The Truth Inside One
One folio, two stages
The final Archive folio supplies one finite two-stage model. Everything used below is written on the page, and its logic belongs to this specific presheaf category.
The two-stage category
Let have two objects:
- : the whole plate;
- : a narrow window.
Besides the identity arrows, there is one arrow .
A presheaf reverses that arrow. For this category, a presheaf is exactly two sets and one restriction function:
A map is a pair of functions
that respects restriction:
A candidate terminal presheaf
A presheaf is terminal if every presheaf has exactly one map .
1. Prove terminality
Use separate paper for your verification.
Let be any presheaf.
- Write the only possible component functions and .
- Verify the restriction equation.
- Conclude that is terminal in the category of these presheaves.
Subpresheaves as global truth values
A subpresheaf of chooses subsets
and must be closed under restriction. Here that means
Following standard categorical language, we call these subobjects of global truth values.
Throughout this folio, truth value means one of these global truth values in this supplied two-stage presheaf category.
2. Check every raw pair
Pairs are always written in the order . Complete both blank columns.
| Restriction condition? | Valid global truth value? | ||
|---|---|---|---|
3. Order the valid values
Complete the chain on this page.
Order global truth values stage by stage:
Put the three valid ordered pairs into the chain from least to greatest. Name the least value , the middle value , and the greatest value .
Carry your chain forward
Use the values that you named in Part 3. Keep your completed chain visible, and continue to write every pair in order.
The logic rules
For global truth values , define AND and OR stage by stage. At either stage ,
Define to be the largest valid global truth value for which
4–7. Test the logic
Use separate paper for your computations and explanations.
-
Flip presence and absence of at both stages of your middle value . Write the apparent complement, then explain why it is not a valid global truth value.
-
Compute , , and . Justify each answer using all three possible values .
-
Compute for each . The classical law of excluded middle says for every global truth value. Which value disproves that law here?
-
Compute . Compare it with .
Scope note: these computations concern the global truth values and logic internal to the one presheaf category defined on page 1. They do not claim that ordinary reasoning—or the Manyfold as a whole—is three-valued.