Introductory LogicPHI 120

Presentation: "Truth Tables – Validity vs. Soundness"

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Homework1. Study Allen/Hand Logic Primer– Sec. 1.1, p. 2: “soundness”– Sec. 2.2, p. 45, “incompatible premises”

2. Ex. 2.2: i-xii

VALIDITY vs. SOUNDNESSP & Q, ~P ⊢ R

Validity: An argument is valid if and only if:• if all of its premises are true• its conclusion is true.

Corollary: It is impossible for a valid argument to have:• all true premises• false conclusion

Validity vs. Soundness

Valid Argument• No invalidating

assignment

Criteria of a Sound Argument1. argument is valid

and

2. all premises are True.

Valid but Unsound no invalidating

assignment not all premises true

Invalidating Assignment (1) conclusion is False (2) all premises are True

Incompatible PremisesP Q R P & R , ~ P ⊢ Q

Incompatible PremisesP Q R P & R , ~ P ⊢ Q

Atomic statements MUST be written in alphabetical order

Incompatible PremisesP Q R P & R , ~ P ⊢ Q

Incompatible PremisesP Q R P & R , ~ P ⊢ QTTTTFFFF

Incompatible PremisesP Q R P & R , ~ P ⊢ QT TT TT FT FF TF TF FF F

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T TT T FT F TT F FF T TF T FF F TF F F

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T TT T FT F TT F FF T TF T FF F TF F F

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T TT T FT F TT F FF T TF T FF F TF F F

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T FT T F FT F T FT F F FF T T TF T F TF F T TF F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T FT T F FT F T FT F F FF T T TF T F TF F T TF F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T FT T F F FT F T FT F F FF T T TF T F TF F T TF F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T FT T F F FT F T FT F F F FF T T TF T F TF F T TF F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T FT T F F FT F T FT F F F FF T T F TF T F TF F T TF F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T FT T F F FT F T FT F F F FF T T F TF T F F TF F T TF F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T FT T F F FT F T FT F F F FF T T F TF T F F TF F T F TF F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T FT T F F FT F T FT F F F FF T T F TF T F F TF F T F TF F F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T T FT T F F FT F T T FT F F F FF T T F TF T F F TF F T F TF F F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T T FT T F F FT F T T FT F F F FF T T F TF T F F TF F T F TF F F F T

Testing for Validity:

Find the Invalidating Assignment

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T T FT T F F FT F T T FT F F F FF T T F TF T F F TF F T F TF F F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T T FT T F F FT F T T F FT F F F FF T T F TF T F F TF F T F TF F F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T T FT T F F FT F T T FT F F F F FF T T F TF T F F TF F T F TF F F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T T FT T F F FT F T T FT F F F FF T T F TF T F F TF F T F T FF F F F T

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T T FT T F F FT F T T FT F F F FF T T F TF T F F TF F T F TF F F F T F

P Q R P & R , ~ P ⊢ QT T T T FT T F F FT F T T FT F F F FF T T F TF T F F TF F T F TF F F F T

No Invalidating Assignment

So the argument is valid

Incompatible Premises

Incompatible PremisesP Q R P & R , ~ P ⊢ QT T T T FT T F F FT F T T FT F F F FF T T F TF T F F TF F T F TF F F F T

p. 45

Validity vs. Soundness

Valid Argument• Impossible for

conclusion to be False and all premises True

Sound Argument• An argument is sound if

and only if it is valid and all its premises are true.

Valid but Unsound

No invalidating assignment

Not all premises true

SEQUENTSTruth Tables

Determine truth-values of:

1. atomic statements2. negations of atomics

3. inside parentheses4. negation of the parentheses

5. any remaining connectives

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ~Q⊢

First, identify the governing connectives.

~P, R v ~P <-> P v Q ~Q⊢

First, identify the governing connectives.

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ~Q⊢

First, identify the governing connectives.

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, (R v ~P) <-> (P v Q) ~Q⊢

The second premise is a complex binary: Φ <-> Ψ

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QThe conclusion is a negation.

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

CONSTRUCT THE TRUTH TABLETruth Tables – Sequents

~P, R v ~P <-> P v Q ~Q⊢Determine the number of rows for the sequent

23 simple statements = 8 rows

~P, R v ~P <-> P v Q ⊢ ~Q~ P , R v ~ P <-> P v Q ⊢ ~ Q _ __

Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ Q _ __

Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ Q _ __

Valid

____Invalid

Alphabetical Sequence!

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ Q _ __

Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QTTTTFFFF

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT TT TT FT FF TF TF FF F

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T TT T FT F TT F FF T TF T FF F TF F F

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F FT T F F FT F T F FT F F F FF T T T TF T F T TF F T T TF F F T T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F F FT T F F F FT F T F F TT F F F F TF T T T T FF T F T T FF F T T T TF F F T T T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F F FT T F F F FT F T F F TT F F F F TF T T T T FF T F T T FF F T T T TF F F T T T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

(R v ~P) <-> (P v Q)

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F F FT T F F F F FT F T F F TT F F F F F TF T T T T FF T F T T FF F T T T TF F F T T T

_ __Valid

____Invalid

(R v ~P) <-> (P v Q)

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F F FT T F F F F FT F T F F TT F F F F F TF T T T T FF T F T T FF F T T T TF F F T T T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F FT T F F F F FT F T F T F TT F F F F F TF T T T T T FF T F T T T FF F T T T T TF F F T T T T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F FT T F F F F FT F T F T F TT F F F F F TF T T T T T FF T F T T T FF F T T T T F TF F F T T T F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F FT T F F F F FT F T F T F TT F F F F F TF T T T T T FF T F T T T FF F T T T T F TF F F T T T F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T FT T F F F F T FT F T F T F T TT F F F F F T TF T T T T T T FF T F T T T T FF F T T T T F TF F F T T T F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T FT T F F F F F T FT F T F T F T TT F F F F F F T TF T T T T T T FF T F T T T T FF F T T T T F F TF F F T T T F F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T FT T F F F F F T FT F T F T F T TT F F F F F F T TF T T T T T T FF T F T T T T FF F T T T T F F TF F F T T T F F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T T FT T F F F F F T FT F T F T F T T TT F F F F F F T TF T T T T T T T FF T F T T T T T FF F T T T T F F TF F F T T T F F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T T FT T F F F F F T FT F T F T F T T TT F F F F F F T TF T T T T T T T FF T F T T T T T FF F T T T T F F TF F F T T T F F T

_ __Valid

____InvalidTesting for Validity:

Find the Invalidating Assignment

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T T FT T F F F F F T FT F T F T F T T TT F F F F F F T TF T T T T T T T FF T F T T T T T FF F T T T T F F TF F F T T T F F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T T FT T F F F F F T FT F T F T F T T TT F F F F F F T TF T T T T T T T FF T F T T T T T FF F T T T T F F TF F F T T T F F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T T FT T F F F F F T FT F T F T F T T TT F F F F F F T TF T T T T T T T FF T F T T T T T FF F T T T T F F TF F F T T T F F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T T FT T F F F F F T FT F T F T F T T TT F F F F F F T TF T T T T T T T FF T F T T T T T FF F T T T T F F TF F F T T T F F T

_ __Valid

____Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T T FT T F F F F F T FT F T F T F T T TT F F F F F F T TF T T T T T T T FF T F T T T T T FF F T T T T F F TF F F T T T F F T

_ __Valid

__Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T T FT T F F F F F T FT F T F T F T T TT F F F F F F T TF T T T T T T T FF T F T T T T T FF F T T T T F F TF F F T T T F F T

Tautology Inconsistency Contingency

_ __Valid

__Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P <-> P v Q ⊢ ~QP Q R ~ P , R v ~ P <-> P v Q ⊢ ~ QT T T F T F T T FT T F F F F F T FT F T F T F T T TT F F F F F F T TF T T T T T T T FF T F T T T T T FF F T T T T F F TF F F T T T F F T

Tautology Inconsistency Contingency

_ __Valid

__Invalid

Truth TablesDirections: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

Homework1. Study Allen/Hand Logic Primer– Sec. 1.1, p. 2: “soundness”– Sec. 2.2, p. 45, “incompatible premises”

2. Ex. 2.2: i-xii