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PHIL512 Agenda

Agenda 4/7

  1. Discuss Project Proposal and Project
  2. Proof Contest
  3. Beyond QL
  4. Identity
  5. Numerical quantification: At least one, at most one, exactly one, exceptives, superlatives
  6. Break
  7. Definite Descriptions
    • DDs as names
    • DDS using the reverse-iota operator
    • Russell's approach
  8. Possessives and functions

Project Proposal

See attached. Proposal is due Monday.

Assignments

  1. Due: Project Proposal: Due 4/13 (Monday) before class.

Translating Identity

Translating "at least 1 Z": \((\exists x)Zx\)

Translating "at least 2 Zs".

  1. Can't do \((\exists x)Zx\wedge (\exists y)Zy\) or \((\exists x)(\exists y)(Zx\wedge Zy)\) since \(x\) and \(y\) could refer to the same Z.
  2. Add identity operator: \((\exists x)(\exists y)(Zx\wedge Zy\wedge x\neq y)\)

Translating "at most 1 Z":

"At most 1 Z" means "Not at least 2 Zs" ("1 or 0 Zs").

So, just translate "at most 1 Z" as "at least 2 Zs", then negate the entire wff.

  • Option 1: "Not at least 2 Zs" or "Not 2 and not more than 2": \(\neg (\exists x)(\exists y)(Zx\land Zy\land x\neq y)\)
  • Option 2: Using equiv rules gives us \((\forall x)(\forall y)(Zx\land Zy\rightarrow x=y)\)

Translating "at most 2 Zs"

  • "At most 2 Zs" means "not at least 3"
  • \((\exists x)(\exists y)(\exists z)(Zx\land Zy\land Zz\land x\neq y \land y\neq z\land x\land z)\)

Applying equiv rules:

\(\forall x\forall y\forall z (Zx\land Zy\land Zz \rightarrow x=y \lor x=z \lor y=z)\)

The disjunction of identities ensures that at least one of the vars is identical to the other, ensuring there are no more than 2.

When translating "Exactly 1 Z", you can combine "at least 1 Z" and "at most 1 Z"

  • At least 1 Z: \(\exists x Zx\)
  • At most 1 Z: \((\forall y)(Zz\land Zy\rightarrow x=y)\)

Then you can combine these wffs as:

\(\exists x (Zx\wedge (\forall y)(Zy\rightarrow x=y))\)

Exactly 2 Zs:

\((\exists x)(\exists y)(Zx\wedge Zy\wedge x\neq y\land (\forall z)(Zz\rightarrow z=x \vee z=y))\)

Definite Descriptions

We have various ways to refer to individuals:

  1. Proper names
  2. Demonstratives
  3. Pronouns
  4. Definite descriptions (DD): describe the thing so your description is only satisfied by one object (give a unique / definition description)

DDs are useful b/c:

  1. Pick out an object w/o knowing their name: The kid in the back of the class who put a piece of gum under their desk
  2. Pick out individual w/o ever having met them: The murderer of Jane Doe

In English, the form of a DD is "the F". Idea is "the F" = DD. But this is not the case because (1) there are DDs not in the "the F" form:

  1. "My father" = DD but not in the form "the F"
  2. Proper names = maybe DDs in strange form

There are expressions in "the F" form but are not DDs:

  1. "John is the man" = "the man" is a predicate
  2. "The whale is a mammal" = "any whale is a mammal" (quantifier)

How to add DDs

There are three proposals for adding DDs to QL:

  1. Treat DDs as names (no alteration necessary)
  2. Add DDs using the iota operator
  3. Russell's approach

DDs as names

If DDs are names, then we translate "the F is K" (the man is kind) as \(Kf\).

  • O1: DDs are syntactically composite but names are not.
  • O2: DDs pick out their objects via description but names appear to pick out via direct reference (convention indexing name to object)
  • O3: Negative existentials. A name is interpreted in terms of an item in the domain. So "the unicorn" (\(u\)) picks out an item in the domain. Consider \(u=u\). By \(\exists I\), \((\exists x)(x=u)\). So, there exists unicorns. But now consider the True proposition "unicorns do not exist" \(\neg (\exists x)(x=u)\). Contradiction!
  • O4: Sentences with DDs seem to permit certain entailments not permitted by names.

Here are some examples ("The king is bald" \(Bk\)):

  1. \(Bk\vdash (\exists x)Bx\) - someone is bald
  2. \(Bk\not\vdash (\exists x)Kx\) - there exists a king
  3. \(Bk\not\vdash (\exists x)(Kx\wedge (\forall y)(Ky\rightarrow y=x))\) - there is a unique king

DDs via iota operator

Two problems with treating DDs as names are (1) DDs are composite and (2) DDs refer by describing. Let's address both problems by adding an operator to QL: reverse-iota operator.

  1. Take some open-wff, e.g., \(Wx\) (x is a woman).
  2. Prefix it with the iota operator, e.g., \((\iota x)Wx\)
  3. The value of \((\iota x)Wx\) is the unique object that satisfies the description.
  4. If there is no unique object that satisfies the description (e.g., no object satisfies it, or multiple objects satisfy it), then the expression is undefined.
  5. The expression is a term (terms are names, vars) and so it can be treated that way syntactically: \(T(\iota x)Wx\) is "The woman is tall."

Two more examples:

  1. The angry man is crazy = \(C(\iota x)(Ax\wedge Mx)\)
  2. The boy is tall = \(T(\iota x)(Bx)\)
  3. The boy is taller than the man = \(T(\iota x)(Bx)(\iota y)(My)\)
  4. The woman is the doctor = \((\iota x)Wx= (\iota y)(Dy)\)

Problems:

  • O1: Violates Bivalence. Suppose there are two beautiful men. The expression "The beautiful man is tall" is neither true nor false b/c "the beautiful man" is undefined.
  • O2: Violates Bivalence. Suppose there are no beautiful men. The expression "The beautiful man is tall" is neither true nor false b/c "the beautiful man" is undefined.
  • O3: Violates LEM (Consequence of rejecting bivalence): If ""The beautiful man" is undefined then LEM is not a tautology since "The beautiful man is tall or The beautiful man is not tall" is neither T nor F.
  • O3: Negative existentials. Consider "Pegasus does not exist". Translate as "the winged horse does not exist." This sentence is true. If true, then \((\exists x)x \wedge \iota x (Wx\wedge Hx)\). That is, the truth of the sentence implies there exists a winged horse. But, the sentence says "Pegasus does not exist". Contradiction!

Russell's approach

Russell's approach makes no modification to QL (no new operator). He offers a way to translate sentences with definite descriptions into QL.

First insight: there is a difference between explicit and contextual definitions

  1. Explicit definition: takes isolated expression \(e\) and gives it its meaning
  2. Contextual definition: looks at the sentence as a whole and tells you what the sentence means if a certain expression \(e\) is in it (tells you how the part impacts the whole but never tells you what the part means in isolation)

This is similar to the meaning of \(\forall\) (all) and \(\exists\) (exists). These words do not have any interpretation on their own. But, we can specify what sentences mean that have these quantifiers in them. That is \(\forall\) and \(\forall x\) do not mean anything but \((\forall x)Fx\) means "everything is F"

Second insight: definite descriptions are defined contextually.

The definite description "the king" does not have meaning. But, "the king is F" does.

Third insight: Sentences w/ definite descriptions make three assertions.

Let's consider an example: "The king is bald."

  1. Existence. There is a king: \((\exists x)Kx\)
  2. Uniqueness. There is one and only king: \((\forall y)(Ky\rightarrow y=x)\)
  3. Predication. That king has a property: \(Bx\)

All of the above combine together to give us the following translation:

\((\exists x)(Kx\wedge (\forall y)(Ky\rightarrow y=x)\wedge Bx)\)

The above says: There exists a king and for any object, if that object is a king, then that king is identical to the existent king, and that king is bald. Or, there exists a unique king who is bald.

Strengths of this approach (to name a few):

  1. DDs are composite
  2. DDs "refer" by describing
  3. No new symbols
  4. We get the entailments we want: \(Bk\vdash (\exists x)Bx\), \(Bk\not\vdash (\exists x)Kx\), \(Bk\not\vdash (\exists x)(Kx\wedge (\forall y)(Ky\rightarrow y=x))\)
  5. Negative existentials (see below)

Consider "the king of US does not exist." Translation:

  1. \(\neg (\exists x)(Kx\wedge (\forall y)(Ky\rightarrow y=x))\)
  2. \((\forall x)(Kx\rightarrow (\exists y)(Ky\wedge x\neq y))\)

Note that when saying "the king of US does not exist" you are denying "there is exactly one king". You are not denying the possibility of more than one king. So,

  1. If there are no Kings, then both are true (since there is not exactly one).
  2. If there is exactly one King, then both are false (since this is what it denies).
  3. If there is more than one King, then both are true (since there is not exactly one).

Possessives and Functions

Consider the following possessive noun phrases (along with a function)

  1. My book ("my" - possessive determiner)
  2. Tek's father (genitive noun phrase)
  3. The sum of x and y (a function)

It does not appear we can treat the above as QL-names or predicates:

  • O1: They pick out individuals by being the value of a function
  • O2: Don't give us desired entailments. Example: Tek's father is Irish \(If\vdash (\exists x)Ix\) but not \(If\vdash (\exists x)(Fx\wedge Ix)\) or \(If\vdash (\exists x)Fx\)
  • O3: Not predicates since the possessives give us terms. Example: Tek's father - gives us an object, but Fxt gives us a set.

Possible solution:

  1. add the \(f^n\) or "function$ symbol, where \(n\) is the number of places that the function takes.
  2. value of the function is an object
  3. syntactically behave like terms so we can add predicates to them to create wffs.

Examples:

  1. \(f^1\) - the father of \(x\) - returns a set of pairs \(\langle x, y\rangle\) where the \(x\) is the child and \(y\) is the father.
  2. \(f^1(t)\) - the father of Tek - "Tek's father" - returns Tek's father
  3. \(f^2\) - sum of \(x\) and \(y\) - returns set of pairs \(\langle \langle x, y\rangle, z\rangle\) where \(x, y\) are input nums and \(z\) is output.
  4. $f^2(1,2) - sum of 1 and 2 - returns 3.

Examples in wffs:

  1. Tek's book is large = \(Lf(t)\). Function term is "x's book". Supply the owner and it returns the book owned by that person.

Current Schedule

  • 4/14 - Metaphysics and Logic: Higher-order logic, Leibniz's laws, Lambda abstraction, definitely operator, Evans on vague objects
  • 4/21 - Modal Propositional Logic
  • 4/28 - Presentations
  • 5/3 - Final Project Due