Chapter 6 : Quantifiers and definite descriptions (first-order logic analysis)

Notes 研究ノート Genre Departmental Bulletin Paper URL https: //koara. lib. keio. ac. jp/xoonips/modules/xoonips/detail. php? koara ᵢd=AN00069467-00000057-0273 慶應義塾大学学術情報リポジトリ (KOARA) に掲載されているコンテンツの著作権は、それぞれの著作者、学会また は出版社/発行者に帰属し、その権利は著作権法によって保護されています。引用にあたっては、著作権法を遵守し てご利用ください。 The copyrights of content available on the KeiO Associated Repository of Academic resources (KOARA) belong to the respective authors, academic societies, or publishers/issuers, and these rights are protected by the Japanese Copyright Act. When quoting the content, please follow the Japanese copyright act. Powered by TCPDF (www. tcpdf. org) ― 273 ― Chapter 6 Quantifiers and Definite Descriptions (First-Order Logic Analysis) Christopher Tancredi Quantifiers Until now we have only looked at propositions made from combining predicates and names. We analyzed sentences like Mary laughs as follows: Mary laughs translates as LAUGH (mary) denotes True iff m ∈LAUGH′ (iff = if and only if. p is True iff q is short for p is True if q is True, and p is True only if q is True. p iff q then says that either p and q are both True, or they are both False. ) Can we use the same kind of analysis for the following sentences? In particular, can we analyze everyone, someone and no one as names? Everyone laughs Someone laughs No one laughs A name picks out an individual. If we analyzed these expressions as names, they would have to denote individuals as well. However, everyone, someone and no one do not denote individuals. We show that for each of the expressions in turn. Suppose no one translates into logic as the name n, which denotes an individual n. Then No one laughs translates as LAUGH (n), which is true iff n is one of the things that laugh. But what individual could n be? It cannot be Mary – she is a someone, and if Mary laughs, then the sentence No one laughs comes out false, not true. Nor Reports of the Keio Institute of Cultural and Linguistic Studies 57 (2026), 273-286 ― 274 ― can it be John, Bill, or Sue, or any other person we can imagine, for the same reason. That means that if no one denotes an individual, it has to be a strange kind of individual to even have a chance of getting the truth conditions to come out right. We might take it to be an individual that does not and cannot exist, for example. Even then, however, analyzing no one as denoting an individual does not work. Here’s why. Mary laughs translates to LAUGH (mary), which is true iff mary′ ∈ LAUGH′. T truth of Mary laughs is independent of the truth of John laughs. It is possible for both Mary laughs and John laughs to be true. Our analysis predicts this fact. The truth conditions for Mary laughs only require mary′ to be a member of LAUGH′. They not put any other restrictions on LAUGH′. In particular, they say nothing abo whether john′ is a member of LAUGH′. Thus, if Mary laughs is true, it is also possibl that John laughs, or that Sue laughs, or that any other person laughs. Now consider no one. If no one laughs translates as LAUGH (n), with n picking out a non-existing individual n, then LAUGH (n) similarly allows it to be possible that John laughs. The truth of LAUGH (n) only requires n to be a member of LAUGH′. does not care about what other individuals are in LAUGH′. Intuitively, however, John laughs, then the sentence No one laughs is false. For No one laughs to be true, it must be that no person is among the things that laugh. That is, for No one laughs to be true, LAUGH′ cannot contain any people. Requiring that n be a member o LAUGH′ does not restrict LAUGH′ in this way, though, regardless of what n is. It follows that no one cannot be analyzed as a name. We can also show that someone cannot be analyzed as a name. Imagine a situation in which John laughs and Mary does not. In this situation, the sentences Someone laughs and Someone does not laugh are both true. If someone is analyzed as a name, this should not be possible. Suppose someone is translated as s, which denotes s. Then Someone laughs is translated as LAUGH (s), and this is true just in case s∈LAUGH′. Someone does not laugh then translates as ~LAUGH (s), and is true just in case it is not the case that s ∈LAUGH′. Since a single individual cannot both be and not be member of a given set, it is not possible for both of these to be true. That is, analyzing someone as a name leads to the prediction that it should be impossible for Someone laughs and Someone does not laugh to both be true. Since we observed that they can both be true, the analysis fails to account for our observation. Therefore the analysis ― 275 ― must be wrong: someone cannot be analyzed as a name. Finally, we show that everyone cannot be analyzed as a name. Imagine the following situation: John is 90cm tall, Mary is 150cm tall, and Pat is 210cm tall. Now consider sentences of the following form with the same name substituting for both occurrences of x: x is over 1 meter tall, or x is under 2 meters tall, or both. Whatever name you put in place of the x’s, the resulting sentence is true. John is not over 1 meter tall, but he is under 2 meters tall, so the sentence is true with John in place of the x’s. Mary is both over 1 meter tall and under 2 meters tall, so she is covered by the or both clause. Finally, Pat is over 1 meter tall but not under 2 meters tall, so the sentence with Pat in place of the x’s is true as well. From this it is clear that no matter what height a person has, this sentence will be true with a name of that person put in place of the x’s. This is because all heights are either over 1 meter, under 2 meters, or both. Now consider what happens when the x’s are replaced by everyone: Everyone is over 1 meter tall, or everyone is under 2 meters tall, or both. While it is possible for this sentence to be true, it is also possible for it to be false. Indeed, in the situation described above with John, Mary and Pat, the sentence is false. If everyone were interpreted as a name, however, it should not be possible for the sentence to be false. We can conclude that everyone is not interpreted as a name. Expressions like everyone, someone and no one are called quantifiers. If quantifie do not denote individuals, then what do they denote? We can start to answer that question by looking at semantic types. Assume that laughs is a 1-place predicate, of type 〈e, t〉. Then for the sentences Everyone/ Someone/ No one laughs, we have the following type information: ― 276 ― S: t | ‐‐‐‐‐‐‐‐‐‐-------------------- | | Everyone:? laughs: 〈e, t〉 Someone:? No one:? We just showed that the quantifiers cannot denote individuals. This means th cannot be of type e, and so they cannot act as the semantic argument of laughs. We saw in Chapter 4, however, that there is another possibility to consider: the quantifie can be of type 〈〈e, t〉, t〉. That is, they can be functions that take laughs as their semantic argument and give back a truth value for the sentence as a whole. To give a full analysis of the quantifiers, we have to assign them a lexical translation into logic. We then have to show how their logical translations get interpreted. To get to this analysis, we look first at the truth conditions of the whole sentence. We th ask how we can get those truth conditions from the parts that we have, namely the quantifier and the word laughs. Everyone Let us start with the quantifier everyone. There are two basic intuitions that semanticists have used to give the truth conditions for sentences like Everyone laughs. The intuition we look at here sees the sentence as looking at individuals one by one, and checking two things: (i) Is that individual a person? and (ii) Does that individual laugh? The sentence is true iff there is no case in which (i) is answered yes and (ii) i answered no. To see how this approach works, consider the sentence Everyone laughs in the following situation: Situation 1: There are 5 individuals, a, b, c, d and e. a and b are people c and d are monkeys e is a pencil ― 277 ― a, b and c laugh, d and e do not. Intuitively, in Situation 1, Everyone laughs is true. This approach correctly predicts this fact. We first ask of the 5 individuals one-by-one whether they are a person. a and b the answer is yes, and for c, d, and e the answer is no. We then ask if that individual laughs. Since a, b and c laugh, the second question is answered yes for each of them. It is answered no for d and e. These results are summarized below: individual person? laughs? a yes yes b yes yes c no yes d no no e no no As can be seen, for no individual is the first question answered yes and the second question answered no. This makes the sentence Everyone laughs true. Turning this idea into a complete analysis requires adding something to our logic. In particular, we need something that picks out individuals one-by-one and checks whether they all have certain properties. For this we use the universal quantifier ∀ and a variable. The logical translation we give to the sentence Everyone laughs is: ∀z (PERSON (z) → LAUGH (z) ) We read this formula as “For all z, if z is a PERSON then z LAUGHs”. We also need to give a denotation for this kind of formula. We start by giving a general denotation for formulas of the form ∀z (ϕ), where ϕ is any formula of logic. In the cases we a interested in, ϕ will contain occurrences of the variable introduced by ∀, here z. In the proposed translation of Everyone laughs, ϕ isthe formula PERSON (z) → LAUGH (z). In order to give the denotation of ∀z (ϕ), we need to be able to interpret variables. variable does not have a fixed denotation. Rather, it acts as a placeh