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Unformatted text preview: CS245 - Winter 2010, Lecture of Feb. 24th Shai Ben-David first Order Logic (FOL) is, in fact, a family of languages, all sharing the same structure. To determine the specific language in first order logic, we have to choose a “vocabulary” for that language, which will consist of three types of objects: 1. Constant symbols eg. a,b,... 2. Relation symbols eg. R 1 ,R 2 ,... 3. Function symbols eg. F 1 ,F 2 ,... Examples: 1. A language for discussing orders L = h∅ ,R ( , ) , ∅i where R ( x,y ) expresses x < y . For example, in this language, the formula ∀ x ∃ y R ( x,y ) expresses the state- ment “there is no largest element”. 2. A language to express statements about numbers L = h ( a,b ) ,R ( , ) , ( F ( , ) ,G ( , ) i where a may denote the number 0, b may denote 1, R ( x,y ) expresses x < y , F ( x,y ) expresses x + y , and G ( x,y ) expresses the multiplication. Under this interpretation, ∀ x ∀ y ∀ z F ( x,G ( y,z )) = G ( F ( x,y ) ,F ( x,z )) ex- presses the equality...
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- Spring '08
- Logic, Quantification, Universal quantification, Existential quantification, Table of mathematical symbols