Truth-Only Logic #2

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.

Semantics

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

The essential property is found in item 3 above: that any interpretation function I will map the set of propositions to T and its totally-ordered subsets.

This diverges from Classical Logic since 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):

4. I(p) = T₀
5. I(f) = T₁
6. I(z) = T₅
7. T₀ < T₁ < T₅
8. I(p) < I(f) < I(z)

Boolean Algebra

Can we recover it? It seems so:

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

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

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

I call this the Transmitting Truth Operator since 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:

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

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⁺.