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# Elementary first order formulations

The impossibility statement is readily formulated as a sentence of the predicate calculus, but I don't see how the parity and counting argument can be translated into a guide to the method of semantic tableaus, into a resolution argument, or into a standard proof.

The word ``elementary'' is used in the sense that the quantifiers range over numbers and not over sets.

The formulas are here for completeness. If you don't like reading them, the rest of this section may be skipped.

We number the rows and columns from 1 to 8 and we introduce predicates S(x,y), L(x,y), E(x,y), , , , , and with the following intended interpretations:

S(x,y) means y = x + 1.

L(x,y) means x < y.

E(x,y) means x = y.

means the square (x,y) and the square (x+1,y) are covered by a domino.

means the square (x,y) and the square (x,y+1) are covered by a domino.

means the square (x,y) and the square (x-1,y) are covered by a domino.

means the square (x,y) and the square (x,y-1) are covered by a domino.

means the square (x,y) is not covered.

We shall axiomatize only as much of the properties of the numbers from 1 to 8 as we shall need.

1. .
2. .
3. .
4. E(x,x).

These axioms insure that all eight numbers are different and determine the values of S(x,y), L(x,y), and E(x,y) for .

5. These axioms insure that every square (x,y) satisfies exactly one .

6. .
7. These axioms insure that the uncovered squares are precisely (1,1) and (8,8).
8. These axioms state the conditions that a pair of adjacent squares be covered by a domino.

These axioms state that the dominoes don't stick out over the edge of the board.

Suppose we had a model of these 15 sentences (in Robinson's clausal formalism, there would be 31 clauses). There would have to be eight individuals satisfying the relations asserted for in the axioms. They would have to be distinct since axioms 1,2, and 3 allow us to prove L(x,y) whenever this is so and axioms 4 and 5 then allow us to show that L(x,y) holds only for distinct x and y.

We then label the squares of a checkboard and place a domino on each square (x,y) that satisfies or sticking to the right or up as the case may be. Axioms 13 and 14 insure that the dominoes don't overlap, axioms 6-12 insure that all squares but the corner squares are covered and axiom 15 insures that no dominoes stick out over the edge.

Since there is no such covering the sentences have no model and are inconsistent.

In a formalism that allows functions and equality we have a briefer inconsistent set of sentences involving

s(x)the successor of x

g(x,y) has value of 1 to 5 according to whether or or

The sentences are

1. s(s(s(s(s(s(s(s(8)))))))) = 8.
2. The sentences insure the existence of 8 distinct individuals using a cyclic successor function.

3. Insures that exactly the corner squares (1,1) and (8,8) are uncovered.

4. Each domino covers two adjacent squares

5. Dominoes don't stick out

6. .
7. .

This identifies the numbers used and ties down the values of g(x,y).

Not only is this language incapable of expressing the colors of the squares. It is also incapable of expressing the counts of the numbers of squares with a given property, the latter being required for expressing the Minsky, Winograd and Stefanuk proofs.

Next: Expressing the creative ideas Up: CREATIVE SOLUTIONS TO Previous: Pinning down the ideas

John McCarthy
Mon Mar 29 15:20:19 PST 1999