Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Parentheses disambiguate:Bad(bob) ᴠ Bewildered(bob) ᴧ Bogeyman(bob)How are the following two interpretations different?(Bad(bob) ᴠ Bewildered(bob)) ᴧ Bogeyman(bob)“Bob is bad or bewildered, and he is a bogeyman.”Bad(bob) ᴠ (Bewildered(bob) ᴧ Bogeyman(bob))“Bob is bad, or he’s a bewildered bogeyman”Complex sentences with negation:¬Bad(bob) ᴧ ¬Bogeyman(bob)“Bob is not bad, and he’s not a bogeyman”¬(Bad(bob) ᴧ Bogeyman(bob))“Bob is not a bad bogeyman” or “It’s not the case that Bob is both bad and a bogeyman” (though he might be just bad, or just a bogeyman)¬(¬Bad(bob) ᴧ ¬Bogeyman(bob))“It’s not true that Bob is neither bad nor a bogeyman”(he might bad or a bogeyman or both)There is sometimes more than one way to logically state the same thing . . .DeMorgan’s Laws:#¬(P ᴧ Q) (¬P ᴠ ¬Q)#¬(P ᴠ Q) (¬P ᴧ ¬Q)Some comments regarding good translations:The minimum requirement of an FOL translation of an English sentence is that the two sentences have the same truth values in all possible circumstances (all possible worlds).The stylistic goal of an FOL translation of an English sentence is that the FOL mirror the English closely enough that a reader could ‘reverse engineer’ it from the FOL back to the English.“It is not true that Claire and Max are both at home”¬(Home(claire) ᴧ Home(max)) better¬Home(claire) ᴠ ¬Home(max)The English words but, however, yet, and nonetheless are, for purposes of FOL, stylistic variants of and. They all carry a sense of ‘unexpectedness’ about what follows, but this extra meaning is irrelevant to the sort of logical analysis we are conducting and can safely be ignored for our purposes. So, for example:Bob is boring but he’s not stupid Boring(bob) ᴧ ¬Stupid(bob)In the instructions to Exercise 3.21, the book mentions a useful technique for checking the accuracy of your translations from English into the blocks language of FOL: Build several blocks worlds that make the English sentence(s) true or false, then check to see if your FOL translations of these sentences have the same truth values (as their English counterparts) in the same worlds (they should).So, how to best translate each of the following (from exercise 3.21)? (put translations on board)1. Either a is small or both c and d are large.2. d and e are both in back of b.3. d and e are both in back of b and larger than it.4. Both d and c are cubes, however neither of them is small.5. Neither e nor a is to the right of c and to the left of b.6. Either e is not large or it is in back of a.If we translated them correctly, then all six of these sentences should be true in Wittgenstein’s World. Let’s check.The books also informs us that if we make the following adjustments to Wittgenstein’s World, then sentences 1, 3, 4 and 10 should become false, while the other sentences remain true:*Make all the large or medium objects small but all the (originally) small objects large.Next, if we rotate this modified Wittgenstein’s World 90 degrees clockwise, sentences 5, 6, 8, 9, and 11 should be the only true ones remaining . . .Any problems with exercise 3.18, writing equivalent sentences but using different predicates? (text p. 83, using Bernay’s Sentences, Ackermann’s World, Bolzano’s World, Boole’s World, Leibniz’s World)Any problems with exercise 3.23 translations?(text p. 88, using Table 1.2 on text p.30)Note that I’ve provided an answer key to both these exercises on Blackboard (answer key for lesson
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