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)

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

Copy the Evaluations beneath the Sentence Letters Q ~ (P ↔ Q) T

Consult Biconditional Truth-Table φ ψ (φ ↔ ψ) T F

Consult Biconditional Truth-Table φ ψ (φ ↔ ψ) T F

Copy Down the Answer P Q ~ (P ↔ Q) T

Consult the Negation Truth-Table φ ~φ T F

Consult the Negation Truth-Table φ ~φ T F

Copy Down the Answer P Q ~ (P ↔ Q) T F

The Answer is “False” P Q ~ (P ↔ Q) T F

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)

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 All sentence letters are WFFs. If φ is a WFF, then ~φ is a WFF. If φ and ψ are WFFs, then (φ & ψ), (φ v ψ), (φ → ψ), (φ ↔ ψ) are also WFFs. 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 WFF Q is a WFF R is a WFF

All Sentence Letters Are WFFs By (i), all sentence letters are WFFs. So: P is a WFF Q is a WFF R 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)

~((P v (Q → R)) & ~P)

All Sentence Letters Are WFFs By (i), all sentence letters are WFFs. So: P is a WFF Q is a WFF R is a WFF

~((P v (Q → R)) & ~P)

~((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)

~((P v (Q → R)) & ~P)

All Sentence Letters Are WFFs By (i), all sentence letters are WFFs. So: P is a WFF Q is a WFF R is a WFF

~((P v (Q → R)) & ~P)

~((P v (Q → R)) & ~P)

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

~((P v (Q → R)) & ~P)

~((P v (Q → R)) & ~P)

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

~((P v (Q → R)) & ~P)

~((P v (Q → R)) & ~P)

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

~((P v (Q → R)) & ~P)

~((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 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))

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? 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)

Copy Down Evaluation P Q R ((P v (Q → R)) & ~ P) T

Copy beneath Sentence Letters Q R ((P v (Q → R)) & ~ P) T

Consult the Negation Truth-Table φ ~φ T F

Consult the Negation Truth-Table φ ~φ T F

Copy Value from Truth-Table Q R ((P v (Q → R)) & ~ P) T F

Consult Truth-Table for Arrow φ ψ (φ → ψ) T F

Consult Truth-Table for Arrow φ ψ (φ → ψ) T F

Copy Down Value P Q R ((P v (Q → R)) & ~ P) T F

Consult Truth-Table for Wedge φ ψ (φ v ψ) T F

Copy Down Value P Q R ((P v (Q → R)) & ~ P) T F

Consult Truth-Table for & φ ψ (φ & ψ) T F

Consult Truth-Table for & φ ψ (φ & ψ) T F

Copy Down Value P Q R ((P v (Q → R)) & ~ P) T F

The Answer is “True” P Q R ((P v (Q → R)) & ~ P) T F

The Answer is “True” P Q R ((P v (Q → R)) & ~ P) T F *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)

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 Q ~ (P v Q) T F

Copy Values beneath Sentence Letters Q ~ (P v Q) T F

Consult Truth Table for Wedge φ ψ (φ v ψ) T F

Insert Values Appropriately Q ~ (P v Q) T F

Consult Truth-Table for Negation φ ~φ T F

Insert Values Appropriately Q ~ (P v Q) T 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 Q (((P → Q) P) T F

Copy beneath Sentence Letters Q (((P → Q) P) T F

Consult Truth-Table for Arrow φ ψ (φ → ψ) T F

Insert Appropriate Values Q (((P → Q) P) T F

Insert Appropriate Values Q (((P → Q) P) T F

Insert Appropriate Values Q (((P → Q) P) 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)

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 (Q → P) T F

Write Down Truth Table For Premises Q (Q → P) T F

Write Down Truth Table For Conclusion P Q (Q → P) T F

Ignore Lines Where Premises Are False Q (Q → P) T F

Make Sure Conclusion is True on the Other Lines P Q (Q → P) T F

Always True with True Conclusion φ ψ (φ → ψ) 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 (Q → P) T F

Write Down Truth Table For Premises Q (Q → P) T F

Write Down Truth Table For Conclusion P Q (Q → P) T F

Ignore Lines with False Premises Q (Q → P) T F

Make Sure Conclusion is All T on Other Lines P Q (Q → P) T F

Sometimes False with True Premise φ ψ (φ → ψ) 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 (P → Q) T F

Write Down Truth-Tables P Q ~P (P → Q) T F

Ignore Lines with False Premises Q ~P (P → Q) T F

Make Sure Conclusion is T if Premises are T Q ~P (P → Q) T F

Always True with False Antecedent φ ψ (φ → ψ) 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 Evaluations, Truth-Tables P Q ~Q (P → Q) T F

Ignore Lines with False Premises Q ~Q (P → Q) T F

Ignore Lines with False Premises Q ~Q (P → Q) T F

Sometimes False with False Consequent φ ψ (φ → ψ) T F