Numerical Quantification

1. Numerical Quantification Debbie Mueller Mathematical Logic Spring 2012

2. English sentences take the form Q A B Q is a determiner expression • the, every, some, more than, at least , no, etc A is a common noun phrase • cube, cat, person, etc B is a verb phrase • is, are, eats, etc

3. Q A B expresses binary relation between A and B • usually the relation is quantitative • can sometimes can be expressed with the universal and existential quantifiers as well as truth-functional connectives • can express: Nothing, Every, Some, All • for those that can’t, supplement FOL with expressions that behave like ∃ and ∀ • Generalized Quantifiers

4. Another quantification type - Numerical Claims • a claim that explicitly uses numbers to say something about the relationship between the A’s and the B’s. • FOL does not allow direct talk about numbers, only about elements in the domain of discourse. • uses universal and existential quantifiers, together with truth-functional connectives and (most importantly) the identity sign. • There are 3 types of claims • At least • At most • Exactly

5. At least “n” • Requires n quantifiers and non-identity clauses joined by conjunction • ex: At least 3 cubes • ∃x∃y∃z (Cube(x) ∧ Cube(y) ∧ Cube(z) ∧ x ≠ y ∧ y ≠ z ∧ x ≠ z)

6. At most “n” • Equivalent to less than or equal to • Allows there to be no object at all • One method: Deny existence of at least n+1 non-identical things • ex: There is at most one large thing • Denial: There does not exist two (non-identical) large things • ~∃x∃y(Large(x) ∧ Large(y) ∧ x ≠ y)

7. At most “n” con’t • Second method: Take n+1 objects. Then at least one pair of them are identical • ex: there are at most three large things • ∀w∀x∀y∀z ((Large(w) ∧ Large(x) ∧ Large(y) ∧ Large(z)) → (w=x ∨ w=y ∨ w=z ∨ x=y ∨ x=z ∨ y=z))

8. Exactly “n” • Similar to no more, no less • Conjunction of at least n and at most n • ex: At least two cubes • ∃x∃y(Cube(x) ∧ Cube(y) ∧ x ≠ y) ∧ ∀x∀y∀z ((Cube(x) ∧ Cube(y) ∧ Cube(z)) → (x = y ∨ y = z ∨ x = z)) • Compact version • ∃x∃y(Cube(x) ∧ Cube(y) ∧ x ≠ y ∧ ∀z (Cube(z) → (y = z ∨ x = z)))

9. Abbreviations for numerical claims Abbreviation scheme: • ∃≥n x P(x) abbreviates the FOL sentence asserting “There are at least n objects satisfying P(x).” • ∃≤n x P(x) abbreviates the FOL sentence asserting “There are at most n objects satisfying P(x).” • ∃!n x P(x) abbreviates the FOL sentence asserting “There are exactly n objects satisfying P(x).” For the special case where n = 1, it is customary to write ∃!x P(x) as a shorthand for ∃!1 x P(x). This can be read as “there is a unique x such that P(x).”

10. The, Both, and Neither • “The” combined with a noun phrase forms an expression that suggests to refer to exactly one object • called a definite description • functions syntactically like names but not semantically • does not guarantee a unique object • “good” description if there is a unique object • can evaluate • “bad” description if not • Bertrand Russell’s famous Theory of Descriptions (1905)

11. The, Both, and Neither con’t • Russell’s Theory of Descriptions • a sentence containing a definite description can be thought of as a conjunction with three conjuncts. • ex: The cube is small. • Russell’s theory: There is at least one cube, and there is at most one cube, and every cube is small. • ∃x Cube(x) ∧ ∀x∀y ((Cube(x) ∧ Cube(y)) → y = x) ∧ ∀x (Cube(x) → Small(x)) • Compact version • ∃x (Cube(x) ∧ ∀y (Cube(y) → y = x) • ∧ Small(x))

12. The, Both, and Neither con’t • Russell’s analysis can be extended to cover “Both” and “Neither”. • “Both” suggests that there are exactly two objects, and each object has the same property. • ex: Both cubes are small. • Russell’s theory: There are exactly two cubes, and each cube is small. • ∃x∃y(Cube(x) ∧ Cube(y) ∧ x ≠ y) ∧ ∀x∀y∀z ((Cube(x) ∧ Cube(y) ∧ Cube(z)) → (x = y ∨ y = z ∨ x = z) ∧ Small(x) ∧ Small(y)) • ∃!2 x Cube(x) ∧ ∀x (Cube(x) → Small(x))

13. The, Both, and Neither con’t • “Neither” suggests that there are exactly two objects, and no object has the property • ex: Neither cube is large • Russell’s theory: There are exactly two cubes, and each of them are not large. • ∃x∃y(Cube(x) ∧ Cube(y) ∧ x ≠ y) ∧ ∀x∀y∀z ((Cube(x) ∧ Cube(y) ∧ Cube(z)) → (x = y ∨ y = z ∨ x = z) ∧ ￢Large(x) ∧ ￢Large(y)) • ∃!2 x Cube(x) ∧ ∀x (Cube(x) → ￢Large(x))

14. Russell’s Theory • Two key features • provides a truth value for every sentence containing a definite description • the introduction of a logical operation such as negation may introduce an ambiguity • ex: the cube is not small • Exactly one cube and it is not small • Not the case that there is exactly one cube and • that it is small

15. Russell’s Theory con’t • Opposing critique by philosopher P.F. Strawson • Russell is mistaken in supposing that one who utters the sentence “the cube is small” makes three claims • person does not even succeed in making a claim unless there is exactly one cube • Presupposition • If the presupposition is fulfilled, then the utterer of the sentence is making a claim • If the presupposition is not fulfilled(bad description), then the speaker has failed to make any claim

16. Russell’s Theory con’t • Consequences of Strawson’s analysis • introduction of truth value gaps • cannot be translated into FOL/weakens it • alternative to presuppositions: implicatures • use cancellability test for validity • Can one conjoin without contradiction?

17. “Numerical” Quantifications not expressible in FOL • Most • indeterminate • implies more than half • disjunction does not end • ex: more than half • [∃xA(x) ∧∀ x~B(x)] ∨ [∃≤2xA(x) ∧ ∃≤1xB(x)] ∨ [∃≤3xA(x) ∧∃≤2xB(x)] ∨ ... • Many, A lot, A few • context dependent

18. References • Barker-Plummer, D. &. (2011). Language, Proof and Logic. Stanford: CSLI Publications. • Cohen, S. (2004). Chapter 14: More on Quantification. http://faculty.washington.edu/smcohen/120/Chapter14.pdf • Cummins, C. &. (n.d.). Numerically Quantified Expressions. www.crcummins.com/CRCNumerically.pdf • Filip, H. (2012, January 18). Lecture 3: Quantification. user.phil-fak.uni-duesseldorf.de/~filip/L3.Tilburg.pdf • Guerts, B. (n.d.). Processing Quantifiers.ncs.ruhosting.nl/bart/talks/paris2005/parislides1.pdf • Johns, R. (n.d.). Translations Involving Complex Quantifiers. http://faculty.arts.ubc.ca/rjohns/notes5.pdf • Shapiro, S. (n.d.). Numerical Quantifiers and Their Use in Reasoning with Negative Information. 128.205.32.53/~shapiro/Papers/sha79b.pdf