Read: Additional Notes, Earlier Notes or the Paper Draft

I'll further explicate this notion.

Below, I'll sketch out the move from a **Comparative Truth Theory** to **Single Value** semantics.

- A set of Truth-Values T = {T₀, T₁, T₂, T₃, ..., Tₓ}.
- T₀ < T₁ < T₂ < T₃ < ... < Tₓ - e.g. a Horn Scale (or ordering) on T.
- < is a
**total ordering**operator but is also intended to be a**proper subset**here (not "less than").

The

essentialproperty is found initem 3 above: that any interpretation function`I`

will map the set of propositions to`T`

and itstotally-ordered subsets.

This diverges from

Classical Logicsince propositions are mapped to singular atomic elements of a set of truth-values`{T, F}`

.

The key insight involves the way truth-values are structured, are interrelated, and define each other. In Fregean semantics,

`T`

and`F`

are discrete, atomic, and dual opposites.

A **Single-Valued Logic** must also support the following features (as expressed in the the following interpretation assignment example):

- I(p) = T₀
- I(f) = T₁
- I(z) = T₅
- T₀ < T₁ < T₅
- I(p) < I(f) < I(z)

Can we recover it? It seems so:

- {T, F}
- F df= {}
- T df= {F} such that F < T (
**Transmitting Truth Operator**). - Or, via a
**Non-Transmitting Truth Operator**.

I can add in the case of a **Transmitting Truth Operator**:

- T⁺ df= F < T (i.e. the
**Truth Recovery Operator**or T⁺) - Giving rise to the truth-table (which is isomorphic to Boolean negation operations):

P |
~P |
---|---|

T⁺ |
F |

F |
T⁺ |

I call this the

Transmitting Truth Operatorsince every`F`

is contained in the definition for`T⁺`

though, strictly speaking,`T⁺`

is identified with the clause and not the logical connectives themselves.

Alternatively, I can define the **Non-Transmitting Truth Operator**:

- ~T⁺ df= ~F (i.e. the
**Truth Recovery Operator**or T⁺) and T⁺ df= T - Which again gives rise to the truth-table:

P |
~P |
---|---|

T⁺ |
F |

F |
T⁺ |

In this case, I map the two Boolean logical connectives to my set

`T`

such that no`F`

is contained in the definition for`T⁺`

.

##### post: 2/28/2018

##### update: 4/2/2018

##### update: 4/24/2021