only if p depends upon the truth of q, the second row is sufficient to show that the corresponding material conditional is necessarily false. However, rows 1, 3, and 4 merely assert necessary (but not sufficient) conditions for the corresponding material conditional to be true. That is:

If p is true and q is true, then p ⊃ q is possibly true.
If p is true and q is true, then it is impossible that p ⊃ q is true.
If p is false and q is true, then p ⊃ q is possibly true.
If both p and q are false, then p ⊃ q is possibly true.

Another way to put it:

If p is true and q is true, then p might depend upon the truth of q to be true.  If p is true and q is false, then p does not depend upon the truth of q to be true.  If p is false and q is true, then p might depend upon the truth of q to be true.  If p is false and q is false, then p might depend upon the truth of q to be true.

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1 Comments:

Blogger Loudogg said...

I pretty much agree.

9:39 PM  

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