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I have 3 statements I want to convert into symbolic logic (P, Q, etc.)

 All men are people.  Some men are clever.  Therefore, all people are clever.

First of all, I know this is illogical, but I'm trying to figure out how to show this using symbolic logic. Can I say:

 P = men
 Q = people
 R = clever

 (All) P -> Q
 (Exist) P -> R
 ----------------
 (All) Q -> R

It's easy to see that Q->R doesn't follow from this.

I realize the statements aren't in the "if, then" format, but it can be modified that way can't it? That is, if you are a man, then you are a person....etc.

  • 1
    What kind of symbolic logic do you mean? Propositional calculus? or predicate logic? The "(P,Q, etc.)" indicates that you wish a conversion into propositional calculus. Your example, however, is somewhat a mixture between the two. In propositional calculus one only considers propositions as a whole, i.e. the variables P, Q, etc. denote propositions. So your example would be "P ∧ Q → R". When you want to look at the structure of the proposition you can use predicate logic as it is demonstrated by Tyler's answer. – AndreasS Nov 8 '12 at 16:50
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∀x (Man(x) → Person(x))
∃x (Man(x) ∧ Clever(x))
--------------------------------
∀x (Person(x) → Clever(x))
1
∀x ∈ Man : x ∈ People
∃x ∈ Man : x ∈ Clever
---------
∀x ∈ People : x ∈ Clever

Typing hint:
- "for every" (HTML: ∀ Unicode: ∀)
- "exists" (HTML: ∃; Unicode: ∃)
- "element of" (HTML: ∈; Unicode: ∈)
: - "such that"; in different Math schools you may find a plain colon as above or (Unicode: ⊩), "forces"

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