“If” is in the map

[Chris_Leong]

Update: See the comment by philh. These logical operations are matters of convention, but the way that it is defined is less arbitary than I thought!

I want to start this sequence with a very basic example—the word “if”.

I expect that most people here already know that there are two definitions of “if”:

• Logical if: “If A then B” is true is equivalent to “B OR NOT A”

• Counterfactual if: “If A was true, then B” means consider a world or a set of worlds like ours, but with A being true. In all these worlds, B is true as well”.

In this post we’ll discuss the logical if and counterfactuals will wait until another post. Someone might naively think that this logical if is in the territory, but it is actually in the map.

Consider the sentence, “If pigs can fly, then Trump is Queen of England”. Both components are false and when this occurs, the if-statement is considered true.

However, this is purely a matter of convention. There’s no reason why we couldn’t consider these sentences where the condition is false to be false or undefined. It’s a mistake to ask why this is the case as though it were necessarily the case, but not a mistake to ask why we chose for it to be the case.

These statements being false isn’t part of the territory (of relations between propositions), but is rather a result of how we’ve drawn our map (of relations between propositions).

Note that I haven’t said what relations between propositions are—whether these are abstract objects or purely exist in our minds. But whatever they are, “if” is in the map of relations between propositions.

[Daniel Kokotajlo]

Consider the sentence, “If pigs could fly, then Trump is Queen of England”. Both components are false and when this occurs, the if-statement is considered false.

However, this is purely a matter of convention. There’s no reason why we couldn’t consider these sentences where the condition is false to be false or undefined.

Sentences of the form “If P, then Q” are true when both P and Q are false. According to standard logical interpretation, at least. (After all, if P is false, then not-P is true, and you yourself said if statements are equivalent to B-or-not-A.) (Also, I think you mean “if pigs can fly” since using “could” makes it a counterfactual rather than logical claim, right?) And there are good reasons for this, if I recall correctly from my logic classes (though I don’t remember what they were, and I agree that ultimately it’s a matter of convention.)

[philh]

Stipulate that we want to choose some boolean function $\to$ on two inputs to represent “if $P$, then $Q$”. Then we want $T\to T=T,T\to F=F$ or what are we even doing with our lives.

So we have four choices for how to define $F\to T$ and $F\to F$. The standard choice says both are true. What about other choices?

Suppose we have $F\to F=F$. Then we’ve translated “if $P$, then $Q$” into something that asserts $Q$ and maybe (depending on $F\to T$) also asserts $P$. That seems like a bad translation. So let’s say $F\to F=T$. (Notably, if we add “undefined”, then $F\to F=U$ has the same problem, when $P$ and $Q$ are defined it lets $P\to Q$ be true only if $Q$ is true.) Without this, we can’t really translate “if $P$, then $Q$. Not $Q$. Therefore, not $P$,” because $(P\to Q)\wedge \neg Q$ is a contradiction. (Which does still give us $\neg P$, but it also gives us $P$, so.)

All that’s left is $F\to T$. If we say this is false, then our “if $P$, then $Q$” is translated into something that means ”$P$ and $Q$ are either both true, or both false”, or $P=Q$. That seems like a bad translation too.

Basically, “if $P$, then $Q$” just doesn’t translate very well into boolean logic, but all the other ways to translate it seem worse.

[Chris_Leong]

Thanks for posting this. I didn’t realise that there were such good reasons for this convention!

[Chris_Leong]

Fixed now. I really should have checked my post for mistakes like this.

[EpicNamer27098]

If I drop this ball, it will bounce back up to me. Is that ‘if’ in the territory? I feel like the potential to bounce when dropped is extrapolable from the physical properties of the ball. Like there’s a mathematics of correct hypothetical situations to be discovered, which is part of the territory as much as e.g. Godel’s theorem is.

[Dagon]

This is a variant (sometimes considered a separate case) of the counterfactual if—the hypothetical if. Or possibly an even more specific variant, the predictive if.

We don’t yet know what the future territory is—you may or may not drop the ball. It may or may not bounce (perhaps there’ll be carpet there when dropped). The map contains a distribution of things that correspond imperfectly to the territory. The conditional statement that, for those imagined territories where I drop the ball, the ball will bounce up, is definitely in the map. Any time you talk about the “imagined” or “possible” or “potential”, you’re describing a map rather than the territory.

[Chris_Leong]

Also, see my post on Natural Structures

[Chris_Leong]

Counterfactuals are actually much harder to define than you might think.