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Introductory Logic PHI 120

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Presentation: "Truth Tables – Validity vs. Soundness"

Introductory LogicPHI 120

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.

- 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

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

Atomic statements MUST be written in alphabetical order

Testing for Validity:

Find the Invalidating Assignment

No Invalidating Assignment

So the argument is valid

p. 45

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

Determine truth-values of:

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

Truth Tables

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.

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.

Truth Tables – Sequents

~P, R v ~P <-> P v Q ⊢ ~Q

Determine the number of rows for the sequent

23 simple statements = 8 rows

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

Alphabetical Sequence!

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

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

_ __

Valid

____

Invalid

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

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

Testing for Validity:

Find the Invalidating Assignment

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

____

Invalid

_ __

Valid

__

Invalid

_ __

Valid

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Invalid

_ __

Valid

__

Invalid

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

- Ex. 2.2: i-xii