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*Symbolic Logic: Syntax, Semantics, and Proof*

6.6 Mixed and Overlapping Quantifiers

__Step 1__: identify all the RL-relevant parts of the sentence and make a translation key for them.

__Step 2__: find the main operator in the English sentence.

__Step 3__: find the subject for each of the two propositions.

__Step 4:__figure out what is being said about the subject in each case.

__Step 5__: Determine the proper quantifiers and their scopes.] He begins with this example.

(1) Someone loves someone.(Agler 277)

(1) | Someone loves someone. | |

Step 1 | Identify and symbolize any English expressions that represent quantifiers (and their bound variables) and propositional operators. (Agler 277) | (∃x) and (∃y) |

(1) | Someone loves someone. | |

Step 2 | Translate any ordinary language predicates into predicates of RL. (Agler 277) | Lxy |

(1) | Someone loves someone. | |

Step 3 | Use the quantifiers from step 1 and the predicates from step 2 and represent the proposition that (1) expresses. (Agler 278) | (∃x)(∃y)Lxy |

(∃x)(∃y)LxyThere exists anxand there exists aysuch thatxlovesy.Someone loves someone.

(Agler 278)

(1) | Someone loves someone. | |

Step 4 | Read the predicate logic wff in English and check to see whether it captures the meaning of the sentence undergoing translation. (Agler 278) | There exists an |

(2) Every zombie loves every human.(Agler 278)

(2) | Every zombie loves every human. | |

Step 1 | Identify and symbolize any English expressions that represent quantifiers (and their bound variables) and propositional operators. (Agler 278) | (∀x) and (∀y) |

The second step says to “Translate any ordinary language predicates into predicates of RL”. Here we have three predicates. We have things with the property of being a zombie, things with the property of being a human, and things with the property of loving other things.

(2) | Every zombie loves every human. | |

Step 2 | Translate any ordinary language predicates into predicates of RL. (Agler 279) | Zx, Hy, Lxy |

(2) | Every zombie loves every human. | |

Step 3 | Use the quantifiers from step 1 and the predicates from step 2 and represent the proposition that (1) expresses. (Agler 279) | (∀x)(∀y)[(Zx∧Hy)→Lxy] |

(2) | Every zombie loves every human. | |

Step 4 | Read the predicate logic wff in English and check to see whether it captures the meaning of the sentence undergoing translation. (Agler 279) | For all |

(∃x)(∀y)((Zx∧Hy)∧Lxy)

(∃x)(∀y)((Zx∧Hy)∧Lyx)

(∃x)(∀y)((Zx∧Hy)→Lyx)

English Sentence | Translation into RL |

Some zombie loves some human. | (∃x)(∃y)((Zx∧Hy)∧Lxy) |

Some zombie loves every human. | (∃x)(∀y)((Zx∧Hy)∧Lxy) |

Every zombie loves some human. | (∀x)(∃y)[(Zx∧Hy)→Lxy] |

Some humans don’t love some zombie. | (∃x)(∃y)((Hx∧Zy)∧¬Lxy) |

No human loves some zombie. | (∀x)(∃y)[(Hx∧Zy)→¬Lxy] |

Examples like those above may give you the impression that the order of the quantifiers does not matter when you are either translating a predicate wff into English or interpreting the expression. This, however, is not the case for the following two wffs:(∃x)(∀y)Lxy(∀x)(∃y)LxyWhile these expressions appear similar, they express different propositions. In English, ‘(∀x)(∃y)Lxy’ expresses the proposition that Someone loves everyone. This proposition is true just in the case that there is at least one person who loves every person. In contrast, ‘(∀y)(∃x)Lxy’ expresses the proposition that Everyone loves someone. This proposition is true just in the case that every individual in the domain of discourse loves at least one person. This shows that the order in which the quantifiers are arranged has an effect on how the formula is interpreted and how we ought to translate the expression from one language into the other.(280)

(∃x)(∀y)Lxyand(∀y)(∃x)Lxy

(∀x)(∃y)Lxyand(∃y)(∀x)Lxy

When dealing with wffs with quantifiers whose scope overlaps, does the order of the quantifiers matter? Consider the following eight wffs (let Lxy express the two-place English expression “x loves y”)1. (∀x)(∀y)Lxy2. (∀y)(∀x)Lxy3. (∃x)(∃y)Lxy4. (∃y)(∃x)Lxy5. (∀x)(∃y)Lxy6. (∃y)(∀x)Lxy7. (∀y)(∃x)Lxy8. (∃x)(∀y)Lxy{I am guessing that 7 and 8 should read:7. (∀y)(∃y)Lxy8. (∃y)(∀y)Lxy}While some of these wffs entail others, only the first two pairs of wffs are equivalent. That is, (∀x)(∀y)Lxy is equivalent to (∀y)(∀x)Lxy and (∃x)(∃y)Lxy is equivalent to (∃y)(∃x)Lxy. (1) and (2) express the proposition that “everyone loves everyone”. In this scenario, every item in the domain of discourse loves every item in the domain of discourse. (3) and (4) express the proposition that “someone loves someone”. In this scenario, at least one item in the discourse loves at least one other. In both cases, the order of the quantifiers does not impact the truth or falsity of the wff.In contrast, (5)-(8) express different propositions. Let’s characterize each in terms of a scenario. (5) is what I will call the “crush” scenario. It says that everyone loves at least one person. It does not say that everyone is loved (there may be some unloved individuals). What it says instead is that for any individual in the domain of discourse, that individual will love at least one other person. In other words, everyone has a crush on someone, even though not | everyone is someone’s crush.(6) is what I will call the “Santa Claus scenario” (I need a better name). It says that there is at least one object who is loved by everyone. This expression is similar to (5) in that it implies that everyone loves at least one person. That is, in (5), every single person loves at least one person, but the loved person can differ from person to person. For example, in a scenario consisting of Jane, John, and Sally, (5) would be true if Jane loves John and John loves Jane and Sally loves herself. In contrast, (6) is true just in the case that there is one person loved by everyone, e.g. John loves Jane, Sally loves Jane, and Jane loves Jane.(7) is what I will call the “Stalker scenario”. It says that everyone is loved by someone. What this says is that if you go through the domain of discourse, pulling people one at a time, you will be able to find at least one other person who loves the selected person. So, if we consider Jane, John, and Sally, (7) is true in the case that, beginning with Jane, we can find at least one other person who loves Jane (e.g. John, but it could be anyone) and one person who loves John and one person who loves Sally. The person doing the loving need not be the same person in each case, nor is it the case that everyone loves someone. (7) differs from (5) in that it doesn’t imply that everyone loves at least one other person. (7), in contrast, can be true if John loves Sally and Jane and himself, but neither Sally nor Jane love anyone. (7) also differs from (6) in that it doesn’t imply that everyone loves at least one object.(8) is what I will call the “Loving God scenario”. It says that someone loves everyone. (8) is true provided there is at least one person who loves every single person in the domain. In contrast to (5), (8) does not imply that everyone loves at least one other person. Rather, it says that there is at least one person who loves all people. In contrast to (6), (8) does not imply that there is at least one person loved by all. It only says that there is one person who loves all. Finally, while (8) implies (7)–for if someone loves everyone, then everyone is loved by at least one person–(7) does not imply (8). This is because (7) can be true in a case where (8) is not, namely in the case where everyone is loved by someone, but everyone is not loved by a single person.1. (∀x)(∀y)Lxy Everyone loves everyone.2. (∀y)(∀x)Lxy Everyone loves everyone.3. (∃x)(∃y)Lxy Someone loves someone.4. (∃y)(∃x)Lxy Someone loves someone.5. (∀x)(∃y)Lxy Everyone loves someone.6. (∃y)(∀x)Lxy Someone is loved by everyone.7. (∀y)(∃y)Lxy Everyone is loved by someone.8. (∃y)(∀y)Lxy Someone loves everyone.{I again guess that 7 and 8 are}7. (∀y)(∃x)Lxy Everyone is loved by someone.8. (∃x)(∀y)Lxy Someone loves everyone.(Agler, Handout 6)

(∃y)(∀x)Lxy(∀x)(∃y)Lxy

(∀x)(∃y)Lxy

(∃y)(∀x)Lxy

(∀y)(∃x)Lxyand(∃x)(∀y)Lxy

(∀y)(∃x)Lxy

(∃x)(∀y)Lxy

The lovers here have the largest scope, and they take the existential quantifier. This means we will begin with one x.

*Symbolic Logic: Syntax, Semantics, and Proof*. New York: Rowman & Littlefield, 2013.

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