Modal Propositions - Amateur Logician

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Modal Propositions

Modal Propositions
— modify copula to indicate necessity, impossibility, or possibility

Modal Propositions

Essentially, in traditional logic, modal propositions modify the copula of a categorical proposition so that it is known how the predicate “belongs” to the subject. There are three possibilities: (1) necessity, (2) impossibility, and (3) possibility.

“Men are mortal” can be analyzed as a categorical proposition. It would be reduced as “all men are mortal.” Yet a modal proposition tells us more. It can declare that “all men are necessarily mortal.” The copula “are” is modified such that the predicate not only belongs to men but necessarily belongs to men.

(Does it necessarily? Maybe not. By nature, it is true, we are born mortal. According to traditional theology, before Original Sin, man had the supernatural gift of immortality. That gift was lost after the fall as one of the four “wounds” from Original Sin.)

Take the proposition “men are virtuous.” It is only possible. That is to say, contingent. The predicate of being “virtuous” can be present or absent. As a modal proposition, we get “it is possible that men are virtuous.”

Modal consequence is a way traditional logic treats modal propositions.
Here modality can be “changed” in the inference.
Necessity implies possibility.
Also, impossibility implies not possible.

George Joyce gives this example (p. 104 in Principles of Logic):
“It is necessary for an equilateral triangle to be equiangular.”
Thus, “It is possible for an equilateral triangle to be equiangular.”

In contemporary modal logic, C. I. Lewis labeled different “strengths” of modal logic. The logical intuition that necessity implies possibility is found in modal system S3. This parallels the Scholastic intuition.

□ P→ ◇ P. That is, “if P is necessarily the case, then P is possibly the case.”

The box (□) represents necessity.
The diamond (◇) represents possibility.

The conditional □ P→ ◇ P is part of S3. (It is a law or axiom.)
We can have this modus ponens argument. . .
(1) □ P premise
(2) □ P→ ◇ P by M3 axiom
(3) Thus, ◇ P by modus ponens

Modal negation provides us with equivalences.

-□∀xFx is equivalent to ◇-∀xFx. That is, “it is not necessary that all of x is an F” implies that “it is possible that not all of x is an F” (and vice versa).

-◇∀xFx is equivalent to □-∀xFx. That is, “it is not possible that all of x is an F” implies that “it is necessary that it is not the case that all of x is an F” (and vice versa). In other words, what is not possible implies impossibility.

**A diagram from**
*Formal Logic* by Jacques Maritain, p. 138

In Scholastic logic, modality has its own “Square of Opposition.” The above result from contemporary logic can be appreciated here, too.

We can easily derive, through “oppositional inference,” one modal proposition from another modal proposition in a similar fashion as categorical propositions.

I particularly like the diagram that the great Thomist philosopher Jacques Maritain provides us.