564 Lecture 4 Sept. 2 1999

First: Truth tables (starting at #26 from last time), tautologies, contradictions and contingencies, and evaluation w/r to a model from last time’s handout.

1. Contraposition:

(P->Q)<=>(~Q->~P)

However, there’s another way to prove logical equivalence; in fact, you can use the given "laws of statement logic" to simplify formulas or mutate formulas into one another (or prove a conclusion from a set of premises). Here’s the same logical equivalence in substitution form, using some of the other logical equivalences given in the table:

2. Contraposition by substitution:

(1) P->Q Starting formula

(2) (~PvQ) Conditional law (a)

(3) (~Pv~~Q) Double negation

(4) (~~Qv~P) Commutative law a

(5) (~Q->~P) Conditional law (a) again, resulting in

logical equivalence captured in conditional law (b).

Note: you can only substitute subformulas (constituents), not non-consitutents. So, for example, you couldn’t do 3. below:

3. Unacceptable substitution of non-constituent:

(1) (P&(Q->R))

This type of operation is not allowed, because only substitution of sub-formulas (constituents) will guarantee that the truth conditions will remain identical, since the sub-formulas in question are truth-conditionally equivalent.

An argument consists of a series of premises, which are assumed true, and a conclusion, which is supposed to follow necessarily from the truth of the given premises. A valid argument is one in which the premises may not be true and the conclusion false at the same time. (If this is possible, it is an invalid argument, or really, not an argument at all.) Another way of saying this is

4. if (P₁ … P_n) are the premises and Q is the conclusion, then (P₁&P₂&… &P_n) -> Q should be a tautology.

Here’s an example of an argument we recognize as valid, in natural language.

5.

If Felicia likes Mindy, then Mindy is happy.

Therefore, Mindy is happy.

6. We can translate this using the following key:

p = Felicia likes Mindy

q = Mindy is happy

p->q

\ q

Here’s a truth table demonstrating that this is a valid argument. What we need to show is that ((p->q)&p)->q) is a tautology. All the possible truth value assignments to the premises should give the result that the value of the whole is 1.

7. ((p->q)&p)->q)

So the above is a valid argument form, or rule of inference. It is called "Modus Ponens", and is one basic rule of inference. Some others are on your handout from last time.

If some premises are not of the right form to apply a rule of inference to get the conclusion, you can apply the laws of statement logic (logical equivalences) to massage the premises into the correct form. Sometimes it’ll take multiple massages and applications of rules of inference followed by more massages to end up with an adequate set of premises from which to infer your conclusion. Here’s an example of how it goes.

8. Prove (p&~(q->r)) from ~(p->~q) and ~r.

(1) ~(p->~q) premise

(2) ~r premise

(3) ~(~(p&~~q)) 1, Conditional law (c)

(4) ~(~(p&q)) 3, Double negation (Complement law (b))

(5) (p&q) 4,Double negation (Complement law (b))

(6) (p&q)&~r 2,5, Conjunction (inference)

(7) p&(q&~r) 6, Associative law (b)

(8) p&~~(q&~r) 7, Double negation (Complement law (b))

(9) p & ~(q->r) 8, Conditional law (c), Q.E.D.

There’s a couple of other types of proof:

9. A conditional proof is one whose conclusion contains a conditional as its main connective (e.g. where the conclusion is (p->q)). Sometimes it’s easiest to prove these by assuming the antecedent of the conclusion as an auxiliary premise (p), and demonstrating that in combination with the other premises, q follows. Then you’ve demonstrated the desired conclusion, i.e. (p->q). (At the point in the proof in which you assume the auxiliary premise, it helps to distinguish the subsequent argumentation from the previous argumentation and the conclusion, e.g. by offsetting it, or placing a bar down the side of the conditional proof part. An example follows:

Prove (p->q) from (p->(qvr)) and ~r.

(1) (p->(qvr)) premise

(2) ~r premise

(3) p auxiliary premise

(4) qvr 1,3, Modus Ponens

(5) q 2,4, Disjunctive Syllogism

(6) p->q 2-5, Conditional Proof.

Finally, there’s a method of proof called indirect proof. If you can show that assuming the falsehood of the conclusion leads to a contradiction, given true premises, you’ve proved your conclusion indirectly — if the premises are true, the conclusion must be true, otherwise it leads to a contradiction. Here’s an example:

10. Indirect Proof

Prove p from (pvq), (q->r) and ~r.

(1) pvq premise

(2) q->r premise

(3) ~r premise

(4) ~p auxiliary premise (negation of desired conclusion)

(5) q 1,4, Disjunctive Syllogism (D.S.)

(6) r 2,5, Modus Ponens (M.P.)

(7) r&~r 3, 6 Conjunction

(8) p 4-7, Indirect Proof