# Representation of Categorical Syllogism in Symbolic Logic

How would I represent a AAA categorical syllogism with symbolic logic?

• What have you tried yourself? Please edit your question to provide some more detail.
– user2953
Mar 12, 2015 at 6:30

How would I represent a AAA categorical syllogism with symbolic logic?

The AAA categorical syllogism looks like this:

``````1. All P is Q.
2. All Q is R.
∴  All P is R.
``````

If by "symbolic logic", you mean propositional logic, then the answer is that you can't (exactly). The closest you can come is the hypothetical syllogism:

``````1. P ⊃ Q
2. Q ⊃ R
∴  P ⊃ R
``````

If by "symbolic logic", you mean quantificational logic, then it would look something like this:

``````1. (x)(Px ⊃ Qx)
2. (x)(Qx ⊃ Rx)
∴  (x)(Px ⊃ Rx)
``````

The above notation is preferred by many philosophers, while mathematicians might prefer notation like this:

``````1. ∀x(P(x) → Q(x))
2. ∀x(Q(x) → R(x))
∴  ∀x(P(x) → R(x))
``````

Both of these notations are semantically equivalent, so use whichever seems more intuitive unless you are operating under additional constraints such as course requirements.

For both notations, it is common to read the first premise symbolically as "for all x, if x has property P, then x has property Q," the second premise as "for all x, if x has property Q, then x has property R," and the conclusion as "for all x, if x has property P, then x has property R."