Introduction to Formal Logic [email protected]

Lecture 6 Another method for proving validity: Trees (Part 1)

Recap Truth tables provide one way of demonstrating that an argument form is valid. They do this by representing, in their rows, each of the possible ways the world could be. Thus, if every row that makes all the premises true, also makes the conclusion true, then there is no possible way the premises could be true and the conclusion false. Thus, the argument form is valid.

Example of truth table test for validity We will show that the following argument form is valid: (p ⊃ ~q); q; therefore, ~p. 1 2

1

p q

(p ⊃ ~ q)

q ~p

T T F F

T F FT

T FT

T T TF F T FT F T TF

F FT T TF F TF

T F T F

Example of truth table test for validity We will show that the following argument form is valid: (p ⊃ ~q); q; therefore, ~p. 1 2

1

p q

(p ⊃ ~ q)

q ~p

T T F F

T F FT

T FT

T T TF F T FT F T TF

F FT T TF F TF

T F T F

Example of truth table test for validity We will show that the following argument form is valid: (p ⊃ ~q); q; therefore, ~p. 1 2

1

p q

(p ⊃ ~ q)

q ~p

T T F F

T F FT

T FT

T T TF F T FT F T TF

F FT T TF F TF

T F T F

Example of truth table test for validity We will show that the following argument form is valid: (p ⊃ ~q); q; therefore, ~p. 1 2

1

p q

(p ⊃ ~ q)

q ~p

T T F F

T F FT

T FT

T T TF F T FT F T TF

F FT T TF F TF

T F T F

Example of truth table test for validity We will show that the following argument form is valid: (p ⊃ ~q); q; therefore, ~p. 1 2

1

p q

(p ⊃ ~ q)

q ~p

T T F F

T F FT

T FT

T T TF F T FT F T TF

F FT T TF F TF

T F T F

Example of truth table test for validity We will show that the following argument form is valid: (p ⊃ ~q); q; therefore, ~p. 1 2

1

p q

(p ⊃ ~ q)

q ~p

T T F F

T F FT

T FT

T T TF F T FT F T TF

F FT T TF F TF

T F T F

Example of truth table test for validity We will show that the following argument form is valid: (p ⊃ ~q); q; therefore, ~p. 1 2

1

p q

(p ⊃ ~ q)

q ~p

T T F F

T F FT

T FT

T T TF F T FT F T TF

F FT T TF F TF

T F T F

Example of truth table test for validity We will show that the following argument form is valid: (p ⊃ ~q); q; therefore, ~p. 1 2

1

p q

(p ⊃ ~ q)

q ~p

T T F F

T F FT

T FT

T T TF F T FT F T TF

F FT T TF F TF

T F T F

Example of truth table test for validity We will show that the following argument form is valid: (p ⊃ ~q); q; therefore, ~p. 1 2

1

p q

(p ⊃ ~ q)

q ~p

T T F F

T F FT

T FT

T T TF F T FT F T TF

F FT T TF F TF

T F T F

Thus, the argument is valid!

Tree proofs also check for ways premises might be true and conclusion false

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid:

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true (p ∨ q)

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true (p ∨ q) ~q

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true along with the negation of the conclusion (p ∨ q) ~q

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true along with the negation of the conclusion (p ∨ q) ~q ~p

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true along with the negation of the conclusion (p ∨ q) ~q ~p

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true along with the negation of the conclusion (p ∨ q) ~q ~p p

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true along with the negation of the conclusion (p ∨ q) ~q ~p p

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true along with the negation of the conclusion (p ∨ q) ~q ~p p

q

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true along with the negation of the conclusion (p ∨ q) ~q ~p p

q

Step 2: Show that each of these ways is impossible.

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true along with the negation of the conclusion (p ∨ q) ~q ~p p

q

Step 2: Show that each of these ways is impossible.

Tree proofs also check for ways premises might be true and conclusion false I want to show that the following argument form is valid: (p ∨ q); ~ q; therefore, p Step 1: Consider all the ways that the premises might be true along with the negation of the conclusion (p ∨ q) ~q ~p p

q

Step 2: Show that each of these ways is impossible.

We need rules! For each connective, ∨, ⊃, ≡, and &, there are two rules: •

One for when the connective occurs unnegated, and

•

One for when the connective occurs negated.

For the connective ~, there is only one rule. The purpose of the rules is to enumerate the possibilities for truth: if the formula on which the rule acts is true, at least one of the branches given by the rule must be true.

The one rule for ~.

The one rule for ~. Negated

~~A

The one rule for ~. Negated

~~A

The one rule for ~. Negated

~~A A

The two rules for ∨.

The two rules for ∨. Unnegated

(A ∨ B)

The two rules for ∨. Unnegated

(A ∨ B)

Negated

~(A ∨ B)

The two rules for ∨. Unnegated

(A ∨ B)

Negated

~(A ∨ B)

The two rules for ∨. Unnegated

(A ∨ B) A

Negated

~(A ∨ B)

The two rules for ∨. Unnegated

(A ∨ B) A

Negated

~(A ∨ B)

The two rules for ∨. Unnegated

(A ∨ B) A

B

Negated

~(A ∨ B)

The two rules for ∨. Unnegated

(A ∨ B) A

B

Negated

~(A ∨ B)

The two rules for ∨. Unnegated

(A ∨ B) A

B

Negated

~(A ∨ B) ~A

The two rules for ∨. Unnegated

(A ∨ B) A

B

Negated

~(A ∨ B) ~A ~B

Examples of proofs that use the ∨ rules: I

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q)

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) q

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) q ~p

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

Unnegated (A ∨ B) A

B

(p ∨ ~q) q ~p

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

Unnegated (A ∨ B) A

B

(p ∨ ~q) q ~p

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) q ~p

Unnegated (A ∨ B) A

B

p

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) q ~p

Unnegated (A ∨ B) A

B

p

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) q ~p

Unnegated (A ∨ B) A

B

p

~q

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) √ q ~p

Unnegated (A ∨ B) A

B

p

~q

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) √ q ~p

Unnegated (A ∨ B) A

B

p

~q

Application of the rule for unnegated occurrences of the connective, ∨.

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) √ q ~p

Unnegated (A ∨ B) A

B

p

~q

Application of the rule for unnegated occurrences of the connective, ∨.

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) √ q ~p

Unnegated (A ∨ B) A

B

p

~q

Application of the rule for unnegated occurrences of the connective, ∨.

Examples of proofs that use the ∨ rules: I (p ∨ ~q); q; therefore, p

(p ∨ ~q) √ q ~p

Unnegated (A ∨ B) A

B

p

~q

Application of the rule for unnegated occurrences of the connective, ∨.

Therefore, the argument is valid: there is no way that the premises could be true while the conclusion is false. If they were, all the formulae on at least one of our branches would have to be true, and this is not possible.

Examples of proofs that use the ∨ rules: II

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

~(p ∨ q)

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

~(p ∨ q) ~~p

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

Negated ~(A ∨ B) ~A ~B

~(p ∨ q) ~~p

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

Negated ~(A ∨ B) ~A ~B

~(p ∨ q) ~~p

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

Negated ~(A ∨ B) ~A ~B

~(p ∨ q) ~~p ~p

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

Negated ~(A ∨ B) ~A ~B

~(p ∨ q) ~~p ~p ~q

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

Negated ~(A ∨ B) ~A ~B

~(p ∨ q) √ ~~p ~p ~q

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

Negated ~(A ∨ B) ~A ~B

~(p ∨ q) √ ~~p ~p ~q

Application of the rule for negated occurrences of the connective, ∨.

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

Negated ~(A ∨ B) ~A ~B

~(p ∨ q) √ ~~p ~p ~q

Application of the rule for negated occurrences of the connective, ∨.

Examples of proofs that use the ∨ rules: II ~(p ∨ q); therefore, ~p

Negated ~(A ∨ B) ~A ~B

~(p ∨ q) √ ~~p ~p ~q

Application of the rule for negated occurrences of the connective, ∨.

Therefore, the argument is valid: there is no way that the premises could be true while the conclusion is false. If they were, all the formulae on at least one of our branches would have to be true, and this is not possible.

The two rules for &.

The two rules for &. Unnegated

(A & B)

The two rules for &. Unnegated

(A & B)

Negated

~(A & B)

The two rules for &. Unnegated

(A & B)

Negated

~(A & B)

The two rules for &. Unnegated

(A & B) A

Negated

~(A & B)

The two rules for &. Unnegated

(A & B) A B

Negated

~(A & B)

The two rules for &. Unnegated

(A & B) A B

Negated

~(A & B)

The two rules for &. Unnegated

(A & B) A B

Negated

~(A & B) ~A

The two rules for &. Unnegated

(A & B) A B

Negated

~(A & B) ~A

The two rules for &. Unnegated

(A & B) A B

Negated

~(A & B) ~A

~B

Examples of proofs that use the & rules: I

Examples of proofs that use the & rules: I (p & q); therefore, p

Examples of proofs that use the & rules: I (p & q); therefore, p

(p & q)

Examples of proofs that use the & rules: I (p & q); therefore, p

(p & q) ~p

Examples of proofs that use the & rules: I (p & q); therefore, p

Unnegated (A & B) A B

(p & q) ~p

Examples of proofs that use the & rules: I (p & q); therefore, p

Unnegated (A & B) A B

(p & q) ~p

Examples of proofs that use the & rules: I (p & q); therefore, p

Unnegated (A & B) A B

(p & q) ~p p

Examples of proofs that use the & rules: I (p & q); therefore, p

Unnegated (A & B) A B

(p & q) ~p p q

Examples of proofs that use the & rules: I (p & q); therefore, p

Unnegated (A & B) A B

(p & q) √ ~p p q

Examples of proofs that use the & rules: I (p & q); therefore, p

Unnegated (A & B) A B

(p & q) √ ~p p q

Application of the rule for unnegated occurrences of the connective, &.

Examples of proofs that use the & rules: I (p & q); therefore, p

Unnegated (A & B) A B

(p & q) √ ~p p q

Application of the rule for unnegated occurrences of the connective, &.

Examples of proofs that use the & rules: I (p & q); therefore, p

Unnegated (A & B) A B

(p & q) √ ~p p q

Therefore, the argument is valid.

Application of the rule for unnegated occurrences of the connective, &.

Examples of proofs that use the & rules: II

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q)

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) p

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) p ~~q

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

Negated ~(A & B) ~A

~B

~(p & q) p ~~q

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

Negated ~(A & B) ~A

~B

~(p & q) p ~~q

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) p ~~q

Negated ~(A & B) ~A

~B

~p

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) p ~~q

Negated ~(A & B) ~A

~B

~p

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) p ~~q

Negated ~(A & B) ~A

~B

~p

~q

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) √ p ~~q

Negated ~(A & B) ~A

~B

~p

~q

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) √ p ~~q

Negated ~(A & B) ~A

~B

~p

~q

Application of the rule for negated occurrences of the connective, &.

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) √ p ~~q

Negated ~(A & B) ~A

~B

~p

~q

Application of the rule for negated occurrences of the connective, &.

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) √ p ~~q

Negated ~(A & B) ~A

~B

~p

~q

Application of the rule for negated occurrences of the connective, &.

Examples of proofs that use the & rules: II ~(p & q); p; therefore, ~q

~(p & q) √ p ~~q

Negated ~(A & B) ~A

~B

~p

~q

Therefore, the argument is valid.

Application of the rule for negated occurrences of the connective, &.

Examples of proofs that use both rules: I

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q)

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q)

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q)

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q

√

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q

√

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q ~p

√

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q ~p

√

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q ~p

~q

√

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q ~p

~q

√ √

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q ~p

~q

√ √

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q ~p

~q

√ √

Examples of proofs that use both rules: I ~(p & q); therefore, ~p ∨ ~q

~(p & q) ~(~p ∨ ~q) ~~p ~~q ~p

~q

Therefore, the argument is valid.

√ √

Examples of proofs that use both rules: II

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p

(p & q) ∨ (p & ~q)

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p

(p & q) ∨ (p & ~q) ~p

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p

(p & q) ∨ (p & ~q) ~p

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p

(p & q) ∨ (p & ~q) ~p (p & q)

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p

(p & q) ∨ (p & ~q) ~p (p & q)

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p

(p & q) ∨ (p & ~q) ~p (p & q)

(p & ~q)

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p

(p & q) ∨ (p & ~q) √ ~p (p & q)

(p & ~q)

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q)

(p & ~q)

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q)

(p & ~q)

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) p

(p & ~q)

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) p q

(p & ~q)

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) √ (p & ~q) p q

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) √ (p & ~q) p q

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) √ (p & ~q) p q

p

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) √ (p & ~q) p q

p ~q

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) √ (p & ~q) √ p q

p ~q

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) √ (p & ~q) √ p q

p ~q

Negated rule for &.

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) √ (p & ~q) √ p q

p ~q

Negated rule for &.

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) √ (p & ~q) √ p q

p ~q

Negated rule for &.

Examples of proofs that use both rules: II (p & q) ∨ (p & ~q); therefore, p Unnegated rule for ∨.

(p & q) ∨ (p & ~q) √ ~p (p & q) √ (p & ~q) √ p q

Therefore, the argument is valid.

p ~q

Negated rule for &.

Examples of proofs that use both rules: III

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r)

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q)

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) (~q ∨ r)

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) (~q ∨ r) ~(~p ∨ r)

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) (~q ∨ r) ~(~p ∨ r)

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) (~q ∨ r) ~(~p ∨ r) ~p

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) (~q ∨ r) ~(~p ∨ r) ~p

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) (~q ∨ r) ~(~p ∨ r) ~p

q

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r) ~p

q

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r)

Unnegated rule for ∨. ~p

q

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r)

Unnegated rule for ∨. ~p

q

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r)

Unnegated rule for ∨. ~p ~q

q

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r)

Unnegated rule for ∨. ~p ~q

q

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r)

Unnegated rule for ∨. ~p ~q

q r

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r)

Unnegated rule for ∨. ~p ~q

q r

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r)

Unnegated rule for ∨. ~p ~q

q r

~q

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r)

Unnegated rule for ∨. ~p ~q

q r

~q

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) ~(~p ∨ r)

Unnegated rule for ∨. ~p ~q

q r

~q

r

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p ~q

q r

~q

r

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

~q

q r

~q

r

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

~q

q r

~q

r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

~q

q r

~q

r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

~q

q r

~q

r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

~q ~~p

q r

~q

r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

~q ~~p ~r

q r

~q

r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

~q ~~p ~r

q r

~q

r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

q

~q

r

~~p ~r

~~p

~q

r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

q

~q

r

~~p ~r

~~p ~r

~q

r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

q

~q

r

~~p ~r

~~p ~r

~q

r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

q

~q

r

~~p ~r

~~p ~r

~q

r ~~p

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r)

Unnegated rule for ∨. ~p Unnegated rule for ∨.

q

~q

r

~~p ~r

~~p ~r

~q

r ~~p ~r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r) √

Unnegated rule for ∨. ~p Unnegated rule for ∨.

q

~q

r

~~p ~r

~~p ~r

~q

r ~~p ~r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r) √

Unnegated rule for ∨. ~p

q

Unnegated rule for ∨.

~q

r

Negated rule for ∨.

~~p ~r

~~p ~r

~q

r ~~p ~r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r) √

Unnegated rule for ∨. ~p

q

Unnegated rule for ∨.

~q

r

Negated rule for ∨.

~~p ~r

~~p ~r

~q Negated rule for ∨.

r ~~p ~r

Unnegated rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r) √

Unnegated rule for ∨. ~p

q

Unnegated rule for ∨.

~q

r

Negated rule for ∨.

~~p ~r

~~p ~r

~q Negated rule for ∨.

r

Unnegated rule for ∨.

~~p Negated ~r rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r) √

Unnegated rule for ∨. ~p

q

Unnegated rule for ∨.

~q

r

Negated rule for ∨.

~~p ~r

~~p ~r

~q Negated rule for ∨.

r

Unnegated rule for ∨.

~~p Negated ~r rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r) √

Unnegated rule for ∨. ~p

q

Unnegated rule for ∨.

~q

r

Negated rule for ∨.

~~p ~r

~~p ~r

~q Negated rule for ∨.

r

Unnegated rule for ∨.

~~p Negated ~r rule for ∨.

Examples of proofs that use both rules: III (~p ∨ q), (~q ∨ r); therefore, (~p ∨ r) (~p ∨ q) √ (~q ∨ r) √ ~(~p ∨ r) √

Unnegated rule for ∨. ~p

q

Unnegated rule for ∨.

~q

r

Negated rule for ∨.

~~p ~r

~~p ~r

~q Negated rule for ∨.

r

Unnegated rule for ∨.

~~p Negated ~r rule for ∨.

Summary

Summary The method of proof trees offers an alternative method by which to demonstrate that an argument form is valid.

Summary The method of proof trees offers an alternative method by which to demonstrate that an argument form is valid. 1) At the top of the tree, we write all the premises and the negation of the conclusion.

Summary The method of proof trees offers an alternative method by which to demonstrate that an argument form is valid. 1) At the top of the tree, we write all the premises and the negation of the conclusion. 2) Then we apply the rules for the connectives to produce branches on the tree.

Summary The method of proof trees offers an alternative method by which to demonstrate that an argument form is valid. 1) At the top of the tree, we write all the premises and the negation of the conclusion. 2) Then we apply the rules for the connectives to produce branches on the tree. 3) If a formula along with its negation occurs in a branch of a tree, then we close that branch with a .

Summary The method of proof trees offers an alternative method by which to demonstrate that an argument form is valid. 1) At the top of the tree, we write all the premises and the negation of the conclusion. 2) Then we apply the rules for the connectives to produce branches on the tree. 3) If a formula along with its negation occurs in a branch of a tree, then we close that branch with a . 4) If we thereby create a tree with all branches closed, then we have shown the argument valid.