1. Logical Inferences:
A set of premises accompanied by a suggested conclusion regardless of whether or not the conclusion is a logical consequence of the premises.
Hence it may be valid inference or faulty inference.
Inference is written as
(conjunction of premises) → (conclusion)
Inference is valid if the implication is tautology otherwise invalid or faulty inference or fallacy
Not a Tautology
Tautology
valid inference
invalid inference (faulty)

2. Rules of Inference – valid: There are 4 fundamental rules
Fundamental rule 1: If the statement in P is assumed as True, and the statement P →Q is accepted as true, then we must accept Q as True. (Modus Ponens Rule)
Symbolically P
P →Q
therefore Q
Fundamental rule 2: Whenever two implications P →Q and
Q →R are accepted as true then we must accept the implication
P →R as true (Hypothetical Syllogism or Transitive Rule)
Symbolically P →Q
Q →R
therefore P → R
Hypothesis / premises
Conclusion
Hypothesis / premises
Conclusion

3. Fundamental rule 3: DeMorgan’s Law ~( P V Q ) = ~P ^ ~Q ~( P ^ Q ) = ~P V ~Q Fundamental rule 4: Law of Contra positive P → Q = ~ Q → ~ P

6. Rules of Inference – Invalid
Logical inference is invalid if the implication is not a tautology
Also called as faulty inference or fallacy
Inference is written as
(conjunction of premises) → (conclusion)
Tautology
Not a Tautology
valid
inference
Invalid
inference (faulty)

7. Rule 1 (Fallacy 1)

8. The fallacy of denying the antecedent takes the form P →Q ~P therefore ~Q
Rule 2 (Fallacy 2)
Rule 3 (Fallacy 3)

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

10. 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)

11. 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)

12. 4 Quantified proposition Rules:

13. 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

14. 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

15. 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)

16. 8 Fundamental Rules:

17. 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.