1. Quantified Propositions
Quantifiers: For all ( x) ,
There exists ( x)

2. Examples: All birds can fly P(x): a bird can fly (proposition)
Quantifier: for all
Symbolic Notation: x P(x)
2. Not all birds can fly
Q(x): x is a bird
a bird can fly (proposition)
Some birds can fly but not all
Quantifier: there exists
Symbolic notation: x Q(x)

3. Examples: 3. If x is a man, then x is a giant P(x): x is a man Q(x): x is giant propositions Quantifier: For some x Symbolic Notation: x P(x) → Q(x) 2. Some men are not giants Symbolic notation: x P(x) → ~Q(x)

4. 4 Quantified proposition Rules:

5. Rules of Inference for Quantified Propositions Fundamental Rule 5: (Universal Specification) If a statement of the form x, P(x) is assumed to be true, then the universal quantifier can dropped to obtain P(c) is true for an arbitrary object c in the universe. x, P(x) therefore P(c) for all c Example: all men are mortal proposition: x is mortal then we conclude that Socrates is mortal

6. Rules of Inference for Quantified Propositions Fundamental Rule 6: (Universal Generalization) If a statement P(c) is true for each element ‘c’ of the universe, then the universal quantifier may be prefixed to obtain x, P(x). In symbols P(c) for all c therefore x, P(x) Fundamental Rule 7: (Existence Specification) If x, P(x) is assumed to be true, then there is an element c in the universe such that P(c) is true. In symbolic notation x, P(x) therefore P(c) for some c

7. Rules of Inference for Quantified Propositions Fundamental Rule 8: (Existence Generalization) If P(c) is true for some element c in the universe, then x, P(x), P(x) is true. In symbols P(c) for some c therefore x, P(x)

8. 8 Fundamental Rules:

9. Problems:
All men are fallible.
all kings are men.
therefore, all kings are fallible.
2. Lions are dangerous animals
There are lions.
therefore, there are dangerous animals.