rules

Question 1

inference rule (introduction of conjunction (∧Intro))

Answer 1

From φ and ψ infer φ ∧ ψ.
Applications of the rule looks like this:

φ
ψ
_____
φ ∧ ψ

Question 2

Inference Rule (elimination of conjunction 1 (∧Elim1))

Answer 2

From φ ∧ ψ infer φ.

Question 3

Inference Rule (elimination of conjunction 2 (∧Elim2))

Answer 3

From φ ∧ ψ infer ψ.

Question 4

Inference Rule (elimination of conjunction 1 (∧Elim1))

Answer 4

From φ ∧ ψ infer φ.
Applications of this rule look as follows:
φ ∧ ψ
\_\_\_\_\_
   φ

Question 5

Inference Rule (elimination of conjunction 2 (∧Elim2))

Answer 5

From φ ∧ ψ infer ψ.
Applications of this rule look as follows:
φ ∧ ψ
\_\_\_\_\_
   ψ

Question 6

Example ((P → R3), Q, R ` (P → R3) ∧ Q ∧ R)

Answer 6

1. P → R3 - premise
2. Q - premise
3. R - premise
4. (P → R3) ∧ Q - ∧Intro 1,2
5. (P → R3) ∧ Q ∧ R - ∧Intro 3,4
   - We write down the premises.
   - We infer the conjunction of previous steps using ∧Intro.
   - We infer the conjunction of previous steps using ∧Intro again.
   - We have a proof of (P → R3) ∧ Q ∧ R from our three premises:
   (P → R3), Q, and R.

Question 7

Example ((P ∨ Q) ∧ (R ∨ Q) ` R ∨ Q)

Answer 7

1. (P ∨ Q) ∧ (R ∨ Q) - premise
2. R ∨ Q - ∧Elim2 1
   - We write down the premise.
   - We infer the right conjunct using ∧Elim2.
   - We have a proof of R ∨ Q from our premise (P ∨ Q) ∧ (R ∨ Q)

Question 8

Example (P ∧ Q, R ` Q ∧ R)

Answer 8

1. P ∧ Q - premise
2. R - premise
3. Q - ∧Elim2 1
4. Q ∧ R - ∧Intro 2,3

Question 9

Inference Rule (introduction of the conditional (→Intro))

Answer 9

From ψ and the assumption that φ infer φ → ψ and discharge this assumption.
Applications of the rule looks like this:

φ
ψ
______
φ → ψ

Question 10

→Intro

Answer 10

→Intro is a rule of inference that requires an assumption that needs to be discharged when we apply the rule.

• This assumption is always the antecedent of the introduced conditional.
• We indicate that an assumption has been discharged and where by enclosing all steps between the assumption and the last step before the discharge in a left square bracket.
• The list of sentences wrapped around the square bracket is a closed subproof. A sentence φ is available in a step of a proof if it occurs before this step outside all closed subproofs.

Question 11

Example (P ∧ Q ` R → P ∧ R)

Answer 11

1. P ∧ Q = premise
2. R = assumption
3. P = ∧Elim1 1
4. P ∧ R = ∧Intro 2,3
5. R → P ∧ R = →Intro 2,4
   - We write down our premise.
   - We assume the antecedent of the conditional we want to prove.
   - We infer the left conjunct using ∧Elim1.
   - We infer the consequent using ∧Intro.
   - We infer the conditional that has our assumption that R as antecedent and a sentence we derived as
   the consequent and discharge our assumption of the antecedent by →Intro.
   - We have a proof of R → P ∧ R from our premise, P ∧ Q.

Question 12

Inference Rule (elimination of the conditional (→Elim))

Answer 12

From φ and φ → ψ infer ψ.
Applications of the rule looks like this:
   φ
φ → ψ
\_\_\_\_\_
   ψ

Question 13

Example ((P ∨ Q) ∧ (R → Q), R ` Q)

Answer 13

1. (P ∨ Q) ∧ (R → Q) = premise
2. R = premise
3. R → Q = ∧Elim2 1
4. Q = →Elim 2,3

Question 14

Inference Rule (introduction of disjunction 1 (∨Intro1))

Answer 14

From φ infer φ ∨ ψ.

Question 15

Inference Rule (introduction of disjunction 2 (∨Intro2))

Answer 15

From ψ infer φ ∨ ψ.

Question 16

Example (¬P ∧ Q, Q ∨ R → ¬R5 ` P ∨ ¬R5)

Answer 16

1. ¬P ∧ Q = premise
2. Q ∨ R → ¬R5 =premise
3. Q = ∧Elim2 1
4. Q ∨ R = ∨Intro1 3
5. ¬R5 = →Elim 2,4
6. P ∨ ¬R5 = ∨Intro2 5

Question 17

Inference Rule (elimination of disjunction (∨Elim))

Answer 17

From φ ∨ ψ, φ → χ, and ψ → χ infer χ.
applications of this rule look like this:
φ ∨ ψ
φ → χ
ψ → χ
\_\_\_\_\_
    χ

Question 18

further explanation of elimination of disjunction (VElim)

Answer 18

From a disjunction, i.e. of the form ‘A or B’, one cannot simply infer A or infer B.
What one can do is show that from each A and B a third claim, C, follows.
Thus, either way, C will be true.

Question 19

Example (P ∨ Q, P → Q, Q → R ` R)

Answer 19

1. P ∨ Q premise
2. P → Q premise
3. Q → R premise
4. P assumption
5. Q →Elim 2,4
6. R →Elim 3,5
7. P → R →Intro 4,6
8. R ∨Elim 1,3,7

We write down all our premises.
We assume the first disjunct of the disjunction we want to eliminate.
We infer Q using →Elim.
We infer R using →Elim.
We infer P → R using →Intro and discharge P.
We infer R using ∨Elim

Question 20

Example (R ∨ Q, R → Q ` Q)

Answer 20

1. R ∨ Q premise
2. R → Q premise
3. Q assumption
4. Q → Q →Intro 3
5. Q ∨Elim 1,2,4