Towards a Modal Syllogistic

How many valid syllogisms do we obtain when the four classical categorical forms (‘All A are B,’ ‘No A are B,’ ‘Some A are B,’ and ‘Some A are not B’) are extended to twenty modal forms by introducing a possibility operator P and a necessity operator N—allowing forms such as N(All A are B), P(Some A are B), ‘Some A are not P(B)’, and ‘All A are N(B)’—and given a semantics that rigidly fixes the actual world, where N(All A are B) means that in all possible worlds all A are B, ‘All A are N(B)’ means that everything that is A in the actual world is B in every possible world in which it exists, and ‘Some A are P(B)’ means that there is something in the actual world that is B in at least one possible world?

A new argument against mathematical Platonism

Ex nihilo nihil fit or from nothing nothing comes (Aristotle-Aquinas). Each truth having an ontological footprint must thus have a sufficient explanation. There are no brute facts in reality. Now, beyond a certain complexity threshold, mathematical truths aren’t provable (Gödel-Chaitin). But then it follows that mathematical truths have no ontological import. It's therefore not true that there exist mathematical abstract objects, so that we arrive at a new argument against mathematical Platonism.

Except Virtue

"And if it be objected that one who uses such power of speech unjustly might do great harm, that is a charge which may be made in common against all good things except virtue, and above all against the things that are most useful, as strength, health, wealth, generalship. A man can confer the greatest of benefits by a right use of these, and inflict the greatest of injuries by using them wrongly." (Aristotle, Rhetoric, Book I)

Fitch, Aristotle, Lovejoy and the semantic argument

The conclusion of my semantic argument has it that there are no contingent universally held positive properties. This comes close to the more general thesis that there are no contingent universally held properties whatsoever, either positive or negative. This thesis results in a collapse of the distinction between universality and necessity. For it entails that if a property is universally held it is necessary, while clearly all necessary properties are universally held. So the semantic argument has an effect that resembles the effect of Fitch’s famous proof that if all actual truths are knowable, then all actual truths are known, leading to a collapse of the difference between possible and actual knowledge. The effect of the semantic argument also resembles the effect of Aristotle’s dictum that each possibility is actualized at some point in time, leading to a collapse of the distinction between modality and time.

Aristotle’s dictum is a temporal version of the principle of plenitude, which holds that all possibilities are actualized. The actual world is so full and diverse that there are no unrealized possibilities. It was Arthur Lovejoy who formulated this principle explicitly for the first time. There is in fact a deep connection between the above mentioned thesis and the principle of plenitude. For each one implies the other. To see that the principle entails the thesis, assume for reductio ad absurdum that there is a contingent universally held property P. Since P is contingent there is a possible world in which there exists something that is not P. But then ‘being not P’ is a possibility. So the principle of plenitude entails that there is something that is not P, which contradicts the fact that P is universally held.

To see that the thesis also entails the principle, let X be some possibility. If ‘being X’ is a necessary property, then there exists something in the actual world that is X. So in that case X is actualized. On the other hand, if ‘being X’ is a contingent property, there exists something in some possible world that is not X. But then ‘not being X’ is a contingent property. Given that there are no contingent universally held properties, it follows that ‘not being X’ is not universally held. But then there is something that is X.

The general thesis that there are no contingent universally held properties is thus in fact nothing more than a different formulation of the principle of plenitude. It simply is the principle of plenitude. As said, this thesis is obtained from the semantic argument’s conclusion by removing the positivity restriction. In other words, the conclusion of the semantic argument is a restricted version of the thesis. But then the semantic argument is in fact an argument for a restricted version of the principle of plenitude. As such it alludes to the principle itself. It suggests its truth.