User:Thepigdog/Implicit Variable Semantics

Implied Variable Semantics is a proposal to resolve certain self referential paradoxes in natural language, and to fix inconsistencies in the Lambda Calculus. In this approach, a self referential, or self recursive expression is deemed to be associated with a variable. This variable, although not explicit in the expression, is implied by the self referential nature of the expression. The variable acts as a container for the value, and moves the paradoxical condition into a separate explicit condition, which resolves the paradox.

The problem
Many paradoxes in natural language are constructed from self reference. For example the Liar Paradox.


 * This statement is false.

This statement is labelled as the variable X then the statement implies the constraint,


 * $$X = \neg X $$

There is no value of X in {true, false} satisfying this equation.

An even more fundamentally difficult problem occurs for the natural language sentence,


 * Four divided by this expression.

Then, $$X = \frac{4}{X} $$ gives $$X^2 = 4$$

So the value of "Four divided by this expression" is in {2, -2}.

Fundamentals of Mathematics
A fundamental principle of mathematics and logic is that an expression, without free variables represents one and only one value. If this principle is violated, mathematics becomes inconsistent.

The Liar Paradox may not be directly stated in mathematical logic because the naming is not valid. However it may be stated if an expression that converts a reference to a sentence into logic. Such an expression might be called an Eval function. Then,


 * s = "not Eval(s)"

then the expression,


 * $$Eval(s) = \neg Eval(s) $$

so Eval(s) is neither true or false.

Some attempted resolutions of the paradox change the definition of assertion of a statement. However the paradox arises from considering the value of the statement, not from the assertion of the statement. So it is not necessary to assert Eval(s) to be true. The problem is that Eval(s) is a Boolean expression, which may be reduced to a form with no free variables, which is neither true nor false. In standard logic all expressions must be either true or false.

Let expressions
The "let" expression will be used below. It is defined,
 * $$\text{let } C \text{ in } E $$

is defined by,
 * 1) $$\exists f: G(\text{let } C \text{ in } E) = \exists f: C \land G(E)  $$ where f is a tuple of the free variables of C
 * 2) $$\text{let } A \land B \text{ in } E) = \text{let } A \text{ in } (\text{let } B \text{ in } E)  $$

Definition of the tag operator
If N is a variable name and E is an expression then,
 * $$ N:E \equiv \text{let } N = E \text{ in } N $$

For normal expressions E the tag operator only sets N = E. But if E is a self referential expression the tag operator effects the semantics of the expression.

Implied Variable
For self referential expressions there is an implied tag expression, with a new unique variable. For example the self referential expression "this statement" in a statement K then "this statement" may be interpreted as (X:K).

Resolution of paradoxes
The Liar Paradox.
 * This statement is false.

Let K be "this statement", then the statement may be written as,
 * $$\neg (X:K) $$

where K is given by,
 * $$K = \neg (X:K) $$

Using the definition of the tag operator,
 * $$\neg (\text{let } X = K \text{ in } X) $$

and,
 * $$K = \neg (\text{let } X = K \text{ in } X) $$

The last statement implies,
 * $$K = \neg X $$

substituting,
 * $$\neg (\text{let } X = \neg X \text{ in } X) $$

and using [1] from the definition of let,
 * $$(X = \neg X) \land \neg X $$

which is false if X is true or false. So the Liar Paradox becomes only a falsehood.

For the multi-valued expression,
 * Four divided by this expression.

Then,
 * $$\frac{4}{X:\frac{4}{X}} $$

gives,
 * $$\frac{4}{\text{let } X = \frac{4}{X} \text{ in } X} $$

which reduces to,
 * $$\text{let } X^2 = 4 \text{ in } X $$

Because the expression has the free variable X then there is no contradiction in the expression having two possible values.

Use in representing multi-value expressions
In natural language expressions such as,
 * The car can be painter red, white or blue.

This form of expression would be useful in mathematics. It may be defined as,


 * $$ X:(A \lor B) \iff \text{let } X = A \lor X = B \text{ in } X $$

For example,


 * $$ (X = 5 \lor X = 8 \lor X = 13) \land (Y = 2 \lor Y = 3) \land Z = X * Y $$
 * $$ \to Z = X:(5 \lor 8 \lor 13) * Y:(2 \lor 3) $$

may be written with implicit variable semantics as,
 * $$ Z = (5 \lor 8 \lor 13) * (2 \lor 3) $$

Note that when the variables are implicit, the expression is not mathematics. In particular the substitution of expressions may not always be applied.

Conclusion
This is a simple translation, or interpretation, of natural language as mathematics. The interpretation is a natural one in that it reflects the intended meaning of self referential statements in a manner which is consistent with mathematics.

The approach may also be used to resolve inconsistencies with Lambda Calculus considered as a part of mathematics.