Logical Argument – An Example

1. Logical Argument – An Example • We considered P1, P2, and Q under a particular (common sense) interpretation: • P1= “If Socrates is human then Socrates is mortal” true • P2 = “Socrates is human” true • Q = “Socrates is mortal” true • Thus, they were merely logical constants to us: • P1=true • P2=true • Q=true

2. Logical Argument – An Example • Consider the two arguments: • P1“If Socrates is human then Socrates is mortal” “If J.B. broke his leg then J.B. is in pain” • P2“Socrates is human” “J.B. broke his leg” • Therefore • Q“Socrates is mortal” “J.B. is in pain” • Both arguments share the same structure: • P1 If X then Y • P2 X • Therefore • Q Y • Then for any interpretation I, as long as I satisfies P1 and P2, interpretation I must satisfy Q.

3. Modus Ponens • The “generalized” argument • P1 = X → Y • P2 = X • Therefore • Q = Y • Because it captures the essence of both arguments and can be used for infinitely many more. “method of affirming” (Lat.)

4. Valid Arguments (Revisited) • Suppose someone makes an argument: • P1,...,PNtherefore Q • The argument is called valid iff: • P1,…,PN logically imply Q • That is: • For any interpretation I that satisfies all Pj, interpretation I must necessarily satisfy Q • Usually: Pj and Q are somehow related statements and P1 ^ … ^ PN can be true or false depending on the interpretation I.

5. Propositional Logic • Method 1: • Go through all possible interpretations and check the definition of valid argument • Method 2: • Use inference rules to get from the premises to the conclusion in a logically sound way • “derive the conclusions from premises”

6. Method #1 • Section 1.3 in the text proves many arguments/inference rules using truth tables. • Suppose the argument is: • P1, …,PN therefore Q • Create a truth table for statement • F = (P1 ^ … ^ PN Q) • Check if F is a tautology.

7. But Why? Recall: • Statement A implies statement Biff (A→B) is a tautology. • In general: • premises P1, …,PN imply Q • iff • Statement F = (P1 ^ … ^ PN→ Q) • is a tautology.

8. Example #1 • P v (Q v R) • ~R • Therefore • P v Q • valid/invalid? • (example 1.3.2 in the book, p. 31)

9. Example #2 • P v Q v R • ~R • Therefore • Q • valid/invalid?

10. Example #3 • P→Q • P • Therefore • Q • valid/invalid? • (Modus ponens)

11. Example #4 • P  Q • Q • Therefore • P • valid/invalid?

12. Example #5 • P  Q • ~Q • Therefore • ~P • valid/invalid? • (Modus tollens)

13. Example #6 • P  Q • Therefore • ~Q  ~P • valid/invalid? • In fact, we proved earlier that: • (P  Q)  (~Q  ~P)

14. Example #7 • P v Q • ~P ^ ~Q • Therefore • P ^ Q • valid/invalid? • Any argument with a contradiction in its premises is valid by default…

15. Pros & Cons • Method #1: • Pro: straight-forward, not much creativity  machines can do • Con: the number of interpretations grows exponentially with the number of variables  cannot do for many variables • Con: in predicate and some other logics even a small formula may have an infinite number of interpretations

16. Method #2 : Inference • To prove that an argument is valid: • Begin with the premises • Use valid/sound inference rules • Arrive at the conclusion

17. Inference Rules • But what are these “inference rules”? • They are simply… • …valid arguments! • Example: • X ^ Y • X ^ Y  Z ^ W • therefore • Z ^ Wby modus ponens

18. Derivations • The chain of inference rules that starts with the premises and ends with the conclusion • … is called a derivation: • The conclusion is derived from the premises. • Such a derivation makes a proof of argument’s validity.

19. Example #1 • (X^Y → Z^W) ^ K • X^Y • Therefore • Z^W • How? • (X^Y → Z^W) ^ K • X^Y → Z^Wby conjunctive simplification • X^Y • Z^Wby modus ponens derivation

20. Pros & Cons • Method #2: • Pro: often can get a dramatic speed-up over truth tables. • Con: requires creativity and intuition (harder to do by machines). • Con: semi-decidable: there is no algorithm that can prove any first-order predicate logic argument to be valid or invalid.

21. Fallacies • An error in derivation leading to an invalid argument • Vague formulations of premises/conclusion • Missing steps • Using unsound inference rules, e.g.: • Converse error • Inverse error

22. Converse Error • If John is smart then John makes a lot of money • John makes a lot of money • Therefore • John is smart • Tries to use this unsound “inference rule”: • A→B • B • Therefore • A

23. Inverse Error • If John is smart then John makes a lot of money • John is not smart • Therefore • John doesn’t make a lot of money • Tries to use this unsound “inference rule”: • A→B • ~A • Therefore • ~B

24. Truth of facts vs. Validity of Arguments • The premises are assumed to be true ONLY in the context of the argument. • The following argument isvalid: • If John Lennon was a rock star then he was a woman. • John Lennon was a rock star. • Therefore: • John Lennon was a woman. • But the 1st premise doesn’t hold under the common sense interpretation.

25. Summary • Equivalence: • A  B • A holds iff B holds • A is a criterion for B • B is a criterion for A • A logically implies (entails) B • B logically implies (entails) A • A and B are “equivalently strong” • Statement F=(A↔B) is a tautology

26. Summary • Implication (Entailment): • A entails (logically implies) B • B follows from A • AB is a valid argument • A is a sufficient condition for B • B is a necessary condition for A • If A holds then B holds • A may be “stronger than” B • Statement F=(A→B) is a tautology

27. The Big Picture • Logic is used to verify validity of arguments. • An argument is valid iff its conclusion logically follows from the premises. • Derivations are used to prove validity. • Inference rules (Table 1.3.1, p40) are used as part of derivations