Chapter 9 Notes: Proofs (Symbolic Logic)

proof

a series of inferences from a set of premises to a conclusion, each one justified by a valid inference rule

to prove the validity of arguments we ALREADY KNOW are valid

What are proofs used for?

valid

Only _____ arguments have proofs.

tells us why an argument is valid

• shows how the conclusion is derived from the premises

Advantage of Proofs

• A proof shows us how the conclusion is derived from the premises

• A truth table only shows us whether or not it’s valid

How does a proof differ from a truth table?

CAN’T be used to prove an argument’s invalidity

• Since it’s not an “effective” method, not finding a proof doesn’t mean that it’s invalid (you might just have not been able to find it)

Disadvantage of Proofs

we write the argument the same way that it’s written normally, except we place the conclusion to the right of the last premise, separated by a slash

How do we set up a proof?

inference rules

represent valid forms of inference

• some are rules of implication and others are rules of replacement

18

How many rules of inference are there?

modus ponens

1. p ⊃ q

2. p

3. So, q

modus tollens

1. p ⊃ q

2. ∼q

3. So, ∼p

disjunctive syllogism

1. p ∨ q

2. ∼p

3. So, q

AND

1. p ∨ q

2. ∼q

3. So, p

Hypothetical Syllogism

1. p ⊃ q

2. q ⊃ r

3. So, p ⊃ r

to reach the conclusion by a series of clearly valid steps

What is the goal of a proof?

start at the conclusion and work backwards

It’s typically easiest to… when solving a proof.

List the inference rule employed and the lines to which you applied that rule

What do you include in your justification?

right

You record each justification to the ____ of the last premise.

any previous

In a (direct) proof, you are allowed to use __ ________ line or set of lines to derive the next line. 

simplification

1. p ● q

2. So, p

AND

1. p ● q

2. So, q

Conjunction

1. p

2. q 

3. So, p ● q

Addition

1. p

2. So, p ∨ q

addition rule

Whenever your conclusion has a letter in it that you don’t find in the premises of the argument, you know you’ll have to use the _______ ____ at some point in the proof

principle of explosion

If the premises of an argument contradict one another, the argument is automatically valid (but unsound because all premises are not true in this case)

Constructive Dilemma (CD)

1. (p ⊃ q) ● (r ⊃ s)

2. p ∨ r

3. So, q ∨ s

argument’s conclusion

The last line of any proof is always the _______ _______.

more than one

Never apply _____ ____ ___ rule per step in a proof.

part of

Never apply any rule of implication to just _____ __ a line.