## Are "All A is B" and "If A then B" always logically equivalent?

5

2

Will "All A is B" always imply "If A then B?"

If so, can we say that the following follows the form of Modus Ponens?

All dogs are mammals.
Toby is a dog.
So, Toby is a mammal.

Can I rewrite the above to be:
If it is a dog, it is a mammal.
Toby is a dog.
So, Toby is a mammal.

When I rewrote it, it seems to follow the textbook definition of Modus Ponens, but I was wondering if it was ok to classify the original one as Modus Ponens as well. I have only very recently started studying philosophy, so I apologize if my terminology is incorrect/making some very fundamental misunderstandings.

2NO; the original argument is a syllogism and not a Modus Ponens. In your rewrited form you have "If it is a dog, it is a mammal" as first premise; in order to apply MP the second premise must be: "It is a dog". Now the conclusion will be "So, it is a mammal.". – Mauro ALLEGRANZA – 2017-06-09T05:56:08.730

Not at all; the first depends on identity, cause is not a factor; in the second, the reverse holds. – Mozibur Ullah – 2017-08-12T04:00:26.813

4

We are talking about expressing statements in two different systems. The first one is the classical syllogistic of Aristotle ("All dogs are mammals"), with categorical syllogisms, whereas the latter ("If it is a dog then it is a mammal") is in the form of a hypothetical syllogism. It is only with the latter that one can speak of something like modus ponens, see Forerunners of Modus Ponens and Modus Tollens. But Aristotle's syllogistic had its own "figures", i.e. rules of inference, instead.

So we are not talking about one "implying" the other, but rather about a conversion of one into another. The syllogistic "All dogs are mammals" is often translated into the predicate logic as ∀x(D(x)→M(x)), which is interpreted as a hypothetical "if x is a dog then x is a mammal", and can be abbreviated to D→M. One worry might be that the domain is empty, e.g. if dogs are replaced by phoenixes. The material conditional would make all statements of the form ∀x(P(x)→M(x)) true, but it is unclear if one wants to go with "All phoenixes are mammals". Nonetheless, the translation above is canonical in elementary textbooks, so your conversion, and the mixing of terminology, are probably safe.

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With predicate logic, the idea of "all" is more accurately represented by the universal quantifier: ∀x[Dx → Mx], which reads "For every x, if x is a dog, then x is a mammal".

• Dx = x is a dog
• Mx = x is a mammal
∀x[Dx → Mx]                       Premise
Dt                                Premise
Dt → Mt                           Universal elimination
Mt                                Modus Ponens


1

Perhaps it is helpful to emphasize the aspect of set theory.

The set of dogs is a subset of the set of mammals. If there is an element of some basic universal set (that has its well-known problems which do not interfere with the present topic however) that has the property dog, then this element has the property mammal too. That means "if dog then mammal" or in short D → M.

In the form "every dog is a mammal" and "if a dog, then a mammal" the statements are nearly identical. Here and also in form used in the original question they express exactly the same information.