Debbie Mueller Mathematical Logic Spring 2012

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

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

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

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)

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)

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))

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)))

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).”

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 )

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))

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))

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))

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

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

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?

“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)] ∨ [ ∃ ≤2 xA(x) ∧ ∃ ≤1 xB(x)] ∨ [ ∃ ≤3 xA(x) ∧ ∃ ≤2 xB(x)] ∨...  Many, A lot, A few  context dependent

 Barker-Plummer, D. &. (2011). Language, Proof and Logic. Stanford: CSLI Publications.  Cohen, S. (2004). Chapter 14: More on Quantification.  Cummins, C. &. (n.d.). Numerically Quantified Expressions.  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.  Shapiro, S. (n.d.). Numerical Quantifiers and Their Use in Reasoning with Negative Information /~shapiro/Papers/sha79b.pdf