564 Lecture 3 Aug. 31 1999

1. The notion of lexical synonymy entails that the sense of a word be identical to the sense of another word. "Sense", though, should have enough content to identify items in the world as belonging to the set denoted by the word or not; if it doesn’t, we can’t say that we really know the meaning of the word. How can we know that "elm" and "maple" are not synonymous when we don’t know enough of the sense of the words "elm" and "maple" to give us a reliable identification of a given tree as belonging to the set of elms or the set of maples?

We can think of the "sense" of an expression as basically an algorithm for determining set-membership, for a given item in the real world. Knowing that an expert exists who can distinguish between elms and maples is essentially giving yourself an algorithm for determining set-membership: elms are a set of trees; to determine set-membership, consult a tree expert; maples are another set of trees; to determine set-membership, consult a tree expert. That way we can know that the sets of elms and maples are not overlapping, and how to tell them apart, without knowing their exact features ourselves.

2. (iii) could be either syntactically ill-formed or semantically ill-formed. If Sara is asking Jimmy to kiss Sara, then the sentence is syntactically ill-formed, since the reflexive is free when it is required to be bound; the pronoun her should have been employed instead. If Sara is asking Jimmy to kiss Jimmy, then it’s semantically ill-formed, because gender is semantic (i.e. natural, corresponding to the sex of the referent) in English and "herself" is not compatible with the male name "Jimmy"; it should be "himself". (Gender in other languages is syntactic, not corresponding to natural gender; in such a case, it could be argued that even a gender mismatch is syntactic, not semantic).

3. This is a fairly debatable problem; different people may have different contexts in which the sense of the given expressions differ. I’ll give some that seem plausible to me, but they’re all fairly subtle.

a) big/large. "Large" is often a size gradation in, e.g., soft drink or coffee cups. Oddly enough, it’s often the smallest size available, contrasting with, e.g. "extra-large" and "jumbo" or "tall" and "grande" or whatever. If the ‘large’ cup is the smallest available, it certainly isn’t "big" — the "big" drink would be the biggest available.

b) fragrance/smell. "Smell" can be employed to denote any odor, pleasant or unpleasant. "Fragrance" has a pleasant connotation. Coming into a room that smelled of rotten vegetables, saying "what’s that smell?" would be appropriate, while "What’s that fragrance?" would not.

c) unmarried / not married. "Unmarried" for me has a connotation of "never been married", while "not married" is more neutral. If Susan is divorced, she’s certainly not married, but she might not be appropriately described as "unmarried".

4. a) He fed her dog biscuits. Ambiguity is in whether "dog" is the head of an NP with possessive "her" (He fed [her dog] biscuits) or an adjective describing biscuits (He fed her [dog biscuits]). This is syntactic ambiguity.

b) I’d like you to take in the dress. In both readings, "take in" is a verb-particle construction, and the syntactic structure is identical. However, in one reading it means "shorten" or "tailor to be smaller" when in the other it means "bring inside from outside (e.g. on the washing line"). This ambiguity is in the semantic entry for "take in".

c) Susan didn’t buy two books. Ambiguous because it could denote a situation in which Susan bought all the books available except for two, or a situation in which Susan bought one or three or any number of books other than two (leaving behind any number of books). This is a scope ambiguity, and we said last time that this type of ambiguity was semantic, not structural (although if you’re familiar with a certain kind of LF treatment of scope ambiguities, you’re forgiven if you said it was structural).

d) Jane told me that she was very unhappy. Ambiguous because "she" could refer to "Jane" or to some other arbitrary person introduced by the discourse. This ambiguity is a combination of syntactic and pragmatic; if it refers to Jane, it’s a syntactic relationship (binding) but if to some other person, it’s pragmatics.

5. i) a) That Felicia was assaulted scared Mary.

b) Felicia was assaulted.

b) is both presupposed and entailed by a). First consider entailment. If a) is true, then b) must be true, right? If Felicia wasn’t assaulted, then a) could not be true. Now presupposition: Consider placing it in an embedded context: "Did the fact that Felicia was assaulted scare Mary?" Or "I regret that the fact that Felicia was assaulted scared Mary". The presupposition seems to hold even in the embedded context, indicating that b) is also a true presupposition of a) (something you have to imagine in order for the a) statement to make sense). Note, of course, that in the embedded context, one can cancel the presupposition (one could answer, "No, it didn’t because Felicia wasn’t assaulted!"). The entailment doesn’t hold in the embedded context (The truth of "I regret that the fact that J. was assaulted scared Mary" does not entail that "Felicia was assaulted" — rather, it entails that I believe that Felicia was assaulted, and that it scared Mary.)

ii) a) Is Felicia not aware that Mary is pregnant?

b) Mary is pregnant.

b) is a presupposition of a), but not an entailment of it. One would usually assume, from the statement a), that b) was true (because a) wouldn’t be too felicitous if b) wasn’t true, or at least if the speaker didn’t believe b) to be true). However, questions, in particular, don’t have truth values that we can ascertain just at the moment, so the idea that the "truth" of a) must entail the truth of b) simply doesn’t apply.

iii) a) Some Italian is a violinist

b) Bernardo is an Italian violinst.

In this case, b) entails a), but does not presuppose it (one need have no beliefs about whether or not any Italian is a violinist in order for b) to make sense. But if Bernardo is Italian, and Bernardo is a violinist, then some Italian must be a violinist, in particular, Bernardo).

iv) a) If I discover that Mary is in New York, I will get angry.

b) Mary is in New York.

Neither b) nor a) presupposes or entails the other.

v) a) It is possible that Clinton will return to Arkansas.

b) Clinton was in Arkansas before.

Tricky. a) certainly presupposes b) (try "Do you think it is possible that Clinton will return to Arkansas?"). Does a) entail b)? If a) is true, then it must be true that b) is true, I think. Otherwise it would not be possible for Clinton to return to Arkansas. Now, what about the next question — does b) entail a)? Discuss.

6. With respect to the bonus question. DeSwart gives "a) Hans is German. b) Hans is a linguist. c) Hans is a German linguist." as an example of entailment: a+b entails c, and c entails a and b. Further, she asserts, that a sentence like "Hans is a German linguist, but he is not German" is inherently contradictory. That would certainly be true if she were using a predicate other than "linguist", but it seems to me that linguists, like language teachers, can be identified as a type of linguist or teacher depending on the language that they work on. So, "Hans is a German teacher (=teacher of German), but he is not German" is perfectly fine; for me, "linguist" behaves this way as well. "Hans is a German skier, but he is not German" would have been a better example.

For our first forays into translating English expressions into a formal language, we’re going to limit ourselves to translating certain connectives, with which you’re all familiar. Essentially, in translating into a formal language, we’re going to start at the top (of a syntactic tree) and work our way down. These connectives have the property of linking together "complete thoughts", or propositions, normally expressed in English by sentences ("kernel sentences" if you remember your early Chomsky; "atomic propositions", to philosophers of language, logicians and semanticists these days).

Recall that we’ve already decided that the reference of a sentence (putting aside questions or imperatives etc. for the moment) is a truth value ("denotation" will for a while be used inconsistently, sometimes referring to a reference, sometimes to a sense). So you could say that the denotation of "Linda likes Sally" is true. The inference that "Either Linda likes Sally or the moon is made of green cheese" is true if "Linda likes Sally" is true is something that we can get from the syntax and semantics of propositional logic. Essentially, propositional logic takes basic sentential connectives, defines them, and enables us to reason about arguments made with atomic propositions.

For example, we know that:

If P, then Q.

P

Therefore Q

is a valid argument. That is, if the first statement is true, and the second statement is true, then the third statement must be true, independent of the content of the propositions. We want our formal language to capture this inference. Propositional logic does this job.

What is a well-formed formula (wff)?

I. Any atomic statement is a wff (p,q,r,s,...)

II. Any wff preceded by ~ (negation - it is not the case that ) is a wff.

III. Any two wffs can be made into another wff by writing the symbol "&" (conjunction - and), "v" (disjunction - or), "-->" (conditional - if-then), or "<-->" (biconditional - if and only if) between them and enclosing the result in parentheses.

Examples of wffs:

1. ~p

2. q'

3. (p v q)

4. ~(p' <-->p')

5. ~r

6. ~~r

7. ((((p&q)v ~q')-->r)<-->s)

Not wffs:

8. pq

9. vp

10. ~vpq

11. p v~q

12. ~(p)

The denotation of an atomic statement is either true or false, which are represented as 1 and 0 respectively.

Connectives: Negation

13. Negation reverses the truth value of the wff to which it is attached.

Truth tables show the necessary truth value output of an operation creating a new wff according to one of our syntactic rules above. The leftmost columns show the truth values of the atomic statement(s). The next column to the right shows the value of the wff created by combining the atomic statement(s) with a connective. For example, the truth table for negation is:

14.

This could be read as, "If the truth value of p is 1, the truth value of ~p is 0. If the truth value of p is 0, the truth value of ~p is 1."

Differences between logical negation and natural language sentential negation:

15. Given the atomic statement "Felicia is here", represented by p, is ~p the same as "Felicia is not here"? How about "It is not the case that Felicia is here."

16. Given the atomic statement "Felicia must leave" represented by q, is ~q the same is "Felicia must not leave"? How about "It is not the case that Felicia must leave?"

The conjunction connective gives a true wff iff both wffs that it conjoins are true; otherwise it gives a false wff.

17. Truth table for conjunction:

Differences between logical conjunction and natural language and.

18. Given "Felicia got dressed" = p and "Felicia took a shower" = q, is (p&q) the same as "Felicia got dressed and Felicia took a shower"? How about (q&p) and "Felicia took a shower and got dressed? Is "Felicia smokes and Felicia smokes" the same as p&p?

We've seen but; other connectives which propositional logic translates as & are however, although, despite the fact that, even though....

19. Note that in propositional logic we can't say "Felicia smokes and drinks" or "Felicia and Mary smoke". Can we treat them as reduced sentential conjunction? What about "Felicia and Mary met in New York"? "At most two children sang and danced"?

The disjunction connective gives a true wff if at least one of its disjuncts are true; it only gives a false wff if both disjuncts are false.

20. Truth table for disjunction:

The logical connective is "inclusive" or. English or is also supposed to have an "exclusive" sense, as in "You can go home or you can stay here" (clearly you can't do both); "All entrees are served with soup or salad".

21. (Truth table for exclusive disjunction "%":

)

deSwart’s take is that exclusive "or" arises as an implicature, but inclusive "or" is the one English uses truth-conditionally. "Sue did not talk to either Felicia or Jane" is the negation of inclusive, not exclusive or, since it’s not true if she did talk to both Felicia and Jane. (Do a truth table for both if you don’t believe it).Problematic natural language cases include examples like "You can get a notarization from a judge or a banker" (although we can't treat those yet – but imagine they are reduced sentential disjunction: "You can get a notarization from a judge or you can get one from a banker", where what you mean is that both judges and bankers can be notaries, not that one of the two might not be a notary).

(deSwart exercise for exclusive/inclusive or).

When two wffs are connected with the conditional --> the resulting wff is false if the antecedent (the first wff) is true and the consequent (the second wff) is false. The resulting wff is true in all other cases.

22. Truth table for conditional:

Big difference between logical conditional and natural language if-then: logical conditionals are true if the antecedent is false. In natural language, it seems more correct to say that a conditional has no truth value or cannot be evaluated when the conditional is false: "If Simin is here, then Andrew is also here" is true right now, when Simin is not here, if we're using the logical conditional.

Note! Watch out for the one-wayness of the conditional. Let's say P-->Q is true. Now, just because Q is always true if P is true doesn't mean that P is necessarily always true if Q is true! "If an animal is placental, then it is a mammal" is true, but "If an animal is a mammal then it is placental" is not.

"If P then Q" = "Q if P" = "P only if Q"

23. a. If an animal gives milk, then it is a mammal.

b. An animal is a mammal if it gives milk.

c. An animal gives milk only if it is a mammal.

24. a. If I can get the car fixed, then I leave tomorrow.

b. I leave tomorrow if I can get the car fixed.

c. I can get the car fixed only if I leave tomorrow. (Strange!)

Now, Pinker test:

All cards have a letter on one side and a number on the other. How do you test to see if the statement:

"If a card has a D on one side, then it has an 18 on the other." is true or not?

Cards showing:

D 16 18 W

Now, consider this: All humans are either legal drinkers or underage. In order for a human to be a legal drinker, she must be 18 years old. Four people are at a bar. Two of them are wearing tags with their age on them (16 and 18). The other two of them have signs on their drinks: one is gin, one is water. (All four are drinking drinks that look exactly the same). What do you have to do to test whether or not this crowd complies with the rule, If a human is a drinker, she is 18 years old?

Biconditional:

When two wffs are connected with the biconditional <--> the resulting wff is false if the antecedent (the first wff) is true and the consequent (the second wff) is false, and false if the consequent is false and the antecedet true. The resulting wff is true when both antecedent and consequent are true or both are false.

25. Truth table, biconditional

(This follows because the biconditional is simply a reduced version of (P-->Q)&(Q-->P). Let's be sure that the two truth tables come out the same, given the same values for P and Q:

(do truth table for (P-->Q)&(Q-->P):

1. Construct columns for atomic statements.

2. Construct columns for innermost bracketed wffs.

3. Construct columns for next embedding of brackets.

4. Continue until final expression is reached.

5. Include rows for all possible combinations of truth values of the atomic statements, in general, 2n, where there are n atomic statements.

)

27. If a statement is always true, simply by virtue of the connectives it has in it, it is a tautology.

For instance, (Pv P). Or, less trivially, (P-->(Q-->P)). (do truth table).

Obvious tautologies are not often used in natural language ("War is war" "A rose is a rose is a rose") except for some special effect. Complex tautologies have the force of arguments, as we will see.

28. If a statement is always false, simply by virtue of its connectives, it is a contradiction. For example, (P&~P), or, less trivially, ~((PvQ)<-->(QvP))(do truth table).

Ditto for contradiction in natural language: "to be or not to be" is a very marked statement.

Otherwise, it is a contingency, because you can't know its truth value without knowing the starting values of the atomic statement – the final truth value is contingent upon the starting truth values.

28. If a biconditional is a tautology, the two constituent statments it connects are logically equivalent. (E.g. natural language example: All men are mortal iff no men are not mortal)."All men are mortal" is logically equivalent to "All men are not mortal"

For example, (~P& ~Q)<-->~(PvQ)

Truth table:

29. Logical equivalence is notated with <=>like so:

(~P&~Q)<==>~(PvQ)

If some wff a is logically equivalent to another wff b, you can substitute b for a in any context, e.g. in a wff c, which has a subformula a, and have a new wff d which is itself logically equivalent (preserves the truth conditions) of c. This is very useful in proofs.

Here's a set of some basic logical equivalences various logicians have worked out that are often useful in proofs, and hence are handy to have available (so you don't have to work through them every time you run into a given subformula). We'll see how these are used in proofs next time. (P 112 of PtMW).

Finally, a note which will become increasingly more relevant as time goes by. When we’re evaluating an expression of English, call it A, if it is a proposition, we evaluate it as true or false, by comparing its truth-conditions to the situation in the real world. We write the interpretation of  as [A|]. This means the valuation or denotation of A. Since A is a proposition, its valuation will be 0 or 1. Now, usually, when we’re speaking and certainly when we’re doing semantic analysis, we’re not usually considering the entire universe when we’re doing our valuation function. For instance, the proper name "Susan" in "Susan snores" doesn’t pick out only one individual in the whole world, but in actual use, it usually identifies exactly one person (or if it doesn’t, we qualify it with something like "Susan who works in the office" until it does). With respect to the discourse situation, or the universe of discourse, Susan uniquely denotes, and we can evaluate the proposition "Susan snores". So the valuation of any proposition is with respect to a certain universe of discourse, usually; in semantic terms, we say it’s with respect to a model. We indicate this by writing the valuation of A as [A]^M. Again, like the outermost parentheses above, the superscript M will often be dropped, but it should always be understood to be there. It’ll get to be particularly important later.

So, we can say now something like the following:

[|Susan snores|]^M = 1 iff Susan snores

and we have made a model-theoretic statement that will provide any interested party or computer or whathaveyou with the algorithm for checking whether or not the proposition denoted by "Susan snores" is true or not.

([| |] is the valuation function, since it maps each proposition onto only one element in its range, 0 or 1. More on this later).

And now, we can provide a fully formal translation of the propositional logic (and insofar as these are the same, natural language) connectives "and", "or", "if…then", "it is not the case that" and "if and only if", instead of the squishy sentences used above:

35. [|~A] =1 iff [|A] = 0

36. [|A & B|]=1 iff [|A|] =1 and [|B|]=1

37. [| AvB |]=1 iff [|A|] =1 or [|B]=1

38. [|A-->B|]=1 iff [|A |] =0 or [B|]=1

39. [|A<-->B|]=1 iff [|A|]=[|B|]