For species, does A=B and B=C imply A=C? I think not.

[cheesemonkey]

Suppose there is a civilization of intelligent beings in a world almost exactly like our own. Despite the intelligence of these beings, there are some areas of their brain that are unintelligent, and thus all members of the civilization are constantly being told by their brains to uncontrollably move in the direction they're born facing until they die. All members of this civilization are born facing the same direction, except for the occasional mutation of members of the civilization; one very common mutation is one that corrupts the direction genes and results in the afflicted members of the civilization being unable to move at all; despite this, if born close enough to one another, these afflicted members may reproduce. In effect, all members of this civilization would constantly be moving forward until they die, except the ones who can't move, who would be left behind and still possibly reproduce. As the civilization moves, it finds itself in places that look more and more different from home as they go along; moving forward begins to be less and less desirable; the variety with the mutation of the direction gene remain where they are, making them more fit for survival than the variety who forge onward no matter what happens. As they reproduce there, the variety with no mutation now extinct, a few of them have directional genes that are corrupted further, allowing them to move at their own will. They prove to be more fit for reproduction than the variety that cannot move at all, and due to this, the new varieties who are capable of movement, remembering the stories that their predecessors passed down about the birth of their variety of beings, begin to search for the other still variety. Instead, in their journey, the beings find one another, and, being overcome with joy, they begin to mate; however, due to their differences from growing up in different climates, not all of the different varieties can reproduce.

Variety A can reproduce with variety B.
Therefore, variety A is the same species as variety B.
Variety B can reproduce with variety C.
Therefore, variety B is the same species as variety C.
Variety C cannot reproduce with variety A.
Therefore, variety C is not the same species as variety A.
Variety C is the same species as a species that is the same species as variety A.
Therefore, if the transitive law of equality held true for species, variety C would be the same species as variety A.
Variety C is not the same species as variety A.
Therefore, the transitive law of equality does not hold true for species; either that, or the situation that I've described is impossible.

Do you think it's possible for three varieties of an animal to coexist, for two varieties to be able to mate with one another, and for the third variety to be able to mate with one of the other two varieties but not the other? If you do, then, by the definition of "species," you accept that the transitive law of equality does not apply to species. If you don't, then clearly one of us is wrong, and it's probably me, and if it is, please tell me why, because I'd really like to know.

 

[CRGreathouse]

Do you think it's possible for three varieties of an animal to coexist, for two varieties to be able to mate with one another, and for the third variety to be able to mate with one of the other two varieties but not the other?

Yes, such species exist. Google "ring species" for some examples.

You chose to interpret this as "'is the same species as' is not transitive"; I tend to favor "'is the same species as' is not well-defined".

 

[farful]

Yes, the definition of "species" poses many problems, such as the one you just described. I'm sure there's an entire wikipedia entry on "species" with all the varying definitions.

In any case, since you're trying to use math, let me give you a more realistic model of how biology works. Suppose a species can reproduce if they are >95% similar. (what does it mean to be x% similar anyway? *evil grin*) A and B are 95% similar, thus they can reproduce. B and C are also 95% similar, thus they can reproduce. This does NOT mean A and C are also 95% similar! (also remember every time a species procreate, they will differ slightly due to random mutations and the such)