INTRODUCTION. Tomoya Sato. Department of Philosophy University of California, San Diego. Phil120: Symbolic Logic Summer 2014

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1 INTRODUCTION Tomoya Sato Department of Philosophy University of California, San Diego Phil120: Symbolic Logic Summer 2014 TOMOYA SATO LECTURE 1: INTRODUCTION 1 / 51

2 WHAT IS LOGIC? LOGIC Logic is the study of formal validity. TOMOYA SATO LECTURE 1: INTRODUCTION 2 / 51

3 WHAT IS LOGIC? LOGIC Logic is the study of formal validity. TOMOYA SATO LECTURE 1: INTRODUCTION 3 / 51

4 WHAT IS VALIDITY? VALIDITY AND ARGUMENT Validity is a property of arguments. An argument is a set of sentences consisting of two parts: (1) the premises; (2) the conclusion. 1. Socrates is a human being. 2. All human beings are mortal. Socrates is mortal. TOMOYA SATO LECTURE 1: INTRODUCTION 4 / 51

5 EXAMPLES OF ARGUMENT 1. Tarski is not a fish. 2. Tarski is either a primate or a fish. Tarski is a primate. 1. x 2 = x < 0. x = 1. TOMOYA SATO LECTURE 1: INTRODUCTION 5 / 51

6 WHAT IS VALIDITY? DEFINITION: VALIDITY def An argument is valid if the premises of the argument are/were true, then the conclusion must be true. def An argument is valid it is impossible that all the premises are true and the conclusion is false. 1. if true 2. if true.. must be true. TOMOYA SATO LECTURE 1: INTRODUCTION 6 / 51

7 EXAMPLES 1. Socrates is a human being. 2. All human beings are mortal. Socrates is mortal. 1. If Tomoya is Japanese, then he can speak Japanese. 2. Tomoya is Japanese. Tomoya can speak Japanese. TOMOYA SATO LECTURE 1: INTRODUCTION 7 / 51

8 EXAMPLES 1. Tomoya is Japanese. Tomoya likes sushi. 1. If Tomoya is French, then he can speak French. 2. Tomoya is French. Tomoya can speak French. TOMOYA SATO LECTURE 1: INTRODUCTION 8 / 51

9 EXAMPLES 1. If Tomoya is Japanese, he can speak Japanese. 2. Tomoya can speak Japanese. Tomoya is Japanese. IMPORTANT POINTS Logic is concerned with validity. Validity is a property of arguments, in particular, a property of the relationship between the premises and conclusion. Actual truth values of the premises and conclusion are irrelevant to whether the argument is valid or invalid. TOMOYA SATO LECTURE 1: INTRODUCTION 9 / 51

10 THE FORMALITY OF LOGIC LOGIC Logic is the study of formal validity. What does "formality" mean? TOMOYA SATO LECTURE 1: INTRODUCTION 10 / 51

11 THE FORMALITY OF LOGIC 1. If Tomoya is Japanese, then Tomoya can speak Japanese. 2. Tomoya is Japanese. Tomoya can speak Japanese. 1. If Tomoya είναι Iαπωνικά, then Tomoya µπoρoύν να µιλoύν ιαπωνικά. 2. Tomoya είναι Iαπωνικά. Tomoya µπoρoύν να µιλoύν ιαπωνικά. TOMOYA SATO LECTURE 1: INTRODUCTION 11 / 51

12 THE FORMALITY OF LOGIC 1. If Tomoya είναι Iαπωνικά, then Tomoya µπoρoύν να µιλoύν ιαπωνικά. 2. Tomoya είναι Iαπωνικά. Tomoya µπoρoύν να µιλoύν ιαπωνικά. WHAT MAKES THE ARGUMENT VALID The function of "If, then." The orange parts are identical. The green parts are identical. The argument is valid due to its logical form. TOMOYA SATO LECTURE 1: INTRODUCTION 12 / 51

13 THE FORMALITY OF LOGIC DEFINITION: FORMAL VALIDITY An argument is formally valid form. An argument is formally valid exactly the same form is valid. def def it is valid due to its logical every argument with THE FORMALITY OF LOGIC Logic is concerned with the logical form of an argument. Logic is not concerned with the content of an argument. TOMOYA SATO LECTURE 1: INTRODUCTION 13 / 51

14 VALIDITY AND FORMAL VALIDITY Arguments Valid Arguments Formally Valid Arguments TOMOYA SATO LECTURE 1: INTRODUCTION 14 / 51

15 COURSE CONTENTS What arguments are formally valid? TWO TASKS Identify the logical form of an argument. Determine whether or not the logical form produces the validity. WHAT WE WILL LEARN 1 How to translate arguments in English into arguments in symbolic language. 2 How to determine whether or not the logical form produces the validity. The semantic method (Truth) The proof-theoretic method (Derivation) TOMOYA SATO LECTURE 1: INTRODUCTION 15 / 51

16 COURSE GOALS GOAL 1 To be able to determine the formal validity or invalidity of a given argument using the semantical method and the proof-theoretic method. 1. Any giraffe that is taller than Harriet is taller than every zebra. 2. Some giraffes aren t taller than some zebras. There is a giraffe that is not taller than Harriet. GOAL 2 To acquire the knowledge and skills that are necessary for further investigations on/with logic. TOMOYA SATO LECTURE 1: INTRODUCTION 16 / 51

17 WHERE ARE WE? Linguistics Advanced Logic Computer Science Philosophy Mathematics First-Order Logic Sentential Logic TOMOYA SATO LECTURE 1: INTRODUCTION 17 / 51

18 ORGANIZATION OF THE COURSE COURSE WEBSITE = "Teaching" Syllabus Lecture Slides Link to the course textbook REQUIRED TEXTBOOK Terence Parsons, An Introduction to Symbolic Logic. Available online TOMOYA SATO LECTURE 1: INTRODUCTION 18 / 51

19 ORGANIZATION OF THE COURSE TOMOYA SATO H&SS 8089 Monday 2:00-4:00. You are responsible for reading s from me. TOMOYA SATO LECTURE 1: INTRODUCTION 19 / 51

20 ORGANIZATION OF THE COURSE COURSE REQUIREMENTS AND EVALUATION 1 Homework (40%, 400 pts): Four homework assignments. Due at the beginning of the Monday lecture of each weak. For each day the assignment is late, 20 points will be deducted from the point total. 2 Midterm Exam (20%, 200 pts): On August 18, 11:00am 11:50pm, in YORK 3050A. 3 Final Exam (40%, 400 pts): On September 5, 11:30am 2:30pm, in (probably) YORK3050A : A, A, or A : B, B, or B : C, C, or C : D 0-599: F. TOMOYA SATO LECTURE 1: INTRODUCTION 20 / 51

21 ADDITIONAL RULES REGARDING MAKE-UP EXAMS MAKE-UP EXAMS Make-up exams will only be given in special circumstances (e.g., serious illness, family emergency, or participation in a university activity). ACADEMIC INTEGRITY You are responsible for knowing and adhering to the policy on academic integrity, which can be found at TOMOYA SATO LECTURE 1: INTRODUCTION 21 / 51

22 CHAPTER 1, SECTION 1 What arguments are formally valid? HOW TO DETERMINE FORMAL VALIDITY 1 Identify the logical form of an argument. 2 Apply the semantic method or the proof-theoretic method. For our purposes, symbolic language is useful. TOMOYA SATO LECTURE 1: INTRODUCTION 22 / 51

23 EXAMPLES 1. If Tomoya is Japanese, then Tomoya can speak Japanese. 2. Tomoya is Japanese. Tomoya can speak Japanese. 1. If the sidewalks are wet, then either it rained recently or the sprinklers are on. 2. The sidewalks are wet. Either it rained recently or the sprinklers are on. TOMOYA SATO LECTURE 1: INTRODUCTION 23 / 51

24 EXAMPLES 1. If Tomoya is Japanese, then Tomoya can speak Japanese. 2. Tomoya is Japanese. Tomoya can speak Japanese. 1. If the sidewalks are wet, then either it rained recently or the sprinklers are on. 2. The sidewalks are wet. Either it rained recently or the sprinklers are on. TOMOYA SATO LECTURE 1: INTRODUCTION 24 / 51

25 EXAMPLES 1. If P, then Q. 2. P. Q. 1. P Q 2. P Q TOMOYA SATO LECTURE 1: INTRODUCTION 25 / 51

26 SYMBOLIC LANGUAGE TWO SYMBOLIC LANGUAGES 1 The symbolic language of sentential logic (Chapters 1 and 2) 2 The symbolic language of first-order logic (Chapters 3 and 4) SYMBOLS OF THE SYMBOLIC LANGUAGE OF SENTENTIAL LOGIC Atomic sentences: P, Q, R,... Z. Logical connectives:,,,, SYMBOLS OF THE SYMBOLIC LANGUAGE OF CHAPTER 1 Atomic sentences: P, Q, R,... Z. Logical connectives:, TOMOYA SATO LECTURE 1: INTRODUCTION 26 / 51

27 SYMBOLIC LANGUAGE SYMBOLIC SENTENCES IN THE 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. EXAMPLES P, P, (P Q), (P Q), ( (P Q) R). P, P P, PQ, (P Q), P Q. TOMOYA SATO LECTURE 1: INTRODUCTION 27 / 51

28 SYMBOLIC LANGUAGE RULES OF SYMBOLIC SENTENCES You can use ( ), [ ], and { }; You can omit the outermost parentheses ( ) EXAMPLES {P Q}, [(P Q) R]. P Q, P Q. EXERCISES 1 P P 2 (Q R) 3 (P Q) (R Q) 4 (P {Q R} Q) 5 ( S R) [( R S) ( S R)] TOMOYA SATO LECTURE 1: INTRODUCTION 28 / 51

29 CHAPTER 1, SECTION 2 THE MEANINGS OF SYMBOLS P, Q, R,..., Z represent atomic sentences in English. " " corresponds to "It is not the case that", "not", "n t". " " corresponds to "If - then", "only if", "given that", etc. ϕ T F ϕ F T ϕ ψ ϕ ψ T T T T F F F T T F F T TOMOYA SATO LECTURE 1: INTRODUCTION 29 / 51

30 SYMBOLIZATION SYMBOLIZATION This salad is not tasty. S : This salad is tasty. S SYMBOLIZATION It is not the case that this salad is not tasty. S : This salad is tasty. S TOMOYA SATO LECTURE 1: INTRODUCTION 30 / 51

31 SYMBOLIZATION SYMBOLIZATION If it rains, the game will be called off. P : It rains. Q : The game will be called off. P Q SYMBOLIZATION If it doesn t rain, the game will not be called off. P : It rains. Q : The game will be called off. P Q TOMOYA SATO LECTURE 1: INTRODUCTION 31 / 51

32 ONLY IF The game will be called off if it rains. The game will be called off only if it rains. "if it rains" "only if it rains" TOMOYA SATO LECTURE 1: INTRODUCTION 32 / 51

33 "IF" AND "ONLY IF" IF If P, then Q. Q if P. P Q. ONLY IF Only if P, Q. Q only if P. Q P. TOMOYA SATO LECTURE 1: INTRODUCTION 33 / 51

34 EXERCISES SCHEME OF ABBREVIATION R : It rains. S : Susan will be late. SYMBOLIZATION 1 Susan will be late only if it rains. 2 Susan will not be late if it doesn t rain. 3 Only if it rains will Susan be late. 4 It doesn t rain only if Susan will not be late. TOMOYA SATO LECTURE 1: INTRODUCTION 34 / 51

35 CHAPTER 1, SECTION 3 AMBIGUOUS SENTENCE 1 It is not the case that Tomoya will leave if Andrew does. It is not the case that Tomoya will leave if Andrew does. A T It is not the case that Tomoya will leave if Andrew does. (A T) AMBIGUOUS SENTENCE 2 If Adam leaves then Bob stays if Carol sings. If Adam leaves then Bob stays if Carol sings. A (C B) If Adam leaves then Bob stays if Carol sings. C (A B) TOMOYA SATO LECTURE 1: INTRODUCTION 35 / 51

36 COMMAS A comma signals a major break in a sentence. EXAMPLES It is not the case that Tomoya will leave if Andrew does. It is not the case that Tomoya will leave, if Andrew does. A T If Adam leaves then Bob stays if Carol sings. If Adam leaves, then Bob stays if Carol sings. A (C B) TOMOYA SATO LECTURE 1: INTRODUCTION 36 / 51

37 EXERCISES SCHEME OF ABBREVIATION V = Veronica will leave. W = William will leave. Y = Yolanda will leave. EXERCISES If Veronica doesn t leave William won t either. William will leave if Yolanda does, only if Veronica doesn t. If Yolanda doesn t leave, then Veronica will leave only if William doesn t. TOMOYA SATO LECTURE 1: INTRODUCTION 37 / 51

38 WHERE ARE WE? The semantic method The proof-theoretic method Symbolization Formal Validity TOMOYA SATO LECTURE 1: INTRODUCTION 38 / 51

39 CHAPTER 1, SECTION 4 THE PROOF-THEORETIC METHOD A method to confirm the validity of a symbolic argument by means of proofs (derivations). 1. P (W T) 2. P T 3. T W TOMOYA SATO LECTURE 1: INTRODUCTION 39 / 51

40 DERIVATION Show W P (W T) P T T P P W T W pr pr pr 3 4 mt 5 dn 2 6 mp 4 7 mt dd 1. P (W T) 2. P T 3. T W TOMOYA SATO LECTURE 1: INTRODUCTION 40 / 51

41 DERIVATION Show W P (W T) P T T P P W T W pr pr pr 3 4 mt 5 dn 2 6 mp 4 7 mt dd DEFINITION: DERIVATION A derivation of an argument is a finite sequence of sentences such that: 1 The first line is "Show (the conclusion of the argument)"; 2 Each sentence is either a premise of the argument or a sentence that can be derived from earlier sentences by an inference rule; 3 Each sentence is followed by a justification. TOMOYA SATO LECTURE 1: INTRODUCTION 41 / 51

42 DERIVATION 1. P 2. P Q 3. Q R R 1. Show R 2. P pr 3. P Q pr 4. Q R pr 5. Q 2 3 mp 6. R 4 5 mp dd TOMOYA SATO LECTURE 1: INTRODUCTION 42 / 51

43 INFERENCE RULES 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 1: INTRODUCTION 43 / 51

44 INFERENCE RULES Double Negation ϕ ϕ ϕ ϕ Repetition ϕ ϕ TOMOYA SATO LECTURE 1: INTRODUCTION 44 / 51

45 CHAPTER 1, SECTION 5 1. P (W T) 2. P T 3. T W 1. Show W 2. P (W T) pr 3. P T pr 4. T pr 5. P 3 4 mt 6. W T 2 5 mp 7. W 4 6 mt 8. W 7 dn dd TOMOYA SATO LECTURE 1: INTRODUCTION 45 / 51

46 RULES OF DERIVATION A show line consists of the word "Show" followed by a sentence. The first step of producing a derivation must be to introduce a show line. Show lines are not given a justification. At any step, any sentence from the set of premises may be introduced, justified with the notation "pr". At any step a line may be introduced if it follows by an inference rule from previous lines in the derivation; it is justified by citing the numbers of those previous lines and the name of the rule. If a line is introduced whose sentence is the same as the sentence in the show line, one may, as the next step, write "dd" at the end of that line, draw a line through the word "Show", and draw a box around all the lines below the show line, including the current line. TOMOYA SATO LECTURE 1: INTRODUCTION 46 / 51

47 DERIVATION AND FORMAL VALIDITY THEOREM If there is a complete derivation of an argument, then the argument is formally valid. TOMOYA SATO LECTURE 1: INTRODUCTION 47 / 51

48 PHIL120 AND PHIL10 The inference rules in Phil120 are different from those in Phil10. The names of some inference rules are different. Phil120 is more rigorous in the application of inference rules than Phil10. DOUBLE NEGATION P Q P Q This is not legitimate! Double Negation and all other inference rules too can be applied to whole sentences, but not to parts of sentences. TOMOYA SATO LECTURE 1: INTRODUCTION 48 / 51

49 PHIL120 AND PHIL10 MODUS TOLLENS 1 1. P Q 2. Q 3. P 1 2 mt 1. P Q 2. Q 3. Q 2 dn 4. P 1 3 mt This is not legitimate! TOMOYA SATO LECTURE 1: INTRODUCTION 49 / 51

50 PHIL120 AND PHIL10 MODUS TOLLENS 2 1. P Q 2. Q 3. P 1 2 mt 1. P Q 2. Q 3. P 1 2 mt 4. P 3 dn This is not legitimate too! TOMOYA SATO LECTURE 1: INTRODUCTION 50 / 51

51 EXERCISES 1. P 2. Q P 3. R Q R 1. Q S 2. V X 3. V S 4. X Q 1. (W Z) (Z W) 2. (Z W) X 3. P X 4. P (W Z) TOMOYA SATO LECTURE 1: INTRODUCTION 51 / 51