Lesson 6 Boolean Connectives Sections 3 1 3 3 Assigned reading pp 67 78 POWERPOINT SLIDE 1 We ve discussed the role of both predicates and terms in an FOL but so far we ve only dealt with atomic sentences i e where a single predicate takes one or more terms as its arguments To create more complex sentences in FOL we need to introduce a third type of component of logical languages Boolean connectives POWERPOINT SLIDE 2 There are three basic Boolean connectives negation conjunction and disjunction Here s a simple table showing how they compare Connective Symbol English equivalent Negation not it is not the case that non un Conjunction and moreover but Disjunction or These connectives connect multiple atomic sentences together or otherwise create a complex FOL sentence out of what without the connective was a simple atomic sentence This is easiest to see with conjunction and disjunction for example POWERPOINT SLIDE 3 Tet a Cube b Between a d e Between a d f translation a is a tet and b is a cube translation a is between d and e or it s between d and f POWERPOINT SLIDE 4 In the case of negation the result of negating an atomic sentence is also a complex sentence but a special sort of simpler complex sentence that we still find useful to sometimes group with atomic sentences as both being literals i e literals can be either atomic sentences OR the negation of atomic sentences Small a Small a translation a is small translation a is not small Negation can also apply to complex sentences Tet a Cube b Small a translation It s not the case that both a is a tet and b is a cube translation It s not the case that a isn t small i e a is small The book mentions some other peculiarities of the Boolean connectives that make them different from their normal English counterparts POWERPOINT SLIDE 5 You just saw that FOL allows multiple negatives stacked together English would normally not allow this I don t not want no more cookies POWERPOINT SLIDE 6 Conjunction and disjunction apply in FOL to entire atomic sentences not just to simple nouns or verbs as in Thom and Jonny play guitar and Fred slipped and cracked his head In FOL PlayGuitar Thom PlayGuitar Jonny Slipped Fred CrackedHead Fred POWERPOINT SLIDE 7 When an object has multiple properties FOL has to break these out as separate predicates in separate atomic sentences whereas English doesn t Intelligent Neil Handsome Neil Young Neil Man Neil vs Neil is an intelligent handsome young man POWERPOINT SLIDE 8 Disjunction always has an inclusive sense in FOL never exclusive Bob or Tony smashed the bug Typical English exclusive sense implies that only one of the men applied fatal pressure to the bug Smashed bob bug Smashed tony bug FOL inclusive sense could be that one man applied the fatal pressure or could be that both men did POWERPOINT SLIDE 9 So there are seven different ways for the sentence Dodec a Dodec b Dodec c to turn out true POWERPOINT SLIDE 10 Note that the negation of a disjunction may be translated with neither nor Tet a Cube a Object a is neither a tet nor a cube POWERPOINT SLIDE 11 There is one other important thing you need to remember from today s reading POWERPOINT SLIDE 12 Truth Tables Truth tables highlight an important fact about the Boolean connectives namely that they are truthfunctional connectives meaning that the truth value of the complex sentence formed when these connectives join together two or more atomic sentences depends strictly on a combination of the truth values of the atomic sentences from which it is built POWERPOINT SLIDE 13 For example here is the truth table for disjunction P Q P Q TRUE TRUE TRUE TRUE FALSE TRUE FALSE TRUE TRUE FALSE FALSE FALSE The way you read such a table is as follows Look at what is under the heading P which is a label for some atomic sentence it doesn t matter exactly what the atomic sentence says it could be anything like Cube a or LeftOf b c it doesn t matter so we ll just label it P here In the column under P there are four truth values representing four different worlds or situations two worlds in which atomic sentence P is True and two worlds in which atomic sentence P is False Now notice that the other atomic sentence labeled Q also has four truth values under it i e the middle column of the table again two worlds for which Q would be true and two worlds for which it would be false Notice also that the truth values in the four rows under P and Q remember that rows run left to right cover all four of the possible combinations of P and Q in terms of their truth values the top row represents the world or situation in which P and Q are both true The row under that represents the world or situation in which P is true but Q is false The third row represents the world where P is false but Q is true Finally the bottom row represents the world where both P and Q are false Keeping all that in mind look at the last column of the table the bottom four cells of which represents the four possible truth values for the entire complex sentence P Q i e P or Q The value of each row under this last column is computed based on the combination of truth values for P and Q in the same row under the first two columns to the lef So for example when P and Q are both true i e as shown in the first row of truth values then the disjunction of P and Q i e P Q will be true And so forth Indeed the only world or situation in which the complex sentence P Q turns out to be false is where both P and Q are each individually false considered as atomic sentences as shown in the bottom row of the table The truth tables for conjunction and negation are as follows each is understood in the same way as above though the resulting truth values for the entire complex sentence represented by the last column of the table are unique for each table POWERPOINT SLIDE 14 Conjunction P Q P Q TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE POWERPOINT SLIDE 15 Negation P P TRUE FALSE FALSE TRUE NOTE I omitted from today s reading a part of the chapter about game rules I am not going to emphasize this in this course and whenever you come across a part of a homework exercise that asks you to play the game you are free to skip that
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