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### Introductory LogicPHI 120

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Presentation: "Truth Tables – Validity vs. Soundness"

This PowerPoint Presentation contains a large number of slides, a good many of which are nearly identical. If you print this Presentation, I recommend six or nine slides per page.

Homework

- Study Allen/Hand Logic Primer
- Sec. 1.1, p. 2: “soundness”
- Sec. 2.2, p. 45, “incompatible premises”
- Ex. 2.2: i-xii

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

P & Q, ~P ⊢ R

Validity vs. soundnessValidity vs. Soundness

Valid Argument

- No invalidating assignment

Criteria of a Sound Argument

- argument is valid and
- 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 Premises

Atomic statements MUST be written in alphabetical order

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

- atomic statements
- negations of atomics
- inside parentheses
- negation of the parentheses
- any remaining connectives

Truth Tables

SequentsTruth 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

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.

~P, (R v ~P)<->(P v Q)⊢ ~Q

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

~P, R v ~P <-> P v Q ⊢ ~Q

The conclusion is a negation.

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Valid

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Invalid

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Invalid

Alphabetical Sequence!

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

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Invalid

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

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Testing for Validity:

Find the Invalidating Assignment

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Invalid

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Invalid

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Invalid

Homework

- Study Allen/Hand Logic Primer
- Sec. 1.1, p. 2: “soundness”
- Sec. 2.2, p. 45, “incompatible premises”
- Ex. 2.2: i-xii

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