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.

I begin with a definition:

- X is possible = X 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

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).

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.

- P-to-A: if A is possible, then A is actual, for some hypothetical A.

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).

- 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).

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
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.

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.

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.

I hope this page serves you.

© Joshua Rasmussen