Questions which have false conditions

Question

There are many "questions" on the internet like

If $$1=5$$ $$2=6$$ $$3=7$$ $$4=8$$ then how many is $5$?

With one "logic" answer is $9$ because $n=n+4$, then $5=9$. With other "logic" answer is $1$ because if $1=5$ then $5=1$. Of course, these answers aren't correct because there are no number $n\in\mathbb{N}$ such that $n=n+4$.
However, is it possible to solve it mathematically? We can write this as a system of equalities: $$1=5\land2=6\land3=7\land4=8\land5=x$$ and it will become $$\operatorname{false}\land\operatorname{false}\land\operatorname{false}\land\operatorname{false}\land5=x$$ which have only solution $x\in\emptyset$.
So, can we say that this question have only solution $x\in\emptyset$, i.e. there is no number which satisfies the conditions? Or this question does not have answer because conditions are false?

Answer 1

I've usually seen the question at the end written as: $$ 5 = \; ? $$

If you take the $=$ symbol literally at its usual meaning in these puzzles, then the puzzle statement is equivalent to solving for $x$ in

$$ (\mbox{false} \wedge\mbox{false}\wedge\mbox{false}\wedge\mbox{false}) \Rightarrow (5 = x)$$

which is vacuously satisfied by any value of $x$, since the left-hand side of the implication is false.

What the puzzle is intended to mean is that we're to interpret the $=$ symbol as a (clearly non-reflexive) relation $\to$, and solve for $x$ in $5\to x.$ Alternatively, since the intended relation is usually a function, we could write $f(1)=5$, etc., and then solve for $x$ in $f(5) = x.$

It can be annoying to see the $=$ symbol abused this way when a more appropriate symbol could just as well have been used. But evidently the people posing these puzzles don't have experience with symbols other than $<$, $>$, or $=$ to represent relations, or they don't expect their audience to have such experience. This has little to do with the quality of the puzzle, which may be trivial, moderately interesting, or ill-conceived regardless of the notation in which it's presented.

Answer 2

I think the best way to understand such questions is that the asker has never properly internalized what "$=$" means. (They have probably been told, but it didn't catch hold).

These askers appear to understand $A=B$ as meaning "$A$ becomes $B$, through some unspecified process". This understanding makes enough sense in many cases -- namely when $A$ is an arithmetic expression and $B$ is the canonical notation for its value -- that it is apparently possible to go through school without noticing that it is wrong.

So what the asker really means by the question is

I'm thinking of some process.
When we apply it to 1 we get 5 back.
When we apply it to 2 we get 6 back.
When we apply it to 3 we get 7 back.
When we apply it to 4 we get 8 back.
What do we get back for 5?

Read this way, the question makes as much sense as any other continue-this-sequence question. It's just the asker who, out of ignorance, misuses notation when he tries to state it.