# [FOM] combining probability and logic

Åke Persson ok.person at swipnet.se
Mon Sep 9 04:21:43 EDT 2002

```Jon Williamson wrote:

> One has to be a bit careful about this. Many-valued logics (such as the
> logics of Lukasiewicz) tend to be truth functional (for example value(a&b)
> is a function of value(a) and value(b)), often employing similar rules to
> fuzzy logic for determining values. Probabilities are not
> truth-functional, so many-valued logic as it is normally developed is not
> a logic of probability.  Many-valued logics can be thought of as logics
> for reasoning about partial truth or vagueness, but if these concepts are
> explicated using many-valued logics, then probability cannot be
> interpreted as partial truth.

It is possible to define a logic that both has the same truth-functionality
properties as probabilities AND is many-valued, if it uses partial truth as
measurable proportion of truth (relative occurency, that is not vagueness).
I want to point to this fact. I have defiened such a truth-value space
(polyvalued truth-value space) and such a logic (polyvalued logic), quite
isomorph to baysian logic, but using polyvalued truth values, with a lot of
more logical possibilities and implications. E.g. is modens tollens ruled
out as a general rule of logic (only valid in certain conditions), and
material implication is replaced by a new implication concept that properly
reflects the conditional property of the implication of everyday speaking.
>From these two important facts some paradoxes of standard logic are avoided.
This polyvalued logic is many-valued, but works quite different from other
traditional many-valued logics. It is not truth functional such stright
forward f(p,q), but is on the other hand sure truth-functional from
conjunctions of p and q, i.e. f(p&q,p&~q,~p&q,~p&~q) (i.e. as baysian logic
is).

For them who are interested in combining probability and logic,
it could be a good idea to give the theory at
http://home.swipnet.se/~w-33552/logic/home/index.htm a try.
As I'm not a professional logican and am not able to fufill my project to
all its academic details. I hope there can be other able to see its
possibilities and to formally work it out. I have probably done many
mistakes in my terminology, but some terms are intensionally redefined
and/or used in another way than usual, to provide more precision.