Notation note, $[]$ are for lists, $()$ are for functions.

Note that many of the words and symbols I am using are made up. When this maths is better understood, someone should reinvent the notation. My notation isn't that great, but its hard to make good notation when you are still working out what you want to describe.

A theory (of my new type, not standard maths) T is formally defined to be,

$T = [ψ, ρ, Ξ, t y p e, a r i t y]$

Where $ψ = {s_{1}, s_{2}, . . .}$ are the theories symbols. These can be arbitrary mathematical objects.

$Ξ = {Ξ_{1}, Ξ_{2}, . . .}$ is the set of types, also arbitrary.

$t y p e : ψ \to Ξ$ is a function.

a r i t y : ψ \to \cup_{i = 0}^{\infty} Ξ \times Ξ \times \dots \times Ξ      i

Is also a function.

An expression $E$ in a theory is defined recursively, it consists of a pair $E = [s, v_{1}, v_{2}, \dots, v_{n}]$ . Here $s \in ψ$ and $\forall 1 \leq i \leq n : v_{i}$ is an expression.

Let $a r i t y (s) = [x_{1}, x_{2}, . . . x_{m}]$

Sometimes we will write $t y p e (E)$ , what we really mean is $t y p e (s)$ , and $a r i t y (E) = a r i t y (s)$ We write $s y m b o l (E) = s$ when we want to refer to just $s$ and ignore $v_{1}, . . ., v_{n}$

Expressions have the restriction that $m = n$ the number of elements of $Ξ$ that $s$ is mapped to is the same as the number of other expressions.

We also insist that for all $i$ , $t y p e (v_{i}) = x_{i}$

The base case happens when $a r i t y (s) = []$ the empty list.

All expressions are strictly finite mathematical objects. There are no self referential expressions.

Expressions can be written $s (v_{1}, . . ., v_{n})$

We can define equality between expressions $e = s (v_{1}, . . .)$ and $f = t (w_{1}, . . .)$ recursively by saying $e = f$ iff $s y m b o l (e) = s y m b o l (f)$ and forall $i : v_{i} = w_{i}$

The distinction between expressions $e, f$ is defined recursively.

$e - f = [e, f]$ if $s y m b o l (e) \neq s y m b o l (f)$

$e - f = [e, f]$ if $\exists i \neq j \in N : v_{i} \neq w_{i} a n d v_{j} \neq w_{j}$

$e - f = N o n e$ if $e = f$

$e - f = v_{i} - w_{i}$ if $\exists_{1} i : v_{i} \neq w_{i}$

These can be uniquely expressed as strings made up of $Ξ \cup {^{'} (^{'},^{'},^{'},^{'})^{'}}$

Lets define $V (n) = V (Ξ_{n})$ to be the set of all possible expressions of type $Ξ_{n}$ .

A function $f : V (n_{1}) \times . . . V (n_{t}) \to V (m)$ is called basic if it is constant, or it is the projection function ( so $f (e_{1}, . . . e_{t}) = e_{k}$ for fixed $k \leq t$ ) or $f$ can be expressed as

$f (e_{1}, . . . e_{t}) = s (v_{1}, . . . v_{n})$ where $s$ is constant and each $v_{i}$ is a basic function of $e_{1}, . . . e_{t}$ .

Note you can cross as much extra junk into $f$ as you like and ignore it. If $f (e_{1}, e_{2})$ is basic, so is $g (e_{1}, e_{3}, e_{2}, e_{4}) = f (e_{1}, e_{2})$ .

Basic functions can be said to have $t y p e (f) = Ξ_{m}$ and $a r i t y (f) = [Ξ_{n_{1}}, . . . Ξ_{n_{t}}]$

Basic functions can be defined in the style of $f (α, β) = s_{1} (c_{1}, α, s_{2} (α, β))$

Finally, $ρ = {[f_{1}, g_{1}$ ... ,

Logical Counterfactuals and Proposition graphs

Logical Counterfactuals and Proposition graphs