# Why is this inference invalid?

So I purchased a book on logic (for beginners) as the subject interests me, and the author presents the following statement as an example of an invalid inference:

"Everyone wanted to win the prize; so the person who won the race wanted to win the prize."

Symbolised as follows, where xP is 'x wanted to win the prize' and xR is 'x won the race': $$\frac{\forall x\;xP}{(|x\;xR)P }$$ where |x means "the object x, such that" (a notation I cannot seem to find anywhere else?).

The author states this is invalid because there is potentially a situation s in which everyone satisfies P but nobody satisfies R. But I do not understand how this could be true, due to the structure of the sentence - surely, xR in the conclusion means the race was ran and there was a winner?

• I think you're right. Maybe there's a logician out there who can tell us why the author is right and we're both wrong. – Ethan Bolker Jul 25 '17 at 0:08
• The race might end in a tie, so "the person who won the race" might not be well-defined. – Qiaochu Yuan Jul 25 '17 at 0:17
• The notation in the conclusion seems to be derived from the "definite descriptions" of Principa Mathematica where it would be written $(^\iota x)Rx$. (Except that the iota should be upside-down). – hmakholm left over Monica Jul 25 '17 at 0:18
• I am assuming that the author's point was to illustrate how error-prone translating from informal English to an unambiguous rigorous notation is since it is rather confusingly written. – Jared Smith Jul 25 '17 at 12:05
• @HenningMakholm Precisely. On the same issue: I recall Russell criticizing Aristotle's logic since it could prove that "since an horse in an animal, the head of the horse is the head of an animal", which becomes false when "horse" is replaced by "hydra", making "the head of the hydra" undefined. – chi Jul 25 '17 at 16:20

There's no guarantee that the race was run, or that someone won - perhaps everyone wanted to win, but the race was called off due to rain. Perhaps every runner crossed the line at the same time, and the officiator decided that no-one would be declared winner. Perhaps no-one actually signed up for the race - typically if a set $S$ is empty, then a statement like $(\forall x \in S) P(x)$ is vacuously true, but any attempt to say something about a particular element of $S$ will fail to hold any meaning.

• So, the conclusion of the inference is a true statement, and the objection of the author is not the right objection. – Ginna Jul 25 '17 at 0:37
• I don't see how vacuous truth has anything to do with this. The second statement to which the restriction you're talking about would presumably apply is not universally quantified over. The first statement is, but it's unrestricted. – spaceisdarkgreen Jul 25 '17 at 0:42
• Well if nobody raced then "Everybody who raced wanted to win" is a vacuous truth. It's true that the wording given allows for "Everybody wanted to win", which doesn't restrict it to racers, but maybe there just isn't anybody in this particular universe at all, in which case every $\forall x$ is a vacuous statement. – ConMan Jul 25 '17 at 1:32
• ok, i misinterpreted the point you were making. The way I learned things the universe is required to be nonempty. – spaceisdarkgreen Jul 25 '17 at 3:52
• @Ginna: Consider the following similar fallacious argument: Every human lives in the solar system. Therefore the human who lives inside Cygnus X-1 lives in the solar system. – user21820 Jul 25 '17 at 10:35

Hmm yes, the problem seems to be with this "notation you can't seem to find anywhere" which I've never heard of either. (Though Henning in the comments seems to have pinned it down).

Referring to 'the person who won the race' as an object is odd from the perspective of mathematical logic since it presumes that person exists and is unique. So translating the second sentence into logic is a bit tricky. Since the author has a notation for it, perhaps it's an abbreviation for a first order statement or we're working in a nonstandard system. If it's an abbreviation, it would probably be defined as: $$(|x\,xR)P = ((\exists!x)xR) \wedge ((\forall x) (xR\rightarrow xP))$$ which is to say that the statement is implicitly asserting that 'the person who won the race' exists.

In the case that there is nobody who won the race, this is false, regardless if $\forall x\, xP$ is true, so that fits with what the author is saying.

• nitpick: Shouldn't it be $(\exists!x)(xR)\wedge((\forall x)xR\Rightarrow xP)$? I think you want a unique $x$ to satisfy $R$, but your current formula is satisfied if many $x$ satisfy $R$ but only one also satisfies $P$. – stewbasic Jul 25 '17 at 1:02
• @stewbasic Thanks! – spaceisdarkgreen Jul 25 '17 at 1:10

The notation is difficult to understand, the way I read it is:

$$\frac{\forall x\;xP}{(|x\;xR)P }$$

is equivalent to:

$$\forall x P(x) \rightarrow \exists x (R(x)\land P(x))$$

Which implies:

$$\forall x P(x) \rightarrow \exists x R(x)$$

And this is indeed not the case as $R(x)$ could never be true, for any x.

On traditional analyses of presupposition, anything that would make this inference wrong would also make the consequent meaningless, since using "the" presupposes the existence of a unique contextually salient individual that won a unique contextually salient race.

Since the logical notation actually contains a symbol corresponding to "the", this should carry over to the truth value of the logical notation, that is, your logical formula contains an open presupposition that must be contextually filled. Traditionally, you'd assume that a sentence that has an unfulfilled presupposition lacks a truth value, that is, it is neither true or false. (The classic case is the sentence "The present king of France is bald", where there is no present king of France.)

So the English sentence "Everyone wanted to win the prize; so the person who won the race wanted to win the prize." is true where the presupposition is fulfilled, i.e. there is an individual that won a certain race, and I would argue that it commits anyone that says it to believing in the existence of the individual, which would make it true once again. Whether that makes the sentence true as an abstract sentence seems debatable to me.

Whether the inference in your logical notation is correct or not would, of course, depend on the exact meaning of your "the"-operator.

• If this is from the Principia Mathematica, however, the sentence should definitely come out wrong, since Russel believed that "The present king of France is bald" is plain wrong. – sgf Jul 25 '17 at 17:47

Your final sentence hints at why this isn't allowed:

"surely, xR in the conclusion means the race was ran and there was a winner?"

Each postulate must mean one thing and one thing only. To say the race was run and there was a winner requires two postulates. This rule exists to avoid paradoxes.

You could quite functionally use some model of logic where this is a non issue, but most logicians agree to use a model where it is an issue. They see it as an instance of the 'Golden Mountain Paradox':

"All golden mountains are mountains, all golden mountains are golden, therefore some mountains are golden"

It's a philosophical conundrum to argue that the winner of the race exists without a postulate to support that. Some operations can be performed on him if you assume he exists: I'm writing about him right now. Others can not, however: We can not ask him to endorse our cereal.

Have you created a winner by introducing them through logic? Is he bald? Can you crown him? Could someone else create someone else just as easily? Is the truth value of that creation consistent (Could they stop existing)?

If you're interested in the subject, you may want to search or ask about 'Existentialism'; the school of thought about how, why, and whether things exist. There's a lot of hokey stuff related, but there are some excellent and entertaining paradoxes. Here are some of my favorites: