Paradox

A Solar Eclipse

1. What is a paradox?

A paradox is a true statement that appears to be self-contradictory.

Paradoxical statements often involve the use of metaphorical language. Such paradoxes work either by equivocating two different meanings through a single label, or by bifurcating a single meaning through two different labels.

Some common examples of paradoxes are:
• The Christian Doctrine of the Triune God
• Identity of the Morning Star and the Evening Star
• The notion of “self-control”
• Socrates’ claim that: he knows that he knows nothing

However, what we realize less often is the fact that a paradox is only possible in a world soundly entrenched in a system of logic.

2. Can logic be paradoxical?

Logic has a strict set of rules that never change. This strictness makes logic very useful to humans. But usefulness is not the only feature of logic. It can be the foundation for creative possibilities — and even fun.

Let’s examine a paradox in formal logic. Below is a diagram of a [customized] “truth-table” for the conditional statement (P→Q). (NB: A conditional statement is read like this: “If P, then Q.”)

This truth-table shows the different conditions under which the conditional statement (P→Q) can be either TRUE or FALSE. Let’s look at each of the lines:
• Line (1) says that when the conditional statement (P→Q) is TRUE, it is POSSIBLE for both P and Q to be TRUE.
• Line (2) says that when the conditional statement (P→Q) is TRUE, it is POSSIBLE for P to be FALSE while Q is TRUE.
• Line (3) says that when the conditional statement (P→Q) is TRUE, it is IMPOSSIBLE for P to be TRUE while Q is FALSE.
• Line (4) says that when the conditional statement (P→Q) is TRUE, it is POSSIBLE for both P and Q to be FALSE.

All of these lines are considered “logically valid.”

But notice that line (3) is different from the rest because it is the only situation in which the suggested values of P and Q are IMPOSSIBLE, given that the conditional statement (P→Q) is TRUE. Therefore, to assert the suggested values of P and Q according to line (3) as being POSSIBLE (given that the conditional statement (P→Q) is TRUE) would be considered paradoxical.

But before we move on, let’s look at a visual pattern of the conditional statement (P→Q). This is because it can be easier to grasp the conditional statement when looking at it as a visual pattern. Below is a modified Euler-Diagram of the conditional proposition (P→Q). (NB: In the modified Euler-Diagram below, the purple area represents an overlap between P and Q.)

In this modified Euler-Diagram (diagram A), there is an area designated for each of the situations in lines (1), (2), and (4) — but not line (3). This is precisely because the modified Euler-Diagram can only show what is logically POSSIBLE. The fact that the situation of line (3) does not even have a place in the modified Euler-Diagram of a world in which the conditional statement (P→Q) is TRUE, is evidence of the IMPOSSIBILITY of a case when P is TRUE while Q is FALSE in such a world.

Again, it would paradoxical to say that despite the visual evidence, that it is still POSSIBLE for P to be TRUE while Q is FALSE.

But suppose that this is just what I am going to claim!

3. Can paradoxes be resolved?

So far, we’ve seen what philosophers like to call the “safe” picture. But now, let’s look at a view that philosophers like to call the “sensitive” picture. Below is a modified Euler-Diagram that does make a place for the situation in line (3) of the truth-table.

In this second modified Euler-Diagram (diagram B), you can see that the red area shows a situation where P is TRUE while Q is FALSE! On this view, what was originally IMPOSSIBLE is now POSSIBLE. (Philosophers sometimes call this view the “sensitive” picture.)

Earlier, it seemed contradictory to say that it is POSSIBLE for P to be TRUE while Q is FALSE, in world where the conditional statement (P→Q) is TRUE. But now we can see that so long as some P are part of Q (P1) while some other P are not a part of Q (P2), we can say that it’s POSSIBLE for there to be cases where P is TRUE while Q is FALSE. The paradox is resolved once we see that both P1 and P2 are both more generally P — the difference between them being that P1 is the part of P that is entrenched in Q, while P2 is the part of P that is not.

In this way, the metaphor can even apply to formal logic to produce paradoxes.

And so, maybe it isn’t so strange to say that what is FALSE is also TRUE, insofar as what we call “false” is simply potential-truth that is yet unactualized.

What is really interesting for me is the possibility that Q might be able to move. If it could move, and it moved just enough to engulf all of P2 as well as P1, then the paradox would disappear as well. Perhaps the more illuminating question is whether Q can move or not, rather than how P can both be and not-be at the same time.

(Question: Has the paradox been sufficiently resolved by this account? Or, is there still more to the phenomenon of paradoxes that demands deeper exploration and also a deeper explanation?)

*If there is any part of this post that is explained badly or is explained in a confusing way, please do not hesitate to email me with your concerns or questions and I will do my best to try to be more satisfactory. It does help me tremendously to know how I could put things better.

• Last updated 2017, July 27 @ 9:49 pm.

For Further Reading:
Hesiod, Theogony (lines 25-35)
Parmenides, Proem
♦ Plato, Parmenides
♦ Bertrand Russell, Wisdom of the West (p.13-31, 36-45)
♦ Robin Waterfield, The First Philosophers (p.49-68)
♦ Kathryn Morgan, Myth and Philosophy from the Presocratics to Plato
♦ Jon Barwise & John Etchemendy, Language, Proof and Logic (p. 178-181)
♦ Ernest Sosa, How to Defeat Opposition to Moore
♦ Robert Nozick, Knowledge and Skepticism
Plato, Theaetetus (201c-d)

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