CONCEPTS AS OBJECTS John McCarthy Computer Science Department jmc@cs.stanford.edu http://www-formal.stanford.edu/jmc/

1. Concepts (including propositions) as objects

2. Functions from objects to concepts of them.

3. Concepts and propositions are not a natural kind.

• There are a variety of useful spaces of concepts.

• Concepts are (usually) approximate entities.

Concepts and propositions—1

“...it seems that hardly anybody proposes to use different variables for propositions and for truth-values, or different variables for individuals and individual concepts.”— (Carnap 1956, p. 113).

Variables for propositions and individuals are written in lower case, e.g. p and x. Variables for propositions and individual concepts are capitalized, e.g. P and X.

This talk is about expressiveness rather than for presenting a theory.

Concepts and propositions–2

We write denotes(Mike, mike) or when functional, mike = denot(Mike) T elephone(Mike) is the concept of Mike’s telephone number denot(T elephone(Mike)) = telephone(mike)

Knowing what and knowing that

knows(pat, T elephone(Mike)) Suppose telephone(mike) = telephone(mary) T elephone(Mike) ≠ T elephone(Mary) Possibly , ¬knows(pat, T elephone(Mary)) Truth values and propositions: man(mike) true(Man(Mike) knows(pat, Man(Mike)) means Pat knows whether Mike is a man. Possibly knows(pat, Man(Mike)) ∧ ¬man(mike) k(pat, Man(Mike)) ≡ true(Man(Mike)) ∧ knows(pat, Man(Mike))

Equality and Existence

true(T elephone(Mike) EqualsC T elephone(Mary), although T elephone(Mike) ≠ T elephone(Mary) telephone(denot(Mike)) = telephone(denot(Mary)) telephone(mike) = telephone(mary) denot(T elephone(Mike)) = denot(T elephone(Mary))

(∀X)(exists(X) ≡ (∃x)denotes(X, x))

ishorseCP egasus W inged(P egasus) ?true(Winged-Horse (P egasus)) true(Greek mythology , Winged-Horse (P egasus)) ¬exists(P egasus)

We can have

(∃X)(exists(Greek Mythology , X) ∧ Winged-Horse (X)),

but most likely, there doesn’t have to be a domain of Greek mythological objects. This suggests that some of the rules of inference of predicate logic be weakened for such theories.

About propositions

true(Not(P)) ≡ ¬true(P) true(P And Q) ≡ true(P) ∧ true(Q) ? P And Q = Q And P ? P And (Q Or R) = (P And Q) Or (P And R)

This way lies NP-completeness and even undecidablity of whether two formulas name the same proposition.

Functions from things to concepts

Numbers can have standard concepts Concept1(n) is a certain standard concept of the number n. Writing Concept1 suggests that there might be another mapping Concept2 from numbers to concepts of them.

We can have ¬knew(kepler, CompositeC(Number(P lanets))), and also knew(kepler, (CompositeC(Concept1(denot(Number(P lanets)))))).

Functions from things to concepts–2

Russell’s example: I thought your yacht was longer thann it is. can be treated similarly, although it requires a function going from the concept Length(Y ouryacht) to what I thought its value was.

denot(I, Length(Y ouryacht)) > length(youryacht)

We may also want a map from things to concepts of them in order to formalize a sentence like, “Lassie knows the location of all her puppies”. We write this

(∀x)(ispuppy(x, lassie) ⊃ knowsd(lassie, LocationdC(Conceptd(x))). Conceptd takes a puppy into a dog’s concept of it, and Locationd takes a dog’s concept of a puppy into a dog’s concept of its location. The axioms satisfied by knowsd, Locationd and Conceptd can be tailored to our ideas of what dogs know.

(∃n2)(k(pat, Concept2(n2) EqualsC T elephone(Mike)))

≡ knows(pat, T elephone(Mike)) or knows(pat, T elephone(Mike))

≡ denot(pat, T elephone(Mike)) = telephone(mike).

Concepts as approximate entities

• Approximate entities occur in human common sense reasoning. They don’t have if-and-only-if definitions, e.g. the rock and ice constituting Mount Everest.

• The set of individual concepts of Greek mythology is another approximate entity. Few of them have denotations.

• The logical way of handling approximate entities is to axiomatize them weakly. Did Pegasus have a mother?

• exists(Greek Mythology , P egasus), ¬exists(Greek Mythology , T hor), ¬exists(Greek Mythology , George Bush ), exists(Greek Mythology , Mother(P egasus))?

Mr. S and Mr. P

Two numbers m and n are chosen such that 2 ≤ m ≤ n ≤ 99. Mr. S is told their sum and Mr. P is told their product. The following dialogue ensues:

Mr. P: I don’t know the numbers. Mr. S: I knew you didn’t know. I don’t know either. Mr. P: Now I know the numbers. Mr S: Now I know them too.

In view of the dialogue, what are the numbers?

“Two puzzles involving knowledge” www-formal.stanford.edu/jmc/puzzles.html

Formalizing Mr. S and Mr. P

knows(person, pair, time), k(person, P roposition, time) persons: s, p, sp ¬knows(p, P air0, 0) knows(s, Sum(P air0), 0) knows(p, P roduct(P air0), 0) (∀pair)(sum(pair) = sum(pair0)

→ ¬k(s, Not(P air0 Equal Concept1(pair)), 0)) k(sp, …, 0)

In the paper A(w1, w2, person, time) means that in world w1, world w2 is possible for person at time.

“Two puzzles involving knowledge” www-formal.stanford.edu/jmc/puzzles.html