Introductory Logic PHI 120 Presentation: "Truth Tables – Sequents" Introductory Logic PHI 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.
Homework Turn to page 40 in The Logic Primer Study Allen/Hand Logic Primer Sec. 1.1, p. 1-2: “validity” Sec. 2.2, p. 43-4, “validity” & “invalidating assignment Complete Ex. 2.1, p. 42: i-x Turn to page 40 in The Logic Primer also take out TTs handout
Truth Value of Sentences Truth Tables Truth Value of Sentences Section 2.1 (quick review)
Atomic sentence P T F Simple
Truth Tables Complex Sentences See bottom of Truth Tables Handout
~Φ False? Φ ~ T F
~Φ False – if the statement being negated (Φ) is True Φ ~ T F
Φ & Ψ False? Φ Ψ & T F
Φ & Ψ False – if one or both conjuncts are False Φ Ψ & T F
Φ & Ψ False – if one or both conjuncts are False Φ Ψ & T F
Φ v Ψ False? Φ Ψ v T F
Φ v Ψ False – only if both disjuncts are False Φ Ψ v T F
Φ v Ψ False – only if both disjuncts are False Φ Ψ v T F
Φ -> Ψ False? Φ Ψ -> T F
Φ -> Ψ False – if antecedent is True and consequent is False Φ Ψ -> T F
Φ -> Ψ False – if antecedent is True and consequent is False Φ Ψ -> T F
Φ <-> Ψ False? Φ Ψ <-> T F
Φ <-> Ψ Φ Ψ <-> T F False – if the two conditions have a different truth value Φ Ψ <-> T F
Φ <-> Ψ Φ Ψ <-> T F False – if the two conditions have a different truth value Φ Ψ <-> T F
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Note the binary structure Identify the main connective. How many atomic sentences are in this WFF?
Determine the number of rows for the WFF or the sequent as a whole (P & ~Q) v ~(P & ~Q) Φ v ~Φ Determine the number of rows for the WFF or the sequent as a whole P Q (P & ~ Q) v
Determine the number of rows for the WFF or the sequent as a whole (P & ~Q) v ~(P & ~Q) Determine the number of rows for the WFF or the sequent as a whole P Q (P & ~ Q) v 1 2 3 4 5 6 7 8 9 10 11 12
TT Method in a Nutshell Determine truth-values of: atomic statements negations of atomics inside parentheses negation of the parentheses any remaining connectives
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 3 on Handout P Q (P & ~ Q) v 1 2 3 4 5 6 7 8 9 10 11 12 Step 3 on Handout Fill in left main column first.
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 3 on Handout P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 3 on Handout Fill in left main column first.
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 3 on Handout P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 3 on Handout Fill in left main column first.
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 4 on Handout P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 4 on Handout Assign truth-values for negation of simple statements
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 4 on Handout P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 4 on Handout Assign truth-values for negation of simple statements
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 4 on Handout P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 4 on Handout Assign truth-values for negation of simple statements
When is a conjunction (an “&” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ When is a conjunction (an “&” statement) false? P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 5 on Handout Assign truth-values for innermost binary connectives
When is a conjunction (an “&” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 When is a conjunction (an “&” statement) false?
When is a conjunction (an “&” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 When is a conjunction (an “&” statement) false?
When is a conjunction (an “&” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 When is a conjunction (an “&” statement) false?
When is a conjunction (an “&” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 When is a conjunction (an “&” statement) false?
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 5 on Handout P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 5 on Handout Assign truth-values for innermost binary connectives
(P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 6a on Handout P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 6a on Handout Assign truth-values for negation of compounds
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 6a on Handout P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 6a on Handout Assign truth-values for negation of compounds
When is a disjunction (a “v” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ When is a disjunction (a “v” statement) false? P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 6b on Handout Assign truth-values for remaining
(P & ~Q) v ~(P & ~Q) Φ v ~Φ Step 6b on Handout P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Step 6b on Handout Assign truth-values for remaining
When is a disjunction (a “v” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 When is a disjunction (a “v” statement) false?
When is a disjunction (a “v” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 When is a disjunction (a “v” statement) false?
When is a disjunction (a “v” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 When is a disjunction (a “v” statement) false?
When is a disjunction (a “v” statement) false? (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 When is a disjunction (a “v” statement) false?
The values under the governing connective are all T’s. (P & ~Q) v ~(P & ~Q) Φ v ~Φ P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 The values under the governing connective are all T’s.
Classifying Sentences TTs: Sentences p. 47-8: “tautology,” “inconsistency & contingent” Classifying Sentences
Φ v Ψ Tautologies P Q (P & ~ Q) v T F Only Ts under main operator Look Under the Main Connective P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Tautologies Only Ts under main operator Necessarily true
Φ v Ψ Tautologies P Q (P & ~ Q) v T F Only Ts under main operator Look Under the Main Connective P Q (P & ~ Q) v T F 1 2 3 4 5 6 7 8 9 10 11 12 Tautologies Only Ts under main operator Necessarily true
~Φ Inconsistencies P Q ~ ((P & Q) v (P Q)) T F Look Under the Main Connective P Q ~ ((P & Q) v (P Q)) T F 1 2 3 4 5 6 7 8 9 10 11 12 13 Inconsistencies Only Fs under main operator Necessarily false
~Φ Inconsistencies P Q ~ ((P & Q) v (P Q)) T F Look Under the Main Connective P Q ~ ((P & Q) v (P Q)) T F 1 2 3 4 5 6 7 8 9 10 11 12 13 Inconsistencies Only Fs under main operator Necessarily false
Φ & Ψ Contingencies P Q & ~ T F Look Under the Main Connective P Q & ~ T F 1 2 3 4 5 6 Contingencies At least one T and one F under main operator Sometime true, sometime false
Φ & Ψ Contingencies P Q & ~ T F Look Under the Main Connective P Q & ~ T F 1 2 3 4 5 6 Contingencies At least one T and one F under main operator Sometime true, sometime false
(conclusion indicator) Truth Tables Section 2.2 Sequents P -> Q, Q ⊢ P Premise(s) ⊢ Conclusion “turnstile” (conclusion indicator)
TTs: Sequents Testing for validity
Φ -> Ψ, Ψ ⊢ Φ Testing for Validity I The Invalidating Assignment Conclusion: False All Premises: True Φ -> Ψ, Ψ ⊢ Φ The TT will contain an invalidating assignment (Invalid form: “Affirming the consequent”)
“Affirming the Consequent” P Q -> , ⊢ Φ -> Ψ , Ψ ⊢ Φ
“Affirming the Consequent” P Q -> , ⊢
“Affirming the Consequent” P Q -> , ⊢
TT Method in a Nutshell Determine truth-values of: atomic statements negations of atomics inside parentheses negation of the parentheses any remaining connectives
“Affirming the Consequent” P Q -> , ⊢ T F
“Affirming the Consequent” P Q -> , ⊢ T F
“Affirming the Consequent” P Q -> , ⊢ T F Always circle the governing connective in each sentence.
“Affirming the Consequent” P Q -> , ⊢ T F
“Affirming the Consequent” P Q -> , ⊢ T F
“Affirming the Consequent” P Q -> , ⊢ T F
“Affirming the Consequent” P Q -> , ⊢ T F If invalidating assignment, then argument is: Invalid
“Affirming the Consequent” P Q -> , ⊢ T F Circle the invalidating assignment!
Homework Study Allen/Hand Logic Primer Complete Ex. 2.1, p. 42: i-x Sec. 1.1, p. 1-2: “validity” Sec. 2.2, p. 43-4, “validity” & “invalidating assignment Complete Ex. 2.1, p. 42: i-x