# Predicate logic Vs propositional logic: a specific example

Consider the following argument.

1) People who write novels are more sensitive than people who play soccer. 2) Alf writes novels. 3) Brian plays soccer.

Conclusion: 4) Alf is more sensitive than Brian.

Here is how you formalise this argument using predicate logic (many thanks to Mauro Allegranza):

1) ∀x∀y((Nx & Sy) ⊃ Mxy) 2) Na 3) Sb Conclusion: 4) Mab

M= more sensitive than N= writes novels S= plays soccer a= Alf b= Brian

If I had to formalise (1)-(4) using propositional logic it would be: 1) p 2) q 3) r Conclusion: 4) s.

That is, we would have 4 different propositions with no links between them. Am I right?

Thank you very much

Fish

Well, no, not really.

A better answer would be to say that propositional logic CANNOT formalize that kind of reasoning.

• Yes of course; but if you had to formalise this reasoning the answer would be the one I propose. – Fishermansfriend Feb 11 '17 at 23:33
• If I had to formalize the reasoning, the answer would be "it's impossible". Just as if I "had to" use the real numbers to solve $x^2+1=0$. – Henning Makholm Feb 12 '17 at 1:13

Something you could do in propositional logic is this:

$A$: Alf plays soccer

$B$: Brian plays soccer

$C$: Alf writes novels

$D$ : Brian writes novels

$E$: Alf is more senitive than Alf

$F$: Alf is more sensitive than Brian

$G$: Brian is more sensitiv than Alf

$H$: Brian is more sensitive than Brian

With these, you could take the first statement, and at least as it relates to Alf and Brian, capture that with:

1. $((A \land C) \to E) \land ((A \land D) \to G) \land ((B \land C) \to F) \land ((B \land D) \to H)$

And of course we also have:

1. $C$

2. $B$

3. $F$

And now you can infer it. Yes, ths statement 1 obviously does not capture the whole universal English statement as it pertains to any two people, but as far as we are interested in Alf and Brian, it does the job.