2

Sometimes, when translating a sentence like "All dogs are animals", I often see it represented like this:

(∀x)(Px ⊃ Qx)

However, I feel like the following is also a good translation:

(∀x)(Px ∧ Qx)

Why do I always see this represented with 'implication' instead of 'and'?

Dictionary:
Px = x is a dog
Qx = x is an animal
  • 2
    Because the second way forces every object to be a dog (and an animal). It's a very common mistake. – Hunan Rostomyan Apr 15 '14 at 0:22
  • Ah, that makes sense. What if you said: (∀x)(Px ∧ Qx); (∀y)Qy – RECURSIVE FARTS Apr 15 '14 at 0:25
  • Forall x(Px and Qx) implies Forall x(Qx), which is identical to Forall y(Qy), modulo the useless change of the bound variable x to y. – Hunan Rostomyan Apr 15 '14 at 0:33
1

I will shamelessly use the opportunity to talk about the distinction between unrestricted vs bounded quantifiers. I suspect our intuitions favor one or the other in different contexts, causing the confusion.

When quantifiers are introduced in logic courses they're usually unrestricted, which means that the x in ∀x and ∃x ranges over the entirety of the domain of discourse. An alternative is to have bounded quantifiers ∀x ∈ A and ∃x ∈ A for every subset A of the domain of discourse. The confusion might be arising because the universal quantifier in "all dogs are animals" is implicitly taken to be a bounded one:

(1) ∀x ∈ Dogs ( Animal (x) ).

Then, since the standard quantifiers are unrestricted, the thought is to express the same idea with this:

(2)* ∀x ( Dog (x) ∧ Animal (x) ),

where Dog(x) is meant to play the role of specifying the subdomain x is supposed to range over. But as I already mentioned in the comments, (2)* will be true iff the whole domain of discourse consists only of dogs, while (1) will be true in every domain where dogs are animals. The 'solution':

(3) ∀x ( Dog (x) → Animal (x) ),

other than being the right translation for (1) using unrestricted quantifiers, is an instance of a general pattern called guarded quantification, which has the following form:

(Guarded Quantification) ∀x ( φ → ψ ).

The φ acts as a "guard" for ψ, meaning that the truth of ψ is tested only for those x in the domain that satisfy φ. In sum, guarded quantification, of which (3) is an instance, is a method of achieving restricted quantification in a language with only unrestricted quantifiers and an unstratified domain of discourse.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.