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1 Transcription from English to Predicate Logic General Principles of Transcription In transcribing an English sentence into Predicate Logic, some general principles apply. A transcription guide must be provided. It should be determined in advance whether you are using restricted quantifiers or are giving a complete transcription. You should symbolize as much of the logical structure of the sentence as possible. If two sentences of Predicate Logic are equivalent, then neither is more adequate than the other, although one of the two may make the logical form of the English sentence more perspicuous. If an English sentence is ambiguous, it may allow for more than one adequate transcription. In different contexts, we will hear the sentence in different ways. The Indefinite Article The indefinite article a can function as a universal and an existential quantity term. A car can go very fast. Any car can go very fast (universal). At least one car can go very fast (existential). Generally, we use the article as a universal quantity term when the subject is taken to be a class of things, e.g. cars. We generally use the article as an existential quantity term when the subject is taken to be a specific thing which we do not wish to or cannot identify. Sometimes how we understand the predicate of the sentence makes a difference: Fast in comparison to the speed of non-motorized vehicles indicates that the universal might be the best reading. Fast is taken in comparison to the highest speeds of modern motorized vehicles, indicates the existential. 1

2 Anyone Generally, anyone indicates a universal quantifier. Anyone has the right to vote. ( x) P Vx. When used in the antecedent of a conditional without cross-reference, anyone can function as an existential quantifier. If anyone is at home, the lights will be on. ( x) P Hx J. If there is cross-reference between the antecedent and the consequent, then the universal quantifier is usually called for. Someone If anyone is at home, it is a boy. ( x) P (Hx Bx). Generally, someone indicates an existential quantifier. Someone is a registered voter. ( x) P Rx. If there is cross-reference between the antecedent and the consequent, then someone can function as a universal quantifier. If someone is at home, it is a boy. ( x) P (Hx Bx). Ambiguity with Anyone and Someone Some uses of anyone in the antecedent of a conditional without cross-reference can be interpreted as universal quantifiers. These will be cases of logical truths in which the conditional expresses universal instantiation. If anyone is a boy, then John is a boy. ( x) P Bx Bj. Some uses of someone which seem to apply to classes of things, and thus call for the universal quantifier, apply to a single thing and thus call for an existential quantifier. Someone who is a registered voter has the right to vote. ( x)(rx & Vx). 2

3 Classes and Members of a Class Generally, we use the universal quantifier when we want to talk about a class of objects. Sometimes in a sentence we have more universal quantity expressions, each being about a class of objects. All the boys kissed all the girls. This is perhaps most naturally taken to relate each member of the class of boys to each member of the class of girls. ( x) B ( y) G Kxy. But it may also taken as stating that the boys, taken as a class, kissed the girls, taken as a class. The boys as a class all gave kisses to girls, and the girls as a class all received kisses from boys. ( x) B ( y) G Kxy & ( x) G ( y) B (Kyx). The Scope of Negation Expressions Ambiguity can arise because of the variability of the scope of natural language operators, including negation, and (in modal logic) words like necessarily and possibly. All the boys are not at home. The word not might have narrow scope, governing the phrase at home. ( x) P (Bx Hx). Or not may have wide scope, governing the whole sentence. ( x) P (Bx Hx). Scope and Quantifiers Scope ambiguities also arise with the use of multiple quantifiers. All the boys kissed a girl. We interpret a as an existential quantifier. Its scope might be the whole sentence. There is at least one girl whom each of the boys kissed. ( x) G ( y) B Kyx. 3

4 Or, the scope might be internal to the sentence. Each of the boys is such that there is at least one girl whom he kissed. ( x) B ( y) G Kxy. Class and Scope Ambiguities Combined When we combine ambiguities of class and scope, even more interpretive possibilities arise. All the boys did not kiss all the girls. The class/member ambiguity led to two different transcriptions of the sentence without negation. ( x) B ( y) G Kxy. ( x) B ( y) G Kxy & ( x) G ( y) B (Kyx). On the second reading (boys and girls as classes), the negation seems to govern the conjunction: [( x) B ( y) G Kxy & ( x) G ( y) B (Kyx)]. On the first reading (boys and girls as members of classes), there are three positions the negation sign could take. ( x) B ( y) G Kxy. ( x) B ( y) G Kxy. ( x) B ( y) G Kxy. Transcribing Only In sentence logic, only reverses the order of a conditional. If Adam is an adult, then he has the right to vote. Aa Va Only if Adam is an adult does he have the right to vote. Va Aa This reversal holds when quantifiers are involved. If a person is an adult, then he has the right to vote (adults have the right to vote). ( x) P (Ax Vx). Only if a person is an adult does he have the right to vote (only adults have the right to vote). ( x) P (Vx Ax). 4

5 Negative Quantifier Expressions Nothing makes a negative claim about everything in a whole class of objects, while not everything negates the universality of a statement about a class of objects. Nothing is furry: ( x) Fx. Not everything is furry: ( x)fx. None and none but are generally used to indicate a restricted class of things, where none but says the same thing as only and reverses the order of subject and predicate terms in the English sentence. None of the boys are registered voters: ( x) B Rx, or ( x)(bx Rx). None but adults are registered voters: ( x) R Ax, or ( x)(rx Ax). No Ps are Qs is transcribed as ( u) P Q(u), or as ( u)(p(u) Q(u)). 5

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Notes and Exercises A sentence is made up of three parts. SUBJECT: What or whom the sentence is about. VERB: What the subject did or is. The REST: Everything else in the sentence. EXAMPLES Rachel talks