Philosophy - Proofs

CH 8 Proofs

· Proof: “A series of steps that leads from the premises of a symbolic argument to its conclusion.”

Eight Rules of Inference

Modus Ponens {MP}

1. P→Q

2. P

3. ∴ Q

Modus Tollens {MT}

1. P→Q

2. ∽Q

3. ∴ ∽P

Disjunctive Syllogism {DS}

1. P ⌵ Q

2. ∽P

3. ∴ Q

Hypothetical Syllogism {HS}

1. P→Q

2. Q→R

3. ∴ P→R

Constructive Dilemma {CD}

1. P ⌵ R

2. P→Q

3. R→S

4. ∴Q ⌵ S

Addition {ADD}

1. P

2. ∴ P ⌵ Q

Conjunction {CONJ}

1. P

2. Q

3. ∴ P ⦁ Q

Simplification {SIMP}

1. P ⦁ Q

2. ∴P / ∴Q

These rules are used to construct proofs which show that an argument is valid.

Addition {ADD}

· Because the “or” operator is inclusive, if you know the truth of some statement A then you know the truth of any disjunctive statement that contains A.

· EX: “It’s raining. So, Either it’s raining or it’s sunny”

· EX: “Eric is learning logic. So, either Eric is learning logic or Jesus is a fire dragon.”

What seems weird about these conclusions isn’t the logic. The logic is fine, and we could construct a truth table showing that. Rather, no one talks like this. It would serve no purpose in everyday conversation and potentially be misleading or confusing. In other words, it violates a norm of communication, even though the inference is valid.

Conjunction {CONJ} and Simplification {SIMP}

· These rules are rather obvious – so obvious that it might be tempting to skip them when solving proofs. Avoid doing this.

· DO NOT make the mistake of thinking you can conjoin or simplify from anything other than the conjunction operator. It does not work for disjunction, conditionals, or biconditionals.

Proofs and Line Justification

We can show that certain arguments are valid using these rules.

For example:

1. (P v Q) → R

2. P

3. So, R v S

4. P v Q

5. R

6. R v S

QED (quod erat demonstrandum) = (which was to be proven)

But we need to show how we proved it, like showing our work. We do this by adding parenthesis or squirrely brackets to the right of the lines we added, like so:

1. P v Q { }

Next, we fill them in with two things:

1. The line(s) we used to attain the new statement

2. The inference rule we used to attain the new statement

So,

4. P v Q {2 ADD}

5. R {1,4 MP}

6. R v S {5 ADD}

Ned’s Wish List Strategy

· Solve proofs by working backwards

1. Where do you want to get to when doing a proof? Answer: the conclusion.

2. Using what you have and one of your inference (or equivalence) rules, can you get the conclusion? If so, great! If not, you’ll need at least one extra step. Let’s keep track of what we want by placing it under a “wishlist”.

EXAMPLE:

1. A ⦁ (B ⦁ C) WISHLIST: -C

2. ∴ C -B ⦁ C

3. B ⦁ C {1, SIMP}

4. C {3, SIMP}

1. [(P ⌵ Q) ⌵ R] ⦁ ∼Q WISHLIST: P, P ⌵ Q, (P ⌵ Q) ⌵ R, ∼R, ∼Q

2. ∼Q → ∼R

3. ∴P

4. (P ⌵ Q) ⌵ R {1 SIMP}

5. ∼Q {1, SIMP}

6. ∼R {2, 5 MP}

7. (P ⌵ Q) {4,6 DS}

8. P {5,7 DS}

Proofs and Invalidity

NOTE: You cannot construct a proof for an invalid argument. Were you to try, you’d get stuck.

HANDOUT

BOOK EXERCISES pg 360-363