# Act and Potency in Logic and Mathematics

Presently, has there been an attempt to incorporate the Aristotelian-Thomistic notions of Act and Potency into Logic and Mathematics?

Also, how would the current interpretation of the principle of excluded middle work in a logic that incorporated act and potency? I ask because I don't think the current interpretation of the principle of excluded middle would work in a logic that incorporated act and potency. Otherwise, the following statement would be a contradiction: X is both actually hot and potentially cold. Or we would have to modify the principle of excluded middle.

• Are you sure that you mean excluded middle rather than non-contradiction? Modal logic with its actual world and possible worlds is somewhat similar to actual/potential in Aristotle. There would be no problem with X being actually hot and possibly cold, it happens in different worlds. However, it has been argued that possible worlds are antithetical to Aristotle's understanding of the potential, see Felt, Impossible Worlds:"potentiality, as an intrinsic character of the actual, has tended to be supplanted by possibilities." Commented Jun 2 at 7:58
• @Conifold, aren’t they logically equivalent though in some logics? Commented Jun 2 at 8:07
• Excluded middle and non-contradiction? Not really, they are dual to each other. Inutitionists keep non-contradiction but drop excluded middle, and paraconsistentists keep excluded middle but drop non-contradiction, so they are independent conditions. Commented Jun 2 at 9:03
• @Conifold, by some logics I meant classical logic or Boolean Algebra though Commented Jun 2 at 9:18
• To be equivalent in the classical logic, each law needs to be derivable from the other. As intuitionism and paraconsistency show, this is not the case. Commented Jun 2 at 11:15

Otherwise, the following statement would be a contradiction: X is both actually hot and potentially cold. Or we would have to modify the principle of excluded middle.

No problem as long as X is not both actually hot and actually cold.

Still, there is no need to "incorporate" actuality and potentiality into logic for they are and always have been crucial aspects of it.

What we may need to do is to incorporate these notions into formal logic, and this is what I take your question to be about.

All we would have to do is to make an algorithm which would be the correct simulation of human logic.

• Yes. I meant formal logic. Commented Jun 2 at 9:20

Presently, has there been an attempt to incorporate the Aristotelian-Thomistic notions of Act and Potency into Logic and Mathematics?

In mathematics, I have only seen it applied to Infinity, where potential infinity is the simply endless, e.g. the pre-theoretical sequence of natural numbers, while actual infinity is infinity realized, as in an infinite sequence with a maximal element, e.g. the natural numbers plus a point at infinity. (Or, e.g., in the distinction between absolute infinity whose reciprocal is (absolute) zero, vs "potential" infinities, such as the infinite surreal numbers, whose reciprocals are infinitesimals, not zero).

In physics there is an analogy with Potential Energy vs the other forms of Energy (do these have a name, as opposed to "potential"?). I am not aware of any specific use in logic.

the following statement would be a contradiction: X is both actually hot and potentially cold. Or we would have to modify the principle of excluded middle.

Logic is not impacted, reasoning remains the same. Rather, that is not a contradiction: not only "actually hot" and "potentially cold" are not opposite (it is not the same as plain "hot" vs "cold"), it seems rather reasonable, for how a bit weird to state, that "a body that is (actually) hot is potentially cold", indeed a hot body can become cold: by cooling.