When logic goes to the dogs

The dog's argument has an implicit inference which has to be made explicit:

All cats have four legs.
All creatures that have four legs are cats.
I have four legs.
Therefore I am a cat.

That argument can also be stated, equivalently, as follows:

If the animal is a cat then it has four legs
I'm an animal with four legs.
Therefore, I'm a cat.

Well our canine shouldn't be heartbroken at all for discovering she's actually a feline. Instead she should be depressed she's committed the fallacy of illicit conversion.

The fallacy occurs when given "all p are q" we infer that the converse "all q are p" is also true. Likewise when  given "if p then q" and we conclude that "if q then p," then we've committed an illicit conversion. The "if - then" case may be more familiar when stated as follows:

If p then q
q
Therefore p.

That's the well known fallacy of affirming the consequent and is an example of an illicit conversion. (p is known as the antecedent and q the consequent.) An argument in this form is invalid.

The thing to remember is that "all s are r" or "if p then q" do not necessarily imply "all r are s" or "if q then p." The converse is is not necessarily implied but it might be true. For instance, in causal arguments, if p causes q, and if p is both  necessary and sufficient to cause q, then the converse is also true:

If a sheet of paper is heated to its combustion temperature and oxygen is present then it will burn.
If paper is burning then it's been raised to its combustion temperature and oxygen is present

Both statements are true. This is called a biconditional--the conditional (i.e., the if - then statement) is true both ways. If p then q and if q then p. To formalize the relationship between temperature/oxygen and paper we would say:

Paper will burn if and only if its temperature is raised to its combustion point and oxygen is present.

The phrase "if and only if" indicates this is a biconditonal statement.


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Print References:

Bennett, Deborah J.. Logic made easy: how to know when language deceives you. New York: W.W. Norton & Co., 2004. p. 108-110, 116-117, 130.

Vaughn, Lewis. The power of critical thinking: effective reasoning about ordinary and extraordinary claims. New York: Oxford University Press, 2005. p. 287-289.