I do not pray. Therefore God exists. "If A then B" is logically equivalent to "if C then D." You're going to have offer a proof that is not the case without equivocating between deductive and inductive logic. I don't see how that can be done. — Hanover
This is quite obviously not logically equivalent. The statements "if A then B" and "if C then D" involve different propositional variables (A, B, C, and D). Unless we have additional information about the relationship between these variables, we cannot assume they have any connection. The truth value of "if A then B" is determined solely by the truth values of A and B, while the truth value of "if C then D" depends only on C and D. These are independent of each other.
Without additional information, there's no reason to believe that the truth value of one statement would always match the other for all possible combinations of truth values. It's therefore entirely possible for "if A then B" to be true while "if C then D" is false, or vice versa, depending on the specific truth values of A, B, C, and D.
This offers an equivocation of the term "true." The sylIogism "If A then B, A, therefore B" is true. The statement "I am at work today" is true. It's the analytic/synthetic distinction. It's for that reason why a statement can be deductively true and inductively false, which is what the OP showed. Analytic validity says nothing about synthetic validity. — Hanover
Yes, you're right to point out some equivocation here but the point I was trying to make stands. If the premisses of a deductive argument are true (and I'm assuming a form of correspondence theory) then a valid argument will have a logically true conclusion
and necessarily correspond with reality.
The definition of "mammal" was arrived at a posteriori as opposed to "bachelor" which, as you've used it, (i.e. there is no probability a bachelor can be married) is a purely analytic statement. That is, no amount of searching for the married bachelor will locate one. On the other hand, unless you've reduced all definitions to having a necessary element for them to be applicable (which would be an essentialist approach), the term "mammal" could be applied to a non-milk providing animal, assuming sufficient other attributes were satisfied. This might be the case should a new subspecies be found. For example, all mammals give birth to live young, except the platypus, which lays eggs. That exception is carved out because the users of the term "mammal" had other purposes for that word other than creation of a legalistic analytic term. — Hanover
While scientific terms do evolve, they do function as relatively fixed definitions within the scientific community. The fact that definitions can change doesn't necessarily mean they are probabilistic or inductive in nature during their period of use and "giving milk" is a rather necessary condition in that definition since the name is derived from breasts because of the mammary gland. So no, nice try but nobody has ever used the term for any animal that doesn't produce milk and they never will.