homework 3. true or false? true or false? for each of the following wffs, determine whether it is...

Homework 3

True or False?

True or false? For each of the following WFFs, determine whether it is true or false on the given evaluation.

Evaluation: P = T, Q = T, R = T

1. ~(P ↔ Q)2. ~((P v (Q → R)) & ~P)

Write Down the Evaluation

P Q ~ (P ↔ Q)T T

Copy the Evaluations beneath the Sentence Letters

P Q ~ (P ↔ Q)T T T T

Consult Biconditional Truth-Table

φ ψ (φ ↔ ψ)T T TT F FF T FF F T

Copy Down the Answer

P Q ~ (P ↔ Q)T T T T T

Consult the Negation Truth-Table

φ ~φT FF T

Copy Down the Answer

P Q ~ (P ↔ Q)T T F T T T

The Answer is “False”

P Q ~ (P ↔ Q)T T F T T T

True or False?



1. ~(P ↔ Q)2. ~((P v (Q → R)) & ~P)

From My Inbox

Dear Michael,

Are there too many or too little parentheses in this formula:

~((P v (Q → R)) & ~P)

Thanks,--Concerned Student

Nope!

~((P v (Q → R)) & ~P)

Set #1

~((P v (Q → R)) & ~P)

Set #2

~((P v (Q → R)) & ~P)

Set #3

~((P v (Q → R)) & ~P)

Definition of WFF

i. All sentence letters are WFFs.ii. If φ is a WFF, then ~φ is a WFF.iii. If φ and ψ are WFFs, then (φ & ψ), (φ v ψ), (φ → ψ), (φ ↔ ψ) are

also WFFs.iv. Nothing else is a WFF.

Demonstration

Using the definition we can show that certain sequences of symbols are WFFs. For example ~((P v (Q → R)) & ~P) is a WFF.

All Sentence Letters Are WFFs

By (i), all sentence letters are WFFs. So:

P is a WFFQ is a WFFR is a WFF

~((P v (Q → R)) & ~P)

By (iii) if φ and ψ are WFFs, then (φ → ψ) is a WFF.

~((P v (Q → R)) & ~P)

By (iii) if φ and ψ are WFFs, then (φ v ψ) is a WFF.

~((P v (Q → R)) & ~P)

By (ii) if φ is a WFF, then ~φ is a WFF.

~((P v (Q → R)) & ~P)

By (iii) if φ and ψ are WFFs, then (φ & ψ) is a WFF.

~((P v (Q → R)) & ~P)

By (ii) if φ is a WFF, then ~φ is a WFF.

~((P v (Q → R)) & ~P)

Scope

Every occurrence of a connective in a WFF has a scope. The scope of that occurrence is the smallest WFF that contains it. For example The scope of “&” in “(~(~P&Q)→P)” is (~P&Q)

• (~(~P & Q) → P) is not a WFF.• (~(~P & Q) → P) is not a WFF.• (~(~P & Q) → P) is a WFF, but is bigger than (~P&Q)

Occurrences

Notice that the same symbol can occur different times in the same formula, and that its different occurrences can have different scopes.

• ~((~P & Q) & (R ↔ Q))• ~((~P & Q) & (R ↔ Q))• ~((~P & Q) & (R ↔ Q))• ~((~P & Q) & (R ↔ Q))

Scope of →

~((P v (Q → R)) & ~P)

Scope of v

~((P v (Q → R)) & ~P)

Scope of &

~((P v (Q → R)) & ~P)

Scope of Main Connective

~((P v (Q → R)) & ~P)

Main Connective

The main connective in a formula is the occurrence of the connective whose scope is widest (the scope of the occurrence of the connective = the formula).

A formula is a conditional if its main connective is →, a conjunction if it’s &, a negation if it’s ~, etc.

The Answer

The answer of an evaluation or truth-table is always located under the main connective.

True or False?



1. ~(P ↔ Q)2. ~((P v (Q → R)) & ~P)

Copy Down Evaluation

P Q R ((P v (Q → R)) & ~ P)T T T

Copy beneath Sentence Letters

P Q R ((P v (Q → R)) & ~ P)T T T T T T T

Consult the Negation Truth-Table

φ ~φT FF T

Copy Value from Truth-Table

P Q R ((P v (Q → R)) & ~ P)T T T T T T F T

Consult Truth-Table for Arrow

φ ψ (φ → ψ)T T TT F FF T TF F T

Copy Down Value

P Q R ((P v (Q → R)) & ~ P)T T T T T T T F T

Consult Truth-Table for Wedge

φ ψ (φ v ψ)T T TT F TF T TF F F

Copy Down Value

P Q R ((P v (Q → R)) & ~ P)T T T T T T T T F T

Consult Truth-Table for &

φ ψ (φ & ψ)T T TT F FF T FF F F

Copy Down Value

P Q R ((P v (Q → R)) & ~ P)T T T T T T T T F F T

The Answer is “True”


The Answer is “True”


*Note: This is because I forgot the ~.

Part 2: Full Truth-Tables

For each of the following WFFs, write out their full truth-table.

3. ~(P v Q)4. (((P → Q) → P) → P)

Write Down ALL POSSIBLE Evaluations

P Q ~ (P v Q)T TT FF TF F

Copy Values beneath Sentence Letters

P Q ~ (P v Q)T T T TT F T FF T F TF F F F

Consult Truth Table for Wedge

φ ψ (φ v ψ)T T TT F TF T TF F F

Insert Values Appropriately

P Q ~ (P v Q)T T T T TT F T T FF T F T TF F F F F

Consult Truth-Table for Negation

φ ~φT FF T

Insert Values Appropriately

P Q ~ (P v Q)T T F T T TT F F T T FF T F F T TF F T F F F

Michael’s Favorite Logic Formula

For each of the following WFFs, write out their full truth-table.

3. ~(P v Q)4. (((P → Q) → P) → P)

Write Down All Possible Evaluations

P Q (((P → Q) → P) → P)T TT FF TF F

Copy beneath Sentence Letters

P Q (((P → Q) → P) → P)T T T T T TT F T F T TF T F T F FF F F F F F

Consult Truth-Table for Arrow


Insert Appropriate Values

P Q (((P → Q) → P) → P)T T T T T T TT F T F F T TF T F T T F FF F F T F F F


P Q (((P → Q) → P) → P)T T T T T T T TT F T F F T T TF T F T T F F FF F F T F F F F


P Q (((P → Q) → P) → P)T T T T T T T T TT F T F F T T T TF T F T T F F T FF F F T F F F T F

Valid or Fails the Test?

Valid or invalid? For each of the following arguments, use the truth-table test for validity to determine whether it’s valid or invalid.5. P Ⱶ (Q → P)6. Q Ⱶ (Q → P)7. ~P Ⱶ (P → Q)8. ~Q Ⱶ (P → Q)

Write Down All Evaluations

P Q P (Q → P)T TT FF TF F

Write Down Truth Table For Premises

P Q P (Q → P)T T TT F TF T FF F F

Write Down Truth Table For Conclusion

P Q P (Q → P)T T T TT F T TF T F FF F F T

Ignore Lines Where Premises Are False


Make Sure Conclusion is True on the Other Lines


Always True with True Conclusion



P Q Q (Q → P)T TT FF TF F

Write Down Truth Table For Premises

P Q Q (Q → P)T T TT F FF T TF F F

Write Down Truth Table For Conclusion

P Q Q (Q → P)T T T TT F F TF T T FF F F T

Ignore Lines with False Premises


Make Sure Conclusion is All T on Other Lines


Sometimes False with True Premise



P Q ~P (P → Q)T TT FF TF F

Write Down Truth-Tables

P Q ~P (P → Q)T T F TT F F FF T T TF F T T

Make Sure Conclusion is T if Premises are T


Always True with False Antecedent


Write Down Evaluations, Truth-Tables

P Q ~Q (P → Q)T T F TT F T FF T F TF F T T


P Q ~Q (P → Q)T T F TT F T FF T F TF F T T

Sometimes False with False Consequent


homework 3. true or false? true or false? for each of the following wffs, determine whether it is...

Documents