The Modal First Way

[punkforchrist]

I started a thread months back that detailed Robert Maydole’s Modal Third Way (bu.edu/wcp/Papers/Reli/ReliMayd.htm), and I became interested in formulating a modal version of Thomas’ First Way.

1. There possibly exists an Unmoved Mover.

Few will doubt this premise, at least at first glance. I think it is much more likely true than its contradictory, or negation.

2. Whatever is possible is either contingent or necessary.

This is a reference to possible worlds semantics (PWS), which states that a thing can be either impossible (existing in no possible worlds), contingent (existing in at least one, but not all possible worlds), or necessary (existing in all possible worlds). Under PWS, premise (2) is true by definition.

3. Whatever is contingent can be actualized.

I think this is the key premise, and it is one that I’m still working on. However, I will say for now that there is much initial plausibility about (3). For, a contingent thing merely needs to be actualized in some possible world, even if not in the actual world, in order to be actualizable.*

4. An Unmoved Mover cannot be actualized.

An Unmoved Mover possesses no potentiality to be actualized, so its actualization from a state of potentiality to actuality is impossible. Given the truth of premises (1) - (4), it necessarily follows that:

5. Therefore, an Unmoved Mover exists necessarily.

We may formally demonstrate the validity of the proof as follows: Let x = entity; y = Unmoved Mover; and z = actualized.

1. ◊ y (x)
2. ◊ (x) → (◊ x & ~ □ x) ^ (□ x)
3. ◊ z (◊ x & ~ □ x)
4. ~ ◊ z
5. .: □ y
   Q.E.D.

Thoughts?

• I may have just invented a new word.

 

[Just_Lurking]

Can you give a definition of actualized and/or actualizable in the Kripke semantics (see here) of modal logics?

 

[punkforchrist]

Well, if you take the K axiom of (□[A → B] → □A → □B]), where A = a thing x can have a transition from potentiality to actuality, and B = x can be actualized, then the necessary truth of A entails the necessary truth of B, such that x’s going from potentiality to actuality in some possible world entails its being necessarily actualizable. The S5 axiom comes to mind, as well.

 

[Just_Lurking]

I still don’t understand what you mean by actualized.

Take the S5 modal logic of Peano arithmetic plus an a constant “c”, where there is a possible world for each possible value of c. Then ◊(c=1) and ~□(c=1), so (c=1) is contingent. What does it mean to say that (c=1) can be actualized? Can you write the statement “(c=1) can be actualized” using standard modal symbols?

 

[punkforchrist]

Maybe it would help if I defined “actualized” by connotation, as opposed to strict denotation. An acorn is potentially an oak tree - that is, it is an oak tree in potentiality. As it becomes an oak tree, it’s oak tree-ness is said to be actualized. Does this help?
Just Lurking:

Can you write the statement “(c=1) can be actualized” using standard modal symbols?

I did that in premise (3). In your example, we might say (where c=[c=1], and z = actualized): ◊ z (◊ c & ~□ c). In English, this means that for any c, c is possibly instantiated via actualization in some possible world z.

 

[Just_Lurking]

Maybe it would help if I defined “actualized” by connotation, as opposed to strict denotation. An acorn is potentially an oak tree - that is, it is an oak tree in potentiality. As it becomes an oak tree, it’s oak tree-ness is said to be actualized. Does this help?

Not really. If “actualized” has such a limited meaning that it only applies living things growing, then I don’t really see any connection between being “actualized” and any type of general modality such as contingency.

I did that in premise (3). In your example, we might say (where c=[c=1], and z = actualized): ◊ z (◊ c & ~□ c). In English, this means that for any c, c is possibly instantiated via actualization in some possible world z.

Is z() a predicate symbol, so that it takes terms in the logic that represent objects and expresses a property of those objects in the current world, or is z a modality, so that it takes propositions and refers to the truth value of those propositions in alternative possible worlds?

Using c=[c=1] is very confusing, because you are using the same symbol “c” for both an object term and a proposition.

 

[punkforchrist]

Just Lurking:

Not really. If “actualized” has such a limited meaning that it only applies living things growing, then I don’t really see any connection between being “actualized” and any type of general modality such as contingency.

That was just one example. In order for something to qualify as contingent, it must exist in at least one, but not all possible worlds. So, contingent things are possible, but not necessary. The mechanism for a contingent thing’s instantiation in some possible world, however, is the result of some kind of actualization. This applies to growing things, yes, but it applies equally to anything that exists first in potentiality.

Is z() a predicate symbol, so that it takes terms in the logic that represent objects and expresses a property of those objects in the current world, or is z a modality, so that it takes propositions and refers to the truth value of those propositions in alternative possible worlds?

Z is indeed a predicate symbol, but it’s a defined predicate such that z = actualization in each possible world in which z.

Using c=[c=1] is very confusing, because you are using the same symbol “c” for both an object term and a proposition.

Yes, I apologize. I just wanted to make the value of “c” unambiguous.

 

[Just_Lurking]

Z is indeed a predicate symbol, but it’s a defined predicate such that z = actualization in each possible world in which z.

Okay, this helps. However, in your (3), “z (◊ x & ~ □ x)” is not a well-formed formula because the argument to z is a modal proposition, not a term in the logic representing an object.

In my Peano arithmetic example, is z(1) true or false? Similarly for z for other specific values of n.

 

[punkforchrist]

Just Lurking, I think I understand what you’re getting at. You’re saying that there’s no necessary connection between logical contingency and metaphysical actualization. If this is what you mean, then I agree. However, even if some logically contingent thing (like a star) exists eternally and is therefore non-actualized in some possible world W, the very same star may have been actualized (having begun to exist) in W*. This makes a logically contingent entity necessarily actualizable, even if not necessarily actualized in the real world.

The same, however, cannot be said of an Unmoved Mover. For, if it were actualized in any possible world, that would entail a contradiction in the very nature of an Unmoved Mover.

 

[punkforchrist]

Just Lurking:

Okay, this helps. However, in your (3), “z (◊ x & ~ □ x)” is not a well-formed formula because the argument to z is a modal proposition, not a term in the logic representing an object.

It’s possible that I made a mistake in the modal version. In any case, one can follow the English translation to see what I’m trying to say.

 

[Just_Lurking]

Thanks for trying to explain. I’m going to accept that actualization is just another concept from metaphysics that I don’t understand, and that it is not any kind of logical concept from modal logic, which I do understand. Thus, I’m going to conclude that the concept of actualization doesn’t apply to most applications of modal logic, such as the Peano arithmetic example that I gave.

 

[punkforchrist]

Unfortunately, I don’t know much about mathematics. In any case, I appreciate your willingness to consider the argument.