Semantics Induction and Abduction Assignment

Part 1. Induction and Abduction [ 10 pts ] In class we discussed valid arguments, arguments that involve logical reasoning patterns (also called deductive inferences). An argument is valid if and only if the following entailment relation holds between its premises and its conclusion: if the premises of an argument are all true, then its conclusion is necessarily true. And we say that the premises entail the conclusion; or simply that the argument is valid. Deductive inferences are not the only kinds of inferences human beings make. For instance, the reasoning pattern in (1) is called abduction and is defined as inference to the best explanation. (1) If it rains, the streets are wet some time afterwards The streets are wet It rained some time before EXPLAIN why abduction is not valid, why abduction is often used in our daily life, even though it is not valid, and how the argument in (1) is related to the inference in (2). (2) The streets are wet It must have rained The reasoning pattern in (3) is called induction and is defined as inference to the whole class (all the carrots) based on evidence from part of the class (1001 carrots). (3) I have dug up 1001 carrots Everyone of the 1001 carrots is orange Therefore, all the carrots are orange EXPLAIN why induction is not valid (the argument in (4) should help) and why induction is often used in our daily life, even though the argument is not valid. (4) I have dug up 1001 carrots Everyone of the 1001 carrots is orange The 1002 carrot I dug up was green Therefore, all the carrots are orange 1 Part 2. Valid Arguments [ 40 pts ] In class, I showed you how to verify whether an argument is valid or not. What I said was this. First translate the arguments into Propositional Logic. Then set up a truth-table that includes each of the premises and the conclusion. Cross out each row which has a false premise (since validity assumes that all the premises are true). Look at the value of the conclusion in the remaining rows: it must be true in these rows. This captures the requirement that whenever the premises are true, so is the conclusion. If there is a row in which the premises are true and the conclusion is false, the argument is not valid. SHOW that each argument in (5) is valid. To do this, you must first translate these arguments into Propositional Logic. You can use truth-tables or the valuation function to do this, but you must show your work. [I have omitted the word therefore throughout.] (5) a. Either the butler or the gardener murdered the countess The butler did not murder the countess The gardener murdered the countess b. If the butler murdered the countess, he must feel guilty now If the butler feels guilty, he will go into hiding If the butler murdered the countess, he will go into hiding c. If it is snowing, then it is cold If it is cold, then it is snowing It is cold just in case it is snowing SHOW that each argument in (6) is not valid. (6) a. If LA is in Canada, then LA is in North America LA is in North America LA is in Canada b. If John or Mary is not guilty, Bob is guilty Bob is not guilty John is not guilty On the last page of the assignment, I have included an alternative way of determining whether an argument is valid or not. This method is more direct but depends on points I did not discuss in class. In fact, you have enough information to show that this alternative method works but the proof is not obvious. If you understand this method, use it; otherwise, stick to what I did in class. [ Translation 3pts + Truth-Table 3pts + Explanation of Truth-Table 2pts = 8pts each ] 2 Part 3. Entailments and Presuppositions [ 40 pts ] Entailments and presuppositions are both implicative relations between propositions. However, they are very different in nature. Entailment is a logical relation in which the truth of one or more propositions guarantees the truth of another proposition. Presupposition is a relation in which one proposition takes for granted the truth of another proposition (or that it is uncontroversial). This exercise requires you to distinguish between entailments and presuppositions in natural language. You will not need to translate anything; instead, simply consider the meaning of each sentence, i.e., what it says about the outside world. There is a relation of presupposition and/or entailment between the pairs of sentences in (9)-(13). (9) a. That John was assaulted scared Mary b. John was assaulted (10) a. Is John not aware that Mary is pregnant? b. Mary is pregnant (11) a. Some Italian is a violinist b. Bernardo is an Italian violinist (12) a. If I discover that Mary is in New York, I will get angry b. Mary is in New York (13) a. It is possible that Clinton will return to Arkansas b. Clinton was in Arkansas before DETERMINE whether an entailment and/or a presupposition relation holds between each pair of sentences in (9) - (13); that is, determine whether or not (a) ⊨ (b) and whether or not (a) >> (b). JUSTIFY each answer you give with a brief explanation. In your explanation, you need to assess whether or not the entailment relation holds and whether or not (b) has the requisite properties of a presupposition, as defined in class. [ Relations 1 pt x 2 + Justification 2pt x 2 = 6pts / pair ] DETERMINE the presuppositions in each sentence in (14)? [There may be more than one.] (14) a. b. c. d. e. The burglar realized that he had been filmed on closed circuit television John forgot to do the washing Doesn’t Hermione like Ron John is off cigarettes again Jane never remarried [ 2 pts / sentence] 3 Part 4. Conversational Implicatures [ 10 pts ] DETERMINE the conversational implicature in each utterance and the Maxim(s) inducing it. (15) a. A: “Can you lend me your bike?” B: “It’s in the garage” b. “John owns three cows” c. “John’s a machine” Children with pragmatic disorders often fail to observe the Gricean maxims. In the conversation below (adapted from Bishop 1997:183), the child violates a maxim. DETERMINE which one and EXPLAIN your reasoning. (16) Speech therapist: So you like ice-cream. What are your favourite flavours? Child [with pragmatic disorder]: Hamburger … fish and chips. 4 AN ALTERNATIVE METHOD TO DETERMINE VALIDITY There is a more direct method to show that an argument is valid. Suppose you want to show that the argument in (17) is valid. The first thing to do is form the proposition in (18).1 (17) Argument (18) ((ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) → ψ) ϕ1 ϕ2 … ϕn ψ The argument in (17) is valid iff the proposition in (18) is always true. (19) Example If it is snowing, then it is cold it is snowing It is cold (p → q) p q If the proposition (((p → q) ∧ p) → q) is always true, the argument is valid. [You can verify this to be sure.] Why does it work? Let me try to convey the sense behind this method. Suppose the proposition ((ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) → ψ) is always true. Then either (i) the propositions (ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) and ψ are both true or else (ii) the proposition (ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) is false. To show that an argument is valid, we only care about instances when the individual premises ϕ1, ϕ2, …, ϕn are all assumed to be true. So we can ignore (ii) because (ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) is false when at least one premise is false. That leaves (i), in which the propositions (ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) and ψ are both true. Now the proposition (ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) is true just in case all the premises are true. So if the proposition ((ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) → ψ) is true (as we originally supposed), it follows that whenever the individual premises ϕ1, ϕ2, …, ϕn are true so is the conclusion ψ, i.e., the argument is valid. It can also be shown that if the argument is valid, the proposition ((ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) → ψ) is always true. If you have trouble with this demonstration, don’t worry. It goes beyond this course and I included it for completeness. 1 You can form the proposition in (18) by conjoining the first two premises (ϕ1 ∧ ϕ2), then conjoining them with the third premise ((ϕ1 ∧ ϕ2) ∧ ϕ3), and then the fourth, and so on until the premises are exhausted. In fact, because of the nature of conjunction, as long as all the premises are conjoined, it does not matter which order you do it in. So the parentheses are not necessary. The result is (ϕ1 ∧ ϕ2 ∧ … ∧ ϕn). Next combine (ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) and ψ using the implication →. The result is the proposition ((ϕ1 ∧ ϕ2 ∧ … ∧ ϕn) → ψ). 5