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The Liar’s paradox

“This statement is false.”

The statement says it is false, and if we believe what it says, then the statement must be false. But if the statement is false, and it says that it’s false, then the statement must be telling the truth.

We can never settle on an answer for very long, and that is deeply disturbing. As we’ll see in a moment, the consequences for logic are dire.

The problem

Logic allows us to reason about ideas. For example, suppose that “It’s raining,” and “If it’s raining, then I should take an umbrella.” Well? Should I take an umbrella, or not?

Let’s start by listing what we know:

1.RAssumption
2.if R then UAssumption

Here we’re using a shorthand: R = “It’s raining,” and U = “I should take an umbrella.”

Now let’s apply one of the rules of inference to reach a new conclusion:

1.RAssumption
2.if R then UAssumption
3.UModus ponens (1 and 2)

So, yes, you should take an umbrella!

Although this was a simple example, logic problems can become tricky very quickly. The rules of inference allow us to continue to reason far beyond the limits of what would otherwise be possible.

Paradoxes

When given valid assumptions, deductive logic is infallible. However, when given paradoxical assumptions, deductive logic fails catastrophically.

Let’s see this for ourselves:

1.AAssumption
2.not AAssumption

There is one statement that we should never be able to derive: the doomsday scenario, X = "Humanity must be destroyed."

Now watch as deductive logic drags us straight into the apocalypse:

1.AAssumption
2.not AAssumption
3.A or XAddition (1)
4.XDisjunctive argument (3 and 2)

A lot of thought has gone into finding ways to protect us from drawing invalid conclusions from a paradox.

However, there is a very simple, and very satisfying, solution.

The solution

In logic, we take a statement, such as “It’s raining,” and we replace it with a symbol like “R.” This seems so reasonable that no one would ever think to challenge it. But it is, in fact, a very strange thing to do.

In mathematics, we would never take an equation, such as “x = 1,” and replace it with a symbol like “X.” We prefer to keep value separate from the variable, so we can reason about the value.

We should be doing the exact same thing for logic.

Let’s take a statement, such as “It’s raining,” and represent it like this:
r = true, where r represents “It’s raining.”

If we don’t know whether it’s raining, then:
r ∈ {true, false}

And, now that we can express logic in terms of variables and equations, we can solve those equations. It turns out this is the same as proving a theorem.

Proofs

Let’s prove the following theorem, but without using any of the rules of inference:

1.not PAssumption
2.if Q then PAssumption
3.if (not Q) then RAssumption
4.if R then SAssumption
5.not QModus tollens using (2) and (1)
6.RModus ponens using (3) and (5)
7.SModus ponens using (4) and (6)

Let’s start by listing the assumptions as equations that we’ll need to solve:
not p = true, if q then p = true, if (not q) then r = true, if r then s = true

These equations are constraints, and we’ll apply them one by one.

First, let’s list all possible values for p. Then cross off any rows that fail to satisfy the first constraint, not p = true:

pnot p
falsetrue
truefalse

That leaves us with:

p
false

And we’ve made our first deduction: p = false. Now let’s apply the rest of the constraints:

pqif q then p
falsefalsetrue
falsetruefalse

pqrif (not q) then r
falsefalsefalsefalse
falsefalsetruetrue

pqrsif r then s
falsefalsetruefalsefalse
falsefalsetruetruetrue

By the time we’ve finished applying the constraints, the solution is ready:

pqrs
falsefalsetruetrue

Not only did we prove that s = true, we also found the value for every other variable as well.

And we now have a way to check the truth of any statement involving p, q, r, or s.

For example, does (P or R) and S logically follow from the four initial assumptions?

The answer is yes, because (p or r) and s = true for every combination of values that satisfies the initial constraints:

pqrs(p or r) and s
falsefalsetruetruetrue

Does Q and S logically follow from the four initial asumptions?

The answer is no, because q and s = false when q = false and s = true.

pqrsq and s
falsefalsetruetruefalse

Not only are we able to determine whether any statement logically follows from the assumptions, we’re also able to provide counterexamples.

Furthermore, we’re now immune to logical paradoxes.

Solving paradoxes

Let’s try to reproduce the failure from earlier, where the rules of inference led to an apocalypse:

1.AAssumption
2.not AAssumption
3.A or XAddition (1)
4.XDisjunctive argument (3 and 2)

If our new way of reasoning is sound, then we’ll be able to spot the flaw in that argument.

Let’s start by listing the constraints that we need to satisfy:
a = true, not a = true

As before, let’s construct a truth table. Then we’ll cross off any rows that fail to satisfy both equations:

anot a
falsetrue
truefalse

And now we can see what was so different about this set of assumptions: it has no solutions!

But that’s not exactly a paradox, is it? Equations may have many solutions, one solution, or no solutions: All of those outcomes are expected, and all of those outcomes are perfectly fine.

Also note that our new way of thinking protected us from making the doomsday deduction. We are able to reason in perfect safety, even when the input is paradoxical.

The Liar’s paradox

Now that we have a system for analyzing paradoxes, let’s return to the Liar’s paradox:

“This statement is false.”

Let’s use a to represent the statement as a whole.

Because we’re asserting the statement, we’re claiming that:
a = true

But the statement itself makes a further claim. The statement claims that:
a = false

Taken together, this gives us those paradoxical assumptions:
a = true, not a = true.

Consequently, the Liar’s paradox is really the apocalypse scenario that we solved earlier.