# Ling 98a: The Meaning of Negation (Week 1)

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1 Yimei Xiang 17 September What is negation? Negation in two-valued propositional logic Based on your understanding, select out the metaphors that best describe the meaning of negation. (1) a. Eraser b. Pressure-lowering drug c. Mirror d. Prison e. Protester f. No-parking sign In natural languages, negation can be very complex. Negation might be composed with a non-propositional constituent. Tautologies in the propositional logic are not always true in natural languages. It is difficult to identify which part of a negative sentence is actually denied.... 1

2 2 Why do we talk about negation? A brief review: propositional logic Connectives and their meanings Table 1: Connectives in propositional logic Connectives Compose proposition with connectives Translation negation p (the negation of p) it is not the case that p conjunction (p q) (conjunction of p and q) p and q disjunction (p q) (disjunction of p and q) p and/or q implication (p q) (implication of p and q) if p, then q equivalence (p q) (equivalence of p and q) p if and only if (iff.) q Syntax of propositional logic: Grouping: negation >> conjunction/disjunction >> implication/equivalence Semantics of propositional logic: Truth tables Table 2: Truth tables p q p p q p q p q p q Logical equivalence: V(p) = V(q) Tautologies, contradictions, contingencies Indirect reasoning (Deduction ad absurdum) 2

3 3 Understanding oppositions 3.1 Warm Up Give an answer that differs from both Mary s and John s. Make sure that your answer cannot be simultaneously true with either of their answers. Case 1: Mary: Edwin likes semantics. John: Edwin dislikes semantics. You: Case 2: Mary: It is raining John: It isn t raining You: 3.2 LNC and LEM in propositional logic Laws toward a formal definition of contradiction and contrariety: LNC (Law of Non-Contradiction): No statement can be true simultaneously with its negation. (2) (φ φ) LEM (Law of Excluded Middle): Every statement is either true or has a true negation. (3) φ φ Defining contrary and contradictory oppositions with laws: (4) a. Contrary opposition only needs to obey LNC: b. Contradictory opposition is governed by LNC as well as LEM: i. A corresponding affirmation and denial cannot both be true, by LNC; ii. But neither can they both be false, by LEM. Question: Consider, as for the negation in propositional logic, is it a contradictory operator, or a contrary one? 3

4 In-class Exercise 1: For signs in each of the following pictures, are they contrary or contradictory to each other? Why? Many things that are said to be contradictory in daily conversations are actually contrary in logic. It is very likely to find an alternative beyond the existing options, although such an alternative can be very... tough. 4

5 In-class Exercise 2: All the following formulas are tautologies in the two-valued propositional logic. The validity of which is debatable in natural languages? (5) a. (φ φ) b. φ φ (6) a. φ φ b. φ φ 3.3 Some tricky cases Case 3: Mary: Harry s daughter is smart. John: Harry s daughter is not smart. You: Case 4: Mary: Edwin likes mushroom. John: Edwin doesn t like mushroom. You: Case 5: Mary: Bob is an old man. John: Bob isn t an old man. Mary and John s mother: Bob is NOT an old man, he is your FAther! Some expressions which appear to be contradictories may in fact be contraries. Mary s and John s statements can be simultaneously denied when: Case 3: The subject fails to refer. (Presupposition failure) Case 4: Negation is read as a contrary operator which takes a narrow scope. (i) It is not the case that Edwin likes mushroom. (Mushroom is not a kind of food that Edwin likes.) (ii) Edwin does not-like mushroom. (Mushroom is a kind a food that Edwin dislikes.) Case 5: Negation applies to the speech-act: metalinguistic negation 5

6 Case 6: Mary: John believes his daughter will pick him up. John: John doesn t believe his daughter will pick him up. You: 4 Term logic of Aristotle 4.1 Basic concepts A proposition consists of two terms, a subject (S) and a predicate (P). A given P can be either affirmed or denied of a given S. There are two negations, and neither of them is propositional. Predicate denial (contradictory negation): S is not P It s a mode of predication, a way of combining subjects with predicates. E.g. Socrates is not ill. (i.e., Socrates [is not] ill) (Predicate) term negation (contrary negation): S is not-p It only says the predicate term is negative. E.g. Socrates is not-ill. (i.e., Socrates is [not ill]) When the given S doesn t actually exist (e.g. Case 3), Both S is P and S is not-p are False (since nothing positive or negative can be truly affirmed of a non-existent subject). Both S is not P and S is not not-p are True. E.g. since there doesn t exist a king of the United States in the actual world, in Aristotle s term logic, (6a-b) are False, and (6c-d) are True: (7) a. The king of the United States is bald. b. The kind of the United States is not-bald. c. The king of the United States is not bald. d. The kind of the United States is not not-bald. 4.2 Square of oppositions (AEIO) Three types of oppositions: (8) for two propositions/statements p and q, they are a. Contradictories iff they cannot both be true and cannot both be false. b. Contraries iff they can both be false, but cannot both be true. c. Subcontraries iff they can both be true, but cannot both be false. 6

7 Appendix: square of opposition for quantificational items ( More about oppositions in term logic ( Homework 1: Write a short paragraph to describe how the term logic of Aristotle addresses the following two issues: (i) the issues of non-existent subject; (ii) the issue of scope ambiguity. 5 Presuppositions, multi-valued logic and the A-operator Negations in the term logic of Aristotle are not propositional. If we follow the classic view and assume that negation is a propositional operator, how to address the tricky cases above? 5.1 What is a presupposition? The given subjects in (8) triggers an existential inference. In normal conversations, this existential inference is taken for granted (namely, part of the background). (9) a. John s daughter will come. b. John s daughter won t come. Existential inference (John has a daughter.) a presupposition Possessive phrase (John s daughter) a presupposition trigger A presupposition of S is a condition that must be met for S to be true or false. (10) S p is defined iff. V (p) = 1 Other presupposition triggers (following a traditional view): Verbs: stop, regret, know, discover, etc. Adverbials: again, too, etc. 7

8 Definite determiners: the Contrastive focus Cleft sentences Gender features (11) a. John has stopped drinking wine for breakfast. b. John used to drink wine for breakfast. (12) a. Mary bakes cookies again. b. Mary has baked cookies before. (13) a. The student is smart. b. There is an unique student in the context. (14) a. JOHN broke the computer. b. Someone broke the computer. (15) a. It was John who broke the computer. b. Someone broke the computer. (16) a. She is cleaver! b. The person pointed at is a female. 5.2 Presupposition projection If φ presupposes p, the presupposition p is inherited by φ, if φ, then ψ, perhaps φ and φ?. (17) a. John s daughter is coming. b. John s daughter is not coming. c. If John s daughter is coming, then we will have a party tonight. d. Perhaps John s daughter is coming. e. Is John s daughter coming? Presupposition: John has a daughter. In those cases, presupposition is projected. the most reliable diagnostic! In-class Exercise 3: Identify whether the presupposition of φ is projected in each of the following complex sentences or not. (φ: John s daughter is coming.) (18) a. If the train arrives on time, then John s daughter is coming. b. If John has a daughter, then his daughter is coming. c. Either John s daughter is coming, or John doesn t have a daughter. 8

9 In-class Exercise 4: It has been argued that John believes φ has an inference that John is opinionated about φ. Identify whether this inference is projected in each of the following sentences. (19) a. Bill doesn t believe that Sue is here. b. If Bill thinks that Sue is here, he will come. c. Perhaps Bill thinks that Sue is here. d. Does Bill think that Sue is here? Distinguish presuppositions from entailments and implicatures: (20) a. Lee kissed Jenny. b. Lee touched Jenny. (21) a. Mary has one child. b. Mary has exactly one child. (22) a. Girl: I m sorry. Let s break up. b. Boy: Who is he? For two statements p and q, p entails q (if p is true, then q is true). p q p presupposes q (q is backgrounded and taken for granted by p) p q p conventionally or conversationally implicates q (q follows from the interaction of the truth conditions of p together with either linguistic conventions on the proper use of p or general principles of conversational exchange). In-class Exercise 5: In each of the following examples, the a sentence presupposes and/or entail the other sentences. Specify which a sentence is a presupposition, which is a simple entailment, and which is both an entailment and a presupposition. (23) a. That John was assaulted scared Mary. b. Mary is animate. c. John was assaulted. d. That John was assaulted caused fear in Mary. (24) a. That John was assaulted didn t scare Mary. b. Mary is animate. c. John was assaulted. d. That John was assaulted didn t cause fear in Mary. 9

10 Homework 2: The picture below is used as an illustration of presupposition in the following web page: ( html) Consider, is it proper to regard the conversation as an illustration of presupposition? (i.e. should we consider vote for and vote again as presupposition triggers?) Why or why not? (Hint: This question is very tricky. You need to consider both the definition of presupposition and relevant diagnostics.) (20) A: Are you going to vote for or against the meeting? B: (What makes you think I m going to the meeting?) 5.3 Presupposition accommodation A presupposition of a sentence must normally be part of the common ground of the utterance context (the shared knowledge of the interlocutors) in order for the sentence to be felicitous. This process of an addressee assuming that a presupposition is true (even in the absence of explicit information that it is), is called presupposition accommodation. If the presupposition is not properly accommodated (namely, the truth of the presupposition is not satisfied in the CG), then we say there is a presupposition failure. (25) a. # John s daughter is coming, and John doesn t have a daughter. b. # If John s daughter is coming, then we will have a party tonight. Although John doesn t have a daughter. A pragmatic approach (Heim 1983, a.o.) A semantic approach (three-valued logic): If one of a sentence s presuppositions is not True, then the sentence is neither True nor False, but has a third truth value #. (Strawson) (26) p is a presupposition of φ (viz. φ p ) iff whenever p is not True, φ is #. φ p p 1 1 or 0 0 # Appendix: representative three-valued logical systems (Gamut 1 section 5.5.2) 10

11 5.4 Presuppositions and negation Presuppositions project under negation φ p φ p # # However, in the following sentence, the presuppositions from the negative clause isn t true. Why this sentence is still felicitous? (27) John s daughter is not coming, since John doesn t have a daughter. - Local accommodation - Negation applies to an operator (assertion operator A) that has a meaning akin to it is the case that/ it is true that.... (28) a. It is not the case that John s daughter is coming, since John doesn t have a daughter. b. It is not the case that John has a daughter and his daughter is coming, since he doesn t have a daughter. φ p Aφ p Aφ p # 0 1 Some characteristic properties of A (Beaver and Krahmer 2001): (29) a. If φ presupposes p, then Aφ p is equivalent with Aφ Ap. b. Aφ is equivalent with φ, if φ is defined. 11

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