Friday, May 14, 2004

Necessitas Consequentiae Versus Necessitas Consequentiis

Take the sentence, 'If I will die tomorrow, then I will die tomorrow.' This has the form If p, then p, where 'p' denotes a proposition. Any sentence of this form is not just true, but logically true, i.e., true in virtue of its logical form. Now every sentence true in virtue of its logical form is necessarily true. (The converse, however, does not hold: there are necessary truths that are not logically true.) Thus we can write,

Necessarily(if p, then p)

or

1. Nec(p-->p).

The parentheses suggest that the necessity attaches to the consequence, represented by the arrow, and not to the consequent, represented by the terminal 'p.' (When speaking of conditionals, logicians distinguish the antecedent from the consequent, or, trading Latin for Greek, the protasis from the apodosis.) Thus the above is an example of the necessitas consequentiae. This, however, must not be confused with the necessitas consequentiis, which is exemplified by

2. p-->Nec p.

In (2) the necessity attaches to the consequent. It should be obvious that (1) does not entail (2), equivalently, that (2) does not follow from (1). For example, although it is necessarily true that if I will die tomorrow, then I will die tomorrow, it does not follow, nor is it true, that if I will die tomorrow, then necessarily I will die tomorrow. Proving fatalism cannot be that easy. For even if I do die tomorrow, that will be at best a contingent occurrence, not something logically necessitated. (Think about it.)

To confuse (1) and (2) is to confuse the necessity of the consequence with the necessity of the consequent. This is an example of what logicians call a fallacy, i.e., a typical error in reasoning, and in particular a modal fallacy in that it deals with the (alethically) modal concepts of necessity and possibility and their cognates.

Class dismissed.