Negation Help

Keane Xavier

Hello, all!

I have a question about the negation of a particular comparative statement that I encountered in the third quiz on negation in the curriculum. Here is that statement: "Small animals can move more rapidly than large animals can." That statement is, of course, negated as others are: "It's not the case that small animals can move more rapidly than large animals can." But the implications of this negated statement confuse me. The implications of this statement are explained to be that either (1) small and large animals move with equal rapidity or (2) large animals move more rapidly than small animals. But why must the entire group of small animals either move in one of these two ways? Don't these implications only account for the negation of the quality on which the two groups are being compared, yet neglect the quantity? Isn't the original statement quantified?

I have a feeling that I'm not being clear, so let me explain further.

Because the author is talking about these animals as sets - small animals and large animals - can we infer that he or she is talking about all small animals and all large animals? Can we thus read this statement as, "All small animals can move more rapidly than large animals can"? If we can, would not the negation of this statement be, "Some small animals cannot move more rapidly than large animals"? From this statement, we would know that there is at least one small animal (yet possibly all) that cannot move more rapidly than large animals. And because we would know that at least one small animal animal (yet possibly all) cannot move more rapidly than large animals, we would also know that there is at least one small animal (yet possibly all) that moves either equally rapidly or less rapidly than larger animals can. Isn't this all that we need to negate the original statement - merely one small animal that moves equally rapidly or less rapidly as large animals? This would deny the truth of "small animals move more rapidly than large animals," wouldn't it? Wouldn't this account for both the quality on which these groups are being compared and the quantity?

Of course, my entire paragraph above relies on an assumption about which I'm unsure that I can make: I took for granted the we can read the original statement as, "All small animals can move more rapidly than large animals." Perhaps we cannot read that statement as such. If so, why? And if that's the case, are we simply to negate the statement in terms of the quality on which the two groups are being compared, as the answer reflects above? (I.e., the negation of this statement would simply mean that small animals, as a group, can either move equally rapidly or less rapidly than large animals, as a group.)

I hope this is clear. If not, I apologize. I'm confused myself.

Any help on how I am to go about negating this statement would be much appreciated. Thank you all for your time!

(I'll be leaving for work soon, so I'll respond to any posts either later this evening or tomorrow morning!)

Cant Get Right

I think you pretty much nailed it. There is an implied “all,” and to negate an “all” statement we just need to convert it to a “some not” statement, which is exactly what you did. And I think your original “it is not the case that” statement works when you add in the implied “all” to get:

It's not the case that all small animals can move more rapidly than large animals can.

So here it’s clear that this is equivalent to a “some not” statement. The meaning is definitely obscured without the all though; the intended meaning becomes a very unnatural way to read the actual statement. You could probably translate it without the “all” and just add “it is not necessarily the case that,” for a more intuitive negation statement.

Great analysis!

quinnxzhang

@"Keane Xavier" said:
Because the author is talking about these animals as sets - small animals and large animals - can we infer that he or she is talking about all small animals and all large animals? Can we thus read this statement as, "All small animals can move more rapidly than large animals can"?

It's not so straightforward. There's an interesting class of statements called "generics", which look like universals on the surface, but are actually quite different. For example, "humans give live birth" might look like a universal, but is in fact a generic. The sentence isn't saying ALL humans give live birth; after all, over 50% of the human population can't give birth at all! There's a lot of debate about how to properly analyze generics, which you can read about here: http://plato.stanford.edu/entries/generics/. The relevant takeaway is that your sentence is probably intended to be read as a generic.

Thus, your sentence, "small animals can move more rapidly than large animals", actually expresses the following proposition: "It's possible that (gen) small animals move more rapidly than large animals". The negation of this is: "It's not possible that (gen) small animals move more rapidly than large animals" (i.e. small animals canNOT move more rapidly than large animals). This is why the explanation tells you small animals must either move more slowly or at the same speed as large animals.

Keane Xavier

Thank you both for your input, @"Cant Get Right" and @quinnxzhang! Both of you provide great advice and thoughtful responses throughout the forums, and I appreciate your responses here.

Let me ensure that I follow, @quinnxzhang. Forgive me if I don't; I don't have a background in logic, although I'm interested in learning as much as I can.

From what I've gathered from your post and from Stanford's discussion of the matter, which is at times esoteric, generic statements express generalizations of individuals of a kind or predicate properties of a kind itself, but these statements aren't quantified - that is, they lack determiners or adverbs of quantification and do no thus provide specific information about the number of members of a kind that have the property or characteristic that is attributed to that kind. Although Gen is an operator that functions like an adverb of quantification, it is not considered a quantifier because it cannot convey how much or how many. In this sense, it cannot be said to share meaning with any of the other quantifiers, such as "all" or "most" or "some," and cannot be equated with them. Generic statements, furthermore, cannot be simply considered universal statements because a generic statement can be true with exceptions - unlike a universal statement.

For these reasons, I cannot infer from the generic statement "small animals move more rapidly than large animals" that all small animals share that ability. The quantity of small animals that move more rapidly than large animals is uncertain. Thus, I cannot equate the original generic statement to the universal statement "all small animals move more rapidly than large animals." So in negating this generic, we simply negate it to "it's not the case that (gen) small animals move more rapidly than large animals," from which we can infer that an unspecified quantity of small animals must either move equally rapidly or less rapidly than large animals.

Am I on the right track? Would you add or correct anything? Thank you for sharing your knowledge! Generic statements are quite interesting, indeed.

quinnxzhang

@"Keane Xavier" said:
For these reasons, I cannot infer from the generic statement "small animals move more rapidly than large animals" that all small animals share that ability. The quantity of small animals that move more rapidly than large animals is uncertain. Thus, I cannot equate the original generic statement to the universal statement "all small animals move more rapidly than large animals."

Right.

@"Keane Xavier" said:
So in negating this generic, we simply negate it to "it's not the case that (gen) small animals move more rapidly than large animals," from which we can infer that an unspecified quantity of small animals must either move equally rapidly or less rapidly than large animals.

This is close, but not quite there. If the original statement were simply "small animals move more rapidly than large animals", then this would be spot on. However, the original generic was a modal statement, i.e. "small animals CAN move more rapidly than large animals". The negation of this negates the possibility, i.e. "small animals CANNOT move more rapidly than large animals" or "IT'S NOT POSSIBLE that small animals move more rapidly than large animals". From this, we are able to infer that NO small animals move more rapidly than large animals. What licenses this inference isn't the (gen) operator, but the negation of the modal operator "can".

It might help to think about a more intuitive analogue. Let's revisit "humans give live birth". This sentence is a generic because, while true, it's not saying that ALL humans give live birth. Now, let's modify this sentence with the modal operator "can", i.e. "humans CAN give live birth". Again, this is a generic because, while true, it's not saying that ALL humans can give live birth (males, children, seniors, infertile women, etc. can't give birth).

The negation of "humans CAN give live birth" is "humans CANNOT give live birth". This negation is saying that it's impossible for humans to give live birth, from which we infer that no humans give live birth. Likewise, "small animals CANNOT move faster than large animals" allows us to infer that no small animals move faster than large animals.

Cant Get Right

That’s really interesting. Totally makes sense. @quinnxzhang , you aware of any LSAT questions hinging on this distinction?

quinnxzhang

@"Cant Get Right" said:
That’s really interesting. Totally makes sense. @quinnxzhang , you aware of any LSAT questions hinging on this distinction?

I'm sure there have been generic sentences in, say, some of the RC passages, but I've never seen a question that you would miss if you didn't know what generics were. The writers are usually pretty explicit in the LR section, and when this distinction would matter, I've only ever seen the writers use universals.

Cant Get Right

Right on.

Keane Xavier

Ah, I did miss the "can" from the original statement, didn't I? That makes sense. It seems that I have more reading to do: modal operators. Thank you, @quinnxzhang, for taking the time to respond at length. I much appreciate it.

Waiting For Grey Day

@quinnxzhang The distinction between modal and generics is incredibly insightful!