DAY 27: THE FOUR ARISTOTELIAN FORMS (9.5); TRANSLATING COMPLEX NOUN PHRASES (9.6); QUANTIFIERS AND FUNCTION SYMBOLS (9.7)Assigned reading (sections 9.5 and 9.6)POWERPOINT SLIDE #1The four Aristotelian forms and how we translate them in modern symbolic logic:All P’s are Q’s ∀x (P(x) Q(x)) Note the conditional.Some P’s are Q’s Ǝx (P(x) ᴧ Q(x)) Note the conjunction.Note that this does NOT equal Ǝx (P(x) Q(x))No P’s are Q’s ∀x (P(x) ¬Q(x)) ¬Ǝx (Px) ᴧ Q(x))Some P’s are not Q’s Ǝx (P(x) ᴧ ¬Q(x))POWERPOINT SLIDE #2Two tips in the “Remember” box on p. 247:1) Translations of complex quantified noun phrases frequently employ conjunctions of atomic predicates. Examples:“A small, happy dog is at home.”Ǝx[(Small(x) ᴧ Happy(x) ᴧ Dog(x)) ᴧ Home(x)]“Every small dog that is at home is happy.”∀x [(Small(x) ᴧ Dog(x) ᴧ Home(x)) Happy(x)]POWERPOINT SLIDE #32) The order of an English sentence doesn’t always correspond to the order of its FOL translation. Examples:“Max owns a small, happy dog.”Ǝx[(Small(x) ᴧ Happy(x) ᴧ Dog(x)) ᴧ Owns(max, x)]Note that the complex noun phrase still comes first in FOL.“Max owns every small, happy dog.”∀x [(Small(x) ᴧ Happy(x) ᴧ Dog(x)) Owns(max,x))POWERPOINT SLIDE #4POTENTIAL PROBLEM CASES:∀x (P(x) Q(x)) in vacuously true cases (i.e., in worlds where the antecedent is False, thus it is impossible to find a counterexample)e.g., ∀x (Tet(x) Small(x)) … vacuously true in a world with no tetrahedraCompare “Every freshman who took the class got an A” if no freshmen took the class!(conversational implicature)e.g., ∀x (Tet(x) Cube(x)) … inherently vacuous: can only be true when asserted of a world with no tetrahedraPOWERPOINT SLIDE #5Another potentially problematic case: “Some P’s are Q’s” does NOT contradict “All P’s are Q’s” … why not?(In English, when we use ‘some’ this usually implies [though it doesn’t rigorously entail] that we include the meaning ‘not all’. Our FOL does not include implicatures of this sort, so the existential quantifier strictly means ‘some’ in the sense of ‘one or more’ without any upper limit .. . and this can include even ‘all’ of the objects in question.)Assigned reading pp.253-255 (9.7)Refer to POWERPOINT SLIDES #6 and #7(I have posted an answer key on Blackboard for exercises 9.23 and
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