Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Quantificationevery cubesome students from USCmost vipers in the playpenthe dodecahedron in the bathroomthree blind miceno student of logicwheneverQuantification is not truth-functional T T T b is a cube and c is a cube. vs. . . . ?Every rich actor is a good actor.termsindividual constantsa, b, clairevariablest, x, y18The Universal Quantifier: ∀means “every” or in some cases “all”∀x = “for every object x …”∀x Home(x) = “For every object x, x is at home” or simply “everything is at home”Binding: In the above expression we say that the universal quantifier ∀ binds the variable x. Without the quantifier present, as in just the expression Home(x), the variable is unbound or free: ∀x Home(x) …… x is bound by the quantifier Home(x) …… x is unbound or free∀x (Doctor(x) Smart(x)) = “Every doctor is smart”(literally: “For every x, if x is a doctor, then x is smart”)Scope: In the above sentence, the scope of the quantifier is indicated by the outer parentheses; that is, the quantifier here binds every x-variable that falls within the entire parenthetical phrase, so it binds the x-variables that are the arguments of both the predicate ‘Doctor’ and the predicate ‘Smart’. Scope of the quantifierIn the earlier example, ∀x Home(x)no parentheses are needed because the scope of the quantifier is simply over the one predicate Home and its variable x (i.e., over the atomic sentence) Scope of the quantifierThe Existential Quantifier: ƎMeans “some” (i.e., at least one)Ǝx = “for some object x …”Ǝx Home(x) = “For some object x, x is at home” or simply “something is at home”Ǝx (Doctor(x) ᴧ Smart(x)) = “Some doctor is smart” or “There is at least one smart doctor”Binding and scope with Ǝ work exactly the same as with ∀ …Ǝx Home(x) Ǝx (Doctor(x) ᴧ Smart(x)) Scope of quantifierScope of quantifierBe careful, though … Ǝx Doctor(x) ᴧ Smart(x) This sentence of FOL looks misleadingly similar to the earlier example, but it’s missing the outer parentheses. This indicates that the scope of the existential quantifier is only over Doctor(x) and NOT over Smart(x). Only the first x-variable argument is bound (the second one here is free and, thus, uninterpreted).Ǝx Doctor(x) ᴧ Smart(x)Scope of quantifierFor an expression containing variables to be a sentence, all of the variables must be bound by some or other quantifier. If there are any unbound variables in the expression, the expression can at most be a well-formed formula (wf), not a sentence.Well-formed formulas (wfs): Expressions that have the proper formal structure of sentences of FOL (i.e., they contain predicates with terms as arguments—either constants or variables—and possibly quantifiers and connectives as well), all generated according to the grammatical rules of FOL, but possibly containing unbound variables. Sentences = those wffs in which all variables (if any) are bound and thus interpretableatomicwfDoctor(x)connectives,quantifierscomplex wfƎx (Doctor(x) ᴧ Smart(x))Cf. pp. 233-234, list of generative rules of FOL. You can reiterate the generative rules over and over to generate increasingly complex sentences of FOL.These four are all wfs but not sentences:1. Cube(x) ᴧ ¬Cube(y)2. ∀x Doctor(x) Smart(x)3. Ǝy RightOf (x,y)4. (Cube(x) ᴧ Small(x)) Ǝy LeftOf (x,y)These four are both wfs and sentences:5. Cube(a) ᴧ Cube(b)“a and b are cubes” 6. ∀x (Doctor(x) Smart(x))“Every doctor is smart”7. ∀x ((Cube(x) ᴧ Small(x)) Ǝy LeftOf (x,y))“Every small cube is to the left of some object”8. Ǝz (Tet(z) ᴧ Large(z))“There is a large tetrahedron”These expressions are not even wffs (i.e., they are ‘ill-formed’). Why?9. SameSize ¬(a, b)10. ∀b SameRow(b,c)11. FrontOf(b,c,d) Cube(x)12. Ǝu RightOf (u,a) Tet(z5) ᴠ Between(u,a,z5)It’s clear that variables don’t ‘refer’ to objects in a world in quite the same way that individual constants do, so we need a new way to talk about the connection of variables to the objects they stand for :SatisfactionƎx (Cube(x) ᴠ Tet(x)) = ‘Some object is a cube or a tetrahedron’This sentence is true iff there is at least one actual object in the world which, if ‘plugged into’ the x-variable placeholder positions of the sentence, would ‘satisfy’ the wff Cube(x) ᴠ Tet(x) by being either a Cube or a Tet.∀x (Doctor(x) Smart(x)) = ‘Every doctor is smart’is true iff every object ‘plugged into’ the wff (Doctor(x) Smart(x)) yields a true result (which is the same as saying that no object would ever yield a true antecedent but a false consequent in this conditional phrase).Exercise 9. 3Use Tarski’s World to identify the expressions in ‘Bozo’s Sentences’ as ill-formed wffs but not sentences, or wffs that are
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