46 | New Scientist | 8 January 2022
two equals four”. This would have a subjective
probability of 1, in other words it is certain.
I would put “The bus will be late” at be 0.75;
more likely than not. “The next coin toss will
be a heads” would get a value of 0.5. And “two
plus two equals seven” would be 0. Subjective
probabilities can vary from person to person,
but not usually by much. In this story, the
values I give are my choices, but they are likely
to be similar to what you would assign.
When we encounter a paradox, we can break
it down into a set of claims and a conclusion
and then evaluate the subjective probability
of each. To see how this works, let’s try it with
a deceptively simple puzzle called the liar
paradox, which comes in the form of a single
sentence: “This sentence is false.” We will
call this sentence “L”. If L is true, then L,
which claims that L is false, must be false.
And if L is false, then it is false that what L
says is false, so it would then be true. And
since L is a declarative sentence, and should
therefore be either true or false, it seems
to follow that L is both true and false. When
we look at the paradox as a set of claims
and give their subjective probabilities in
parentheses we get the following:- If L is true, then it is false (1.0)
- If L is false, then it is true (1.0)
- L is either true or false (0.9)
Conclusion: Therefore, L is both true and false (0)
The subjective probability of the parts of
this paradox are high, yet the conclusion
seems like nonsense. The first claim, that if the
sentence is true, then it is false, follows from
the meaning of the sentence. So does the
second claim. As for the third, it is an oft-cited
feature of declarative sentences that they
are true or false, and L takes the form of a
declarative sentence. Yet the conclusion
is an outright contradiction.
One advantage of my method is that
it helps us see how strong a paradox is.
Assuming we are using valid logical reasoning,
then the greater the disparity between the
subjective probabilities of the claims and
the final conclusion, the stronger the paradox.
The liar paradox is a strong paradox.
Another benefit of this approach is that itIt feels like common sense to say that all
statements must be true or false. Aristotle
called it “the most certain of principles”. But
is it? Some philosophers hold the radical view
that statements can be both true and false,
which is called dialetheism. This view is
gaining traction among logicians because
of the way it can help with paradoxes.
What are we to make of this strange idea?
One big problem for dialetheism is that it
allows direct contradictions to exist and
this leads to a well-known difficulty in logic
called the problem of explosion. If it is fine
to say it is raining and not raining, then our
entire basis for belief and action blows up.
One reason this is so tricky is that you can
use a contradiction to prove anything you like.
To see how this works, let’s take the sentence
“Alice is pregnant” and call it A. According
to dialetheism, we can say that both A and
Not A are true. Now we can construct a
statement of logic in which two options are
present, A and B, where A is the sentence
about Alice and B can be absolutely anything,
such as “bread is expensive”. We can say that
either A or B must be true, because we have
already assumed that A is true as part of
our starting assumptions. If A or B is true
and so is Not A (which, again, was part of
our starting point) then, according to a
rule of inference called disjunctive syllogism,
we can conclude that B is true.
All this seems like an indictment of
dialetheism. Surely, we shouldn’t be able to
use a contradiction to prove some unrelated
fact chosen more or less at random? But on
the other hand, strong mind-benders like the
liar paradox (see main story) force us to give
ideas like this a chance. Plus, dialetheists
have developed clever ways of denying that
disjunctive syllogism is always valid.
Toying with the foundations of logic in
this way is a worthwhile pursuit. Much of
science is based on logical reasoning and
there is no guarantee that the classical
rules are the perfect or only tools we need.
The human brain works in a more malleable
fashion most of the time, with grey areas
and contradictions. Perhaps our logic would
be better if it did likewise.Both true
and false
“ Paradoxes
don’t exist in a
vacuum, they
are puzzles that
take shape in
our minds”