From Possibility to Actuality

The Argument from S5 The purpose of this page is to show how to deduce an actual truth from a possible truth. Initially, it may seem absurd to think we could deduce an actuality from a possibility. After all, possibilities are possibilities, not actualities. However, twentieth century advances in logic discovered a special type of case. I will show that case here.

Definitions

I begin with a definition:

X is possible = X is consistent with the truths of reason.
For example, a cubical object is possible if a cubical object is consistent with the truths of reason.

A truth of reason is any truth that can be seen to be true by the light of reason (that is, a priori). For example, by reason, we can see the truth of the law of non-contradiction: not, both A and not A, for any A.

You might wonder why I have defined "possibility" in terms of consistency with truths of reason. It is because this notion of possibility is very weak. Its weakness makes the task of deducing an actuality from a (weak) possibility all the more challenging—and significant.

Next, we define "necessity" in terms of possibility:

X is necessary = not X is not possible.
For example, if it is not possible that not, 2 + 2 = 4, then it is necessary that 2 + 2 = 4.

The next step is to give an axiom that characterizes possibility:

Axiom 5: whatever is possible (consistent with reason) cannot be impossible (inconsistent with reason).
To illustrate, imagine a cubical object. If this object is consistent with reason, then this object is possible (by our definition of "possible"). Moreover, it is impossible for this object to be inconsistent with reason. That is because the principles of reason are themselves invariable. Consider, by contrast, a square circle. That shape is not consistent with reason. Its inconsistency doesn't just happen today; a square circle cannot be consistent with reason. Possibilities, on the other hand, cannot be inconsistent with reason.

A couple notes about Axiom 5 are in order. First, in the present context, we can treat this axiom as part of the definition of "possible." That is because we are concerned with a certain weak, logical notion of "possibility." Axiom 5 puts the scope on logical possibilities that cannot contradict reason. Anything that can contradict reason fails to count as "possible" in the sense here.

Second, I call it "Axiom 5" because it is equivalent to the characteristic axiom of S5 modal logic. But note that this axiom is not the same as everything associated with S5. It is important to distingsuish Axiom 5 from theorems that are deducible from it. Many theorems in mathematics and logic are not obvious on their face. In particular, the theorem I am about to deduce from Axiom 5 is not obvious on its face. But it would be a mistake to dismiss the theorem just because it is not obvious on its face.

The Possibility-To-Actuality Conjecture

Consider the following conjecture:
P-to-A: if A is possible, then A is actual, for some hypothetical A.
This conjecture is not about all As. Rather, it is about a certain type of A. In particular, it is about any A that is a hypothesis about what's necessary. For example, A could be the hypothesis that the foundation of the universe has necessary existence. For the sake of neutrality, I will let A be the hypothesis that necessarily, P, where 'P' is some unspecified statement.

I give a proof of the P-to-A conjecture below. For the sake of efficiency, I first define some symbols. I abbreviate "possibly" and "necessarily" as "◊" and "□," respectively. And I use "¬" to express negation (or "not"). We may now state our conjecture as follows:

If ◊A, then A (where A = □P).
Here is the proof (adapted from a proof given to me by Chris Menzel):
1. Let Q = ¬P.
2. If ◊Q, then □◊Q (Axiom 5).
3. If ◊¬P, then □◊¬P (Definition of Q).
4. If ◊¬P, then ¬◊¬◊¬P (Definition of '□').
5. If ¬¬◊¬◊¬P, then ¬◊¬P (Contraposition).
6. If ◊¬◊¬P, then ¬◊¬P (Double Negation).
7. If ◊□P, then □P (Definition of '□').
8. If ◊A, then A (Definition of A).
Q.E.D.

We have just demonstrated the P-to-A conjecture.

We can also give an intuitive presentation of the deduction in terms of possible worlds (i.e., complete ways things could have been). Imagine □P is possible. Then □P is true in at least one of the possible worlds. But □P cannot be true in any world unless P is true in all worlds, since □ means truth in every possible world. Therefore, P is true in the actual world. This demonstration assumes, via Axiom 5, that possible worlds cannot be impossible worlds.

Note: the symbolic proof does not require possible worlds. While "possible world" language can help us express certain claims, it is not necessary.

Significance

The P-to-A principle has some interesting applications. For example, it features famously in an ontological argument for the existence of God. Consider, for example, the following premises:

Premise 1: A perfect being is possible.
Premise 2: Necessarily, if a perfect being exists, then that being has necessary existence.
These premises may seem innocuous on their face. Yet, when we conjoin these premises with the P-to-A principle, we can deduce something striking. We can deduce that a perfect being actually exists. Here is how. First, from (1) and (2), it follows that possibly, necessarily, a perfect being exists. Then, applying P-to-A, possibly, necessarily P reduces to P. In other words, it follows that a perfect being actually exists.

It is too early to celebrate, however. For we can reverse the argument to show that a perfect being does not exist. Consider the following:

Premise 1: A world without a perfect being is possible.
Premise 2: Necessarily, if a perfect being exists, then that being has necessary existence.
This time logic works in another direction. Here's the proof against God. From (1) and (2), it follows that possibly, necessarily, there is no perfect being. Once again, possibly, necessarily P reduces to P. Hence, no perfect being exists.

You might think this pair of arguments shows that the P-to-A principles is pointless. However, P-to-A still helps us see something significant. It helps us see that certain propositions, including certain famous propositions about ultimate reality, are either necessary or impossible. There is no middle ground. This result was not so clear before the late twentieth century.

Here are a couple reasons why the discovery is significant. First, an open area of research involves discovering ways to break the symmetry between the possibility claims in parallel arguments (such as one I explore here). There are many avenues open for exploration and discovery. Maybe you will discover new ones yourself.

Second, the P-to-A principle has caused an advancement in arguments for a necessary foundation. Alexander Pruss and I mark out such arguments in our book, Necessary Existence. What's especially interesting about those arguments is that they are not cancelled out by reverse, parallel arguments.

You can find more applications of the P-to-A principle at my website, necessarybeing.com. This website uses a version of the P-to-A principle in some pathways to a necessary, causal foundation.