Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15So far we’ve dealt only with atomic sentences.It’s time now to get our hands dirty with . . .Complex sentences!!!!Boolean connectivesConnective Symbol English equivalentNegation ¬ not, it is not the case that, non-, un-Conjunction ᴧ and, moreover, butDisjunction ᴠ orTet(a) ᴧ Cube(b) atomic atomic how to translate these? complexBetween(a,d,e) ᴠ Between(a,d,f) atomic atomic complexA ‘literal’ = either an atomic sentence or the negation of an atomic sentence Small(a)¬Small(a) how to translate these?Negation can apply to complex sentences: ¬(Tet(a) ᴧ Cube(b))¬¬Small(a) how to translate these?Other peculiarities of Boolean connectives:FOL allows multiple negatives stacked together. ¬¬¬¬Small(a)English would normally not allow this:“I don’t not want no more cookies”= ungrammaticalConjunction and disjunction apply in FOL only to entire atomic sentences, not to simple nouns or verbs. The latter is possible, however, in English: “Thom and Jonny play guitar” “Fred slipped and cracked his head.”PlayGuitar(Thom) ᴧ PlayGuitar(Jonny)Slipped(Fred) ᴧ CrackedHead(Fred)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) ᴧ Handsome(Neil) ᴧ Young(Neil) ᴧ Man(Neil) “Neil is an intelligent, handsome, young man.”Disjunction always has an inclusive sense in FOL, never exclusive. Compare: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) ᴠ Smashed(tony,bug)FOL inclusive sense of ᴠ . . . could be that one man applied the fatal pressure, or could be that both men did.Note, then, that there are seven different ways for the following complex sentence to be true! Dodec(a) ᴠ Dodec(b) ᴠ Dodec(c)The negation of an entire disjunction may be translated with “neither . . . nor”: ¬(Tet(a) ᴠ Cube(a))“Object a is neither a tet nor a cube.”There is one other very important thing to remember from today’s reading . . .Truth tables:Truth tables show that the Boolean connectives are truth-functional, meaning that the truth-value of the complex sentence formed by the use of the connectives depends on a combination of the truth-values of the atomic sentences from which complex sentence is built.P QP ᴧ QTRUE TRUE TRUETRUE FALSE FALSEFALSE TRUE FALSEFALSE FALSE FALSETruth table for DisjunctionP QP ᴠ QTRUE TRUE TRUETRUE FALSE TRUEFALSE TRUE TRUEFALSE FALSE FALSETruth table for ConjunctionP QP ᴧ QTRUE TRUE TRUETRUE FALSE FALSEFALSE TRUE FALSEFALSE FALSE FALSETruth table for NegationP ¬PTRUE FALSEFALSE
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