DouglasReay/TheToposOfParadox

Terminology

I'm going to use isolated non-bold capital letters as labels for statements.
I'm going to use paired normal quotation marks to indicate the contents of a statement.
I'm going to use colon-equals to define a statement by allocating a contents string to a label.
Example: A := "1 + 1 = 2"

I'm going to use isolated bold capital letters as truth values
I'm going to use paired back-tick symbols to convert a statement label to the truth value it evaluates to
Example: `A` = T

Introduction

In binary logic, every statement is either true or false.

If it is true, then its contradiction is false.

If it is false, then its contradiction is true.

If statement A is true, and statement B is true, then the statement
"(statement A is true) and (statement B is true)" is also true.

These operators AND, OR, NOT, NAND, NOR, etc can all be defined mathematically as functions which given various truth states represented as numbers (0, 1) as input, will return a number (0, 1) as an output representing the truth of the resulting statement.

There exist sets of operators that work in trinary (0, 1, 2) or (-1, 0, 1), that can also be used to generate 'logic tables'. Sometimes the third value is taken to mean 'undecided'.

Thus

: not 1 = 0
: not 0 = 1
: not undecided = undecided
: 1 or undecided = 1
: 1 and undecided = undecided
: 0 and undecided = 0
: 0 or undecided = undecided
: 1 and 1 = 1
: 1 or 1 = 1
: 0 and 0 = 0
: 0 or 0 = 0
: 1 and 0 = 0
: 1 or 0 = 1
: undecided and undecided = undecided
: undecided or undecided = undecided

Which leads to results such as

: undecided and (not undecided) = undecided

Alternatives to "undecided" include "undecidable", "mu" and "self-referential"

The rules thus defined can be encoded as a [Topos]

4 Value Logic

Suppose instead of 2 or 3 truth values, we have 4. To avoid preconceptions or prematurely allocating numbers to them, I shall refer to these truth values as M, N, Q, R

IDENTITY:

IF the statement

: A

has truth value M, THEN the statement

: "`A` = M"

also has truth value M

NEGATION

The negation of the statement

: A

is the statement

: "`A` =/= M"

IF the statement

: A

has truth value M, THEN its negation has truth value N

Importantly, it is also required for a statement to have truth value M that its negation have truth value N, and it is required for a statement to have truth value N that its negation have truth value M.

Which would normally lead to problems when you get statements such as

: "This statement is true"

(Which is true if it is true, and false if it is false)

That statement has the negation

: "This statement is not true"

(which is true if it is false and false if it is true)

However we still have two truth values to play around with:

POSITIVE SELF-REFERENCE

IF statement A says

: "`A` = M"

THEN statement A has truth value Q

NEGATIVE SELF-REFERNCE

IF statement A says

: "`A` =/= M"

THEN statement A has truth value R

Which solves that little difficulty. We then need to think about the truth value of the following statements:

: Statement A := "`A` = Q"
: Statement B := "`B` =/= Q"
: Statement C := "`C` = R"
: Statement D := "`D` =/= R"

Are these describable within the 4 value system without leading to a paradox, or do you need to keep adding truth values each time?

Workings

Let's think about that first one: A := "`A` = Q"
Its negation (!A) is "`A` =/= M"

If A had truth value M, it would mean that

: "A = Q" had truth value M
: "(A = Q) =/= M" had truth value N

If A had truth value N, it would mean that

: "A = Q" had truth value N
: "(A = Q) =/= M" had truth value M

If A had truth value Q, it would mean that

: ???

If A had truth value R, it would mean that

: ???

Let's think about that second one: B := "`B` =/= Q"
Its negation (!B) is "`B` =/= M"

If B had truth value M, it would mean that

: ???

If B had truth value N, it would mean that

: ???

If B had truth value Q, it would mean that

: ???

If B had truth value R, it would mean that

: ???

Let's think about that third one: C := "`C` = R"
Its negation (!C) is "`C` =/= M"

If C had truth value M, it would mean that

: ???

If C had truth value N, it would mean that

: ???

If C had truth value Q, it would mean that

: ???

If C had truth value R, it would mean that

: ???

Let's think about that fourth one: D := "`D` =/= R"
Its negation (!D) is "`D` =/= M"

If D had truth value M, it would mean that

: ???

If D had truth value N, it would mean that

: ???

If D had truth value Q, it would mean that

: ???

If D had truth value R, it would mean that

: ???

See also: [Saptabhangi]

From [Jain Epistemology] describes the saptabhanginaya or sevenfold predication.

   1. syadasti = Perhaps or maybe, ... it is.
   2. syatnasti = Perhaps or maybe, ... it is not.
   3. syadasti nasti ca = Perhaps or maybe, ... it is, it is not.
   4. syadavaktavyah = Perhaps or maybe, ... it is indeterminate or indescribable.
   5. syadasti ca avaktavya sca = Perhaps or maybe, ... it is and also indeterminate or indescribable.
   6. syatnasti ca avaktavyasca = Perhaps or maybe, ... it is not and also indeterminate or indescribable.
   7. syadasti nasti ca avaktav-yasca = Perhaps or maybe, ... it is and it is not and also indeterminate or indescribable.