DERIVATIONS AND TRUTH TABLES
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1 DERIVATIONS AND TRUTH TABLES Tomoya Sato Department of Philosophy University of California, San Diego Phil120: Symbolic Logic Summer 2014 TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 1 / 65
2 WHAT IS LOGIC? LOGIC Logic is the study of formal validity. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 2 / 65
3 REVIEW SENTENCES IN SYMBOLIC LANGUAGE Any capital letter between "P" and "Z" is a symbolic sentence (an atomic sentence). If "ϕ" is a symbolic sentence, so is " ϕ" (a molecular sentence). (called a negation of ϕ) If "ϕ" and "ψ" are symbolic sentences, so is "(ϕ ψ)" (a molecular sentence). (called a conditional) Nothing is a symbolic sentence of Chapter 1 unless it can be constructed by means of these provisions. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 3 / 65
4 REVIEW Modus Ponens ϕ ψ ϕ ψ Modus Tollens ϕ ψ ψ ϕ TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 4 / 65
5 REVIEW Double Negation ϕ ϕ ϕ ϕ Repetition ϕ ϕ. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 5 / 65
6 REVIEW THREE TYPES OF DERIVATIONS Direct derivation; Conditional derivation; Indirect derivation. THEOREM If there is a complete derivation for an argument, then the argument is formally valid. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 6 / 65
7 CHAPTER 2, SECTION 1 SYMBOLS OF SYMBOLIC LANGUAGE OF CHAPTER 1 Atomic sentences: P, Q, R,... Z. Logical connectives:, SYMBOLS OF SYMBOLIC LANGUAGE OF CHAPTER 2 Atomic sentences: P, Q, R,... Z. Logical connectives:,,,, TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 7 / 65
8 SYMBOLIC LANGUAGE SYMBOLIC SENTENCES IN SYMBOLIC LANGUAGE Any capital letter between "P" and "Z" is a symbolic sentence (an atomic sentence). If "ϕ" is a symbolic sentence, so is " ϕ" (a molecular sentence). (called a negation of ϕ) If "ϕ" and "ψ" are symbolic sentences, so are "(ϕ ψ)", "(ϕ ψ)", "(ϕ ψ)", "(ϕ ψ)" (molecular sentences). (called a conditional, a conjunction, a disjunction, a biconditional) Nothing is a symbolic sentence for purposes of Chapter 2 unless it can be generated by means of these provisions. EXAMPLES ( P Q), ((Q R) P), (P (Q R)). (P Q), (P Q R), ( PQ). TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 8 / 65
9 CONVENTIONS RULES OF SYMBOLIC SENTENCES You can use ( ), [ ], and { }; You can omit the outermost parentheses ( ) INFORMAL CONVENTIONS P Q R is an informal notation for (P Q) R; P Q R is an informal notation for P (Q R). TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 9 / 65
10 CHAPTER 2, SECTION 2 THE MEANINGS OF NEW SYMBOLS " " corresponds to "and", "but", etc. " " corresponds to "or", "unless", etc. " " corresponds to "if and only if", etc. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 10 / 65
11 "BUT" AND "UNLESS" "BUT" AND "AND" "But" and "and" are equivalent connectives from a logical point of view. Tomoya is Japanese but he loves junk food. Tomoya is Japanese and he loves junk food. "UNLESS" AND "OR" "Unless" and "or" are equivalent connectives from a logical point of view. I won t go to the party unless there is junk food. I won t go to the party or there is junk food. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 11 / 65
12 SYMBOLIZATION SYMBOLIZATION Tomoya is Japanese but he loves junk food. P : Tomoya is Japanese. Q : Tomoya loves junk food. P Q SYMBOLIZATION I will have pizza for lunch or I will have a hamburger for lunch. R : I will have pizza for lunch. S : I will have a hamburger for lunch. R S SYMBOLIZATION Number n is odd if and only if n is not divisible by 2. T : Number n is odd. U : n is divisible by 2. T U TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 12 / 65
13 CHAPTER 2, SECTION 3 SCHEME OF ABBREVIATION P : Peter is a UCSD student. Q : Quincy is a UCSD student. SYMBOLIZATION Peter and Quincy are UCSD students. Peter is a UCSD student and Quincy is a UCSD student. P Q SYMBOLIZATION Peter, but not Quincy, is a UCSD student. Peter is a UCSD student but Quincy is not a UCSD student. P Q TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 13 / 65
14 SYMBOLIZATION "NOT BOTH" Not both Peter and Quincy are UCSD students. It is not the case that both Peter and Quincy are UCSD students. (P Q) Peter is not a UCSD student or Quincy is not a UCSD student. P Q "EITHER OR" Either Peter or Quincy is a UCSD student. Peter is a UCSD student or Quincy is a UCSD student. P Q TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 14 / 65
15 SYMBOLIZATION "NEITHER NOR" Neither Peter nor Quincy is a UCSD student. It is not the case that either Peter is a UCSD student or Quincy is a UCSD student. (P Q) Peter is not a UCSD student and Quincy is not a UCSD student. P Q TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 15 / 65
16 EXERCISES SCHEME OF ABBREVIATION W = Wilma attends. S = Sally attends. R = Robert will be bored. P = Peter will be bored. If neither Wilma nor Sally attends, either Robert or Peter will be bored. (W S) (R P) TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 16 / 65
17 EXERCISESS SCHEME OF ABBREVIATION V = Veronica will leave. W = William will leave. Y = Yolanda will leave. EXERCISESS 1 Veronica won t leave if and only if William won t leave. 2 William and Veronica will both leave if Yolanda does, given that Veronica doesn t. 3 Veronica or William will leave unless Yolanda leaves. 4 Either Yolanda leaves and Veronica doesn t, or Veronica leaves and William doesn t. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 17 / 65
18 CHAPTER 2, SECTION 4 THE FIRST FOUR INFERENCE RULES mp: Modus Ponens; mt: Modus Tollens; dn: Double Negation; r: Repetition. Modus Ponens ϕ ψ ϕ ψ Modus Tollens ϕ ψ ψ ϕ TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 18 / 65
19 REVIEW Double Negation ϕ ϕ ϕ ϕ Repetition ϕ ϕ TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 19 / 65
20 INFERENCE RULES Simplification (s) ϕ ψ ϕ / ψ Adjunction (adj) ϕ ψ ϕ ψ 3. P (Q R).. 7. P Q R 3. S T.. 7. U 8. (S T) U U (S T) TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 20 / 65
21 INFERENCE RULES Simplification (s) ϕ ψ ϕ / ψ Both Peter and Quincy are UCSD students Peter is a UCSD student. Quincy is a UCSD student. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 21 / 65
22 INFERENCE RULES Adjunction (adj) ϕ ψ ϕ ψ Peter is a UCSD student. Quincy is a UCSD student. Both Peter and Quincy are UCSD students TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 22 / 65
23 INFERENCE RULES Addition (add) ϕ (ϕ ψ) / (ψ ϕ) Modus tollendo ponens (mtp) ϕ ψ ψ / ϕ ϕ / ψ 3. P.. 7. (Q R) P R ((Q R) P) 3. S T.. 7. S T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 23 / 65
24 INFERENCE RULES Addition (add) ϕ (ϕ ψ) / (ψ ϕ) Peter is a UCSD student. Either Peter or Quincy is a UCSD student. Either Peter or Roger is a UCSD student. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 24 / 65
25 INFERENCE RULES Modus tollendo ponens (mtp) ϕ ψ ψ / ϕ ϕ / ψ Either Peter or Quincy is a UCSD student. Peter is not a UCSD student. Quincy is a UCSD student. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 25 / 65
26 INFERENCE RULES bc ϕ ψ (ϕ ψ) / (ψ ϕ) cb ϕ ψ ψ ϕ ϕ ψ 3. P (Q R).. 7. P (Q R) (Q R) P 3. S T.. 7. T S S T 11. T S TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 26 / 65
27 INFERENCE RULES Biconditional-to-Conditional (bc) ϕ ψ (ϕ ψ) / (ψ ϕ) n is an odd number if and only if n is not divisible by 2. If n is an odd number, then n is not divisible by 2. If n is not divisible by 2, then n is an odd number. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 27 / 65
28 INFERENCE RULES Conditional-to-Biconditional (cb) ϕ ψ ψ ϕ (ϕ ψ) / (ψ ϕ) If n is an odd number, then n is not divisible by 2. If n is not divisible by 2, then n is an odd number. n is an odd number if and only if n is not divisible by 2. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 28 / 65
29 EXERCISES 1. S P 2. (P Q) R 3. Q R Q 1. R P 2. Q R P Q TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 29 / 65
30 CHAPTER 2, SECTION 5 STRATEGY P or ϕ: Use id. ϕ ψ: Use cd, unless dd is easier or you can get the consequence easily. ϕ ψ: Show ϕ ψ. Use cdj, then dd. ϕ ψ: Show ϕ, then show ψ. Then, adj and dd. ϕ ψ: Show both ϕ ψ and ψ ϕ. Then, cb and dd. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 30 / 65
31 CHAPTER 2, SECTION 6 NOTE Abbreviating derivations is not allowed in this course. You can use one inference rule for each line. 1. P 2. Q P 3. Q R R 1. Show R 2. Q pr1 dn pr2 mt 3. R 2 dn pr3 mp dd TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 31 / 65
32 CHAPTER 2, SECTION 7 NOTE Using theorems as rules is not allowed in this course. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 32 / 65
33 CHAPTER 2, SECTION 8 Conditional as Disjunction (cdj) ϕ ψ ϕ ψ ϕ ψ ϕ ψ ϕ ψ ϕ ψ ϕ ψ ϕ ψ TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 33 / 65
34 INFERENCE RULES Conditional as Disjunction (cdj) ϕ ψ ϕ ψ ϕ ψ ϕ ψ 3. P (Q R).. 7. P (Q R) 3. (S T) U.. 9. (S T) U TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 34 / 65
35 INFERENCE RULES Conditional as Disjunction (cdj) ϕ ψ ϕ ψ ϕ ψ ϕ ψ 3. P (Q R).. 7. P (Q R) 3. (S T) U.. 9. (S T) U TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 35 / 65
36 INFERENCE RULES Conditional as Disjunction (cdj) ϕ ψ ϕ ψ Unless you know the American culture, you cannot understand American jokes. If you don t know the American culture, you cannot understand American jokes. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 36 / 65
37 INFERENCE RULES DEMORGAN (DM) A negation of a conjunction is equivalent to the disjunction of the opposites of its conjuncts. A negation of a disjunction is equivalent to the conjunction of the opposites of its conjuncts. (ϕ ψ) ϕ ψ (ϕ ψ) ϕ ψ ( ϕ ψ) ϕ ψ ( ϕ ψ) ϕ ψ TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 37 / 65
38 INFERENCE RULES (ϕ ψ) ϕ ψ "NOT BOTH" Not both Peter and Quincy are UCSD students. It is not the case that both Peter and Quincy are UCSD students. (P Q) Peter is not a UCSD student or Quincy is not a UCSD student. P Q TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 38 / 65
39 INFERENCE RULES (ϕ ψ) ϕ ψ "NEITHER NOR" Neither Peter nor Quincy is a UCSD student. It is not the case that either Peter is a UCSD student or Quincy is a UCSD student. (P Q) Peter is not a UCSD student and Quincy is not a UCSD student. P Q TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 39 / 65
40 INFERENCE RULES NOTE "nc", "sc", and "nb" are not allowed in this course. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 40 / 65
41 THE LIST OF INFERENCE RULES 1 Modus Ponens (mp) 2 Modus Tollens (mt) 3 Double Negation (dn) 4 Repetition (r) 5 Simplification (s) 6 Adjunction (adj) 7 Addition (add) 8 Modus tollendo Ponens (mtp) 9 Biconditional-to-Conditional (bc) 10 Conditional-to-Biconditional (cb) 11 Conditional as Disjunction (cdj) 12 DeMorgan (dm) TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 41 / 65
42 CHAPTER 2, SECTION 10 The semantic method The proof-theoretic method Symbolization Formal Validity TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 42 / 65
43 TRUTH TABLE P Q R R Q R P (Q R) T T T F T T T T F T T T T F T F F F T F F T T T F T T F T T F T F T T T F F T F F T F F F T T T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 43 / 65
44 TRUTH CONDITIONS When is a negation true? TRUTH CONDITION OF NEGATIONS P : Tommy loves In-N-Out french fries. P : Tommy doesn t love In-N-Out french fries. P is true if P is false. P is false if P is true. ϕ T F ϕ F T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 44 / 65
45 TRUTH CONDITIONS When is a conjunction true? TRUTH CONDITION OF CONJUNCTIONS P : Peter loves In-N-Out french fries. Q : Quincy loves In-N-Out french fries. P Q : Peter and Quincy love In-N-Out french fries. P Q is true if both P and Q are true. P Q is false otherwise. ϕ ψ ϕ ψ T T T T F F F T F F F F TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 45 / 65
46 TRUTH CONDITIONS When is a disjunction true? TRUTH CONDITION OF DISJUNCTIONS P : You can ask for soup. Q : You can ask for salad. P e Q : You can ask for soup or salad. P e Q is true if you can ask for either soup or salad (but not both). P e Q is false otherwise. ϕ ψ ϕ e ψ T T F T F T F T T F F F TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 46 / 65
47 TRUTH CONDITIONS When is a disjunction true? TRUTH CONDITION OF DISJUNCTIONS P : You can ask for cream. Q : You can ask for sugar. P i Q : You can ask for cream or sugar. P i Q is true if you can ask for cream or sugar or both. P i Q is false otherwise. ϕ ψ ϕ i ψ T T T T F T F T T F F F TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 47 / 65
48 INCLUSIVE "OR" AND EXCLUSIVE "OR" TWO TYPES OF "OR" "Soup or Salad?" (Exclusive "or") "Cream or Sugar?" (Inclusive "or") ϕ ψ ϕ e ψ T T F T F T F T T F F F ϕ ψ ϕ i ψ T T T T F T F T T F F F TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 48 / 65
49 TRUTH CONDITIONS When is a conditional true? TRUTH CONDITION OF CONDITIONALS P : It rains. Q : The game will be called off. P Q : If it rains, the game will be called off. P Q is true if both P and Q are true. P Q is false if P is true but Q is false. ϕ ψ ϕ ψ T T T T F F F T T F F T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 49 / 65
50 TRUTH CONDITIONS Why is a conditional true when its antecedent is false? ϕ ψ ϕ ψ T T T T F F F T T F F T EQUIVALENT SENTENCES P Q P Q When P is false, P is true. P Q is true. Therefore, P Q is also true. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 50 / 65
51 TRUTH CONDITIONS When is a biconditional true? TRUTH CONDITION OF BICONDITIONALS P : n is an odd number Q : n + 1 is divisible by 2. P Q : n is an odd number if and only if n + 1 is divisible by 2. P Q is true if both P and Q are true, or both of them are false. P Q is false otherwise. ϕ ψ ϕ ψ T T T T F F F T F F F T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 51 / 65
52 TRUTH TABLES OF LOGICAL CONNECTIVES ϕ ϕ T F F T ϕ ψ ϕ ψ T T T T F F F T F F F F ϕ ψ ϕ ψ T T T T F F F T T F F T ϕ ψ ϕ i ψ T T T T F T F T T F F F ϕ ψ ϕ ψ T T T T F F F T F F F T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 52 / 65
53 TRUTH TABLE (P Q) Q HOW TO CONSTRUCT A TRUTH TABLE 1 Add columns for each atomic sentence letter. 2 Add columns for each molecular statement, including the full statement itself. 3 Fill in the "T"s and "F"s for the atomic sentences one column at a time, from left to right. (for n atomic sentences, you will need 2 n rows of "T"s and "F"s) 4 Fill in the "T"s and "F"s for the molecular sentences. P Q P Q (P Q) Q T T T T T F T F F T T T F F F T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 53 / 65
54 TRUTH TABLE P (Q P) HOW TO CONSTRUCT A TRUTH TABLE 1 Add columns for each atomic sentence letter. 2 Add columns for each molecular statement, including the full statement itself. 3 Fill in the "T"s and "F"s for the atomic sentences one column at a time, from left to right. (for n atomic sentences, you will need 2 n rows of "T"s and "F"s) 4 Fill in the "T"s and "F"s for the molecular sentences. P Q P Q P P (Q P) T T F T F T F F F F F T T F F F F T T T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 54 / 65
55 EXERCISES EXERCISES Construct truth tables for the following sentences. 1 (P Q) 2 P (Q Q) 3 P (Q R) EXERCISES Construct truth tables for the following sentences. 1 (P Q) R 2 ( P Q) P TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 55 / 65
56 TAUTOLOGY DEFINITION: TAUTOLOGY def A sentence is a tautology no matter what truth values are assigned to its simple parts, the definitions of the connectives used in the sentence determine that the sentence is true. A sentence is a tautology def its column contains only Ts. (P Q) P P Q P Q (P Q) P T T T T T F F T F T F T F F F T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 56 / 65
57 EXERCISES EXERCISES Construct truth tables for the following sentences to determine whether they are tautologies or not. 1 P (Q P) 2 ( P Q) (Q P) EXERCISES Construct truth tables for the following sentences to determine whether they are tautologies or not. 1 [P (Q R)] [(Q P) R] 2 (P Q) ( R Q) TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 57 / 65
58 CHAPTER 2, SECTION P Q 2. R Q 3. P R Q DEFINITION: TAUTOLOGICAL IMPLICATION A set {ϕ 1, ϕ 2,, ϕ n } of sentences tautologically implies a given sentence ψ def there is no assignment of truth values to the atomic parts which makes the sentences in the set all true and the given sentence false. A set {ϕ 1, ϕ 2,, ϕ n } tautologically implies ψ there is no row in the truth table in which ϕ 1, ϕ 2,, ϕ n are all true and ψ is false. A set {ϕ 1, ϕ 2,, ϕ n } tautologically implies ψ ϕ 1, ϕ 2,, ϕ n are true, then ψ has to be true. def def if all TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 58 / 65
59 EXAMPLE 1. P Q 2. R Q 3. P R Q P Q R Q P Q R Q P R T T T F F F T T T F F F T T T F T T T T T T F F T T T T F T T F T F T F T F F T T F F F T T T T T F F F T T T F TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 59 / 65
60 EXAMPLE 1. P Q 2. R Q 3. P R Q P Q R Q P Q R Q P R T T T F F F T T T F F F T T T F T T T T T T F F T T T T F T T F T F T F T F F T T F F F T T T T T F F F T T T F TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 60 / 65
61 EXAMPLE 1. P Q 2. Q R R P P Q R P Q Q R R P T T T T T T T T F T F T T F T F T T T F F F T T F T T T T F F T F T F T F F T T T F F F F T T T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 61 / 65
62 EXAMPLE 1. P Q 2. Q R R P P Q R P Q Q R R P T T T T T T T T F T F T T F T F T T T F F F T T F T T T T F F T F T F T F F T T T F F F F T T T TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 62 / 65
63 EXAMPLE 1. P Q 2. P Q P Q P Q Q P Q P Q P Q T T F T F T T F T F T T F T F T F T F F T T F F IMPORTANT FACT If there is no row in which the premises are all true, then the argument is formally valid. TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 63 / 65
64 EXERCISES 1. P Q 2. Q Q P Q 1. P Q 2. Q P TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 64 / 65
65 EXERCISES 1. P Q 2. R Q 3. P R Q 1. (U V) X 2. V U 3. (X V) U V U TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 65 / 65
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