8 January 2022 | New Scientist | 47This might seem abstract, but it tells us
two interesting things. First, sometimes it
is helpful to separate out logical arguments
to avoid self-reference problems. Second,
it hints that we may need to question our
intuition about the fundamental nature
of truth (see “Both true and false”, left).
Paradoxes aren’t solely relevant to
semantics and logic, nor are they always
ancient puzzles. As I have explored paradoxes,
I have seen how they occur in everyday life
and draw attention to recurring issues with
how we think about the world.
One puzzle that keeps rearing its head is
known as the Jevons paradox. It was discovered
in the 1850s by the economist and logician
William Stanley Jevons in the context of coal.
It seems to show that by increasing the fuel
efficiency of machines, you end up using more
fuel overall. Here is how it looks, point by
point, together with subjective probabilities:- Increasing the fuel efficiency of a piece
of technology will allow for less consumption
of fuel for the same amount of work (0.9) - A piece of technology that will allow for
less consumption of fuel for the same amount
of work will be used more often (0.8) - Increasing the use of a piece of technology
will increase the amount of overall fuel
consumption (0.9)
Conclusion: Therefore the increasing fuel
efficiency of a piece of technology will increase
the amount of overall fuel consumption (0.3)
Even on a first read, this paradox feels
surprising, but less deep than the liar paradox.
It turns out there is an way to increase the
subjective probability of the conclusion. Think
of what has happened as internet speeds have
gone up over the past two decades. We might
have assumed that faster speeds would mean
that people take less time to complete tasks
and so overall time spent online would
decrease. However, the increased speed
actually opened up new opportunities and
people ended up spending more time online.
This paradox teaches us that changes to part
of a system don’t necessarily get multiplied
at a constant rate as that improvement rolls
out across the whole system. >suggests a way to tackle paradoxes – by finding
a way to lower the subjective probability of
some of the claims. In the liar paradox, we
can see that premise 3 has the lowest subjective
probability. Premises 1 and 2 are true as a direct
result from the meaning of L. Premise 3,
though, assumes that every statement is
either true or false. This seems intuitively
right, but there is that word – intuition.
Perhaps we are on to something here.
One legitimate response to this paradox
is to say it highlights a problem with our
intuition about truth and falsehood. The
philosopher Alfred Tarski pointed out that apeculiar feature of languages like English is
that they don’t separate out the expressions
used to speak about everyday facts, such as
“the book is on the table”, from the expressions
that refer to the language itself, like “the
sentence about the book is true”. Tarski didn’t
think this would be a practical problem for
most of us. But he did realise that it would
cause trouble in certain circumstances, such as
when a linguist is analysing the structure of a
language. He proposed that in those situations
we should employ a separate artificial meta-
language, typically based in mathematics, to
help us talk about the language being analysed.SH
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