<Note: This is in effect an intermediate research report. >

Introduction

There is an older purely algebraic and bivalent system in Logic - EPL (that dates back to the eighties and was part of my M.A.-thesis).

It turns out this may be generalized considerably, and with a nice abbreviated probabilistic notation.

The following is a condensed explanation, that is later to be extended.

What follows is probabilistic. There now also is a new purely algebraic approach. This is probably easier.

I will use several notations that make the following nice syntactical distinctions possible:

"q"      for classical propositional logic
"q=1"   for a brief non-personal probabilistic verstion
"q_a=1"  for a brief personal probabilistic verstion

while the last may also be written as

"aB(q)" or "aBq" or "aB(p(q)=1)" or "p(a,q)=1"

i.e. resp. "a believes q" (bracketed and not) and "a believes the probability of q is 1" and "a's personal probability of q is 1", all as longer versions of "q_a=1".

Here "aBq" - alternative notation "q_a=1" - is notation of LPA for propositional attitudes.

Also, we extend classical propositional logic with three new unary operators:

"-p_a=1" for "a believes p is false"
"?p_a=1" for "a believes p is uncertain"
"?!p_a=1" for "a is not acquainted with p".

Now we adopt the following, while assuming that for any proposition q

q=0 or q=1 or 0 < q < 1

Here it may be noted that this is like saying "x is small or x is tall or x is neither small nor tall", which also can be dealt with intuitively and logically without introducing a third truth-value.

Rules for simple propositions:

 p_a=1    IFF    p_a=1 & ?p_a=0 & ?!p_a=0          IFF  aBp  & -aB?p & -?!aBp
-p_a=1    IFF    p_a=0 & ?p_a=0 & ?!p_a=0            IFF  aB-p & -aB?p & -?!aBp   
?p_a=1    IFF    p_a>0 & -p_a>0 & ?!p_a=0          IFF -aB-p & -aBp  & -?!aBp 
?!p_a=1   IFF    p_a<1 & -p_a<1 & ?p_a<1           IFF -aB-p & -aBp  & -aB?p

~p_a=1   IFF    -p_a=1 V ?p_a=1  V ?!p_a=1        IFF  aB-p  V aB?p  V ?!aBp
~-p_a=1  IFF     p_a=1 V ?p_a=1   V ?!p_a=1        IFF  aBp   V aB?p   V ?!aBp
~?p_a=1  IFF     p_a=1 V -p_a=1  V ?!p_a=1        IFF  aBp   V aB-p  V ?!aBp
~?!p_a=1 IFF     p_a=1 V -p_a=1  V ?p_a=1         IFF  aBp   V aB-p  V aB?p

And one writes standard classical Propositional Logic as normal: "p&q" etc. while one does the same using Personal Probabilities with "p&_aq=1 IFF p_a=1 & q_a=1" using the "=" to indicate assigned probabilities, personal or not.

Rules for compound propositions:

Using these, there are the following stipulations for strong (S) and weak (W) conjunction (&) and disjunction (V):

S&.     p&_aq=1    IFF    p_a=1 & q_a=1   
IFF      aB(p&q)   IFF    aBp & aBq
W&.    p&_aq=1    IFF    p_a>0 & q_a>0   
IFF      aB(p&q)   IFF    -aB-p & -aB-q

SV.    pV_aq=1     IFF     p_a=1 & q_a>0 V q_a=1 & p_a>0  
IFF    aB(pVq)      IFF     aBp&-?!aBq V aBq&-?!aB-p
WV.   pV_aq=1     IFF     p_a>0 & ?q_a=0 V q_a>0 & ?p_a=0  
IFF    aB(pVq)     IFF     -aB-p&-?!aBq V -aB-q&-?!aBp 

while also noting strong and weak negation (-):

S-.    -p_a=1       IFF      p_a=0 & ?p_a=0 & ?!p_a=0 
IFF    aB-p         IFF      ~aBp & ~aB?~p & ~?!aBp
W-.   ~p_a=1       IFF     -p_a>0 V ?p_a>0 V ?!p_a>0          
IFF    aB~p        IFF      ~aBp V ~aB?p V ~?!aBp  

This suggests the following &-i rules

p_a=1 & q_a=1     |- p&_aq=1           IFF   aBp&aBq        |- aB(p&q)       
p_a>0 & q_a>0     |- p&_aq=1           IFF   -aB-p&-aB-q   |- aB(p&q)

and &-e rules

p&_aq=1          |- p_a=1              IFF    aB(p&q)        |- aBp             
p&_aq=1          |- q_a=1              IFF    aB(p&q)        |- aBq  

p&_aq=1          |- p_a>0              IFF    aB(p&q)        |- -aB-p
p&_aq=1          |- q_a>0              IFF    aB(p&q)        |- -aB-q

And this suggests the following V-i rules

p_a=1 & q_a>0   |- pV_aq               IFF    aBp&-aB-q        |- aB(pVq)
q_a=1 & p_a>0   |- pV_aq               IFF    aBq&-aB-p        |- aB(pVq)

p_a>0 & ?q_a=0  |- pV_aq               IFF    -aB-p&-?!aB?q    |- aB(pVq)
q_a>0 & ?p_a=0  |- pV_aq               IFF    -aB-q&-?!aB?p    |- aB(pVq)

and V-e rules

pV_aq=1 & -q_a>0  |- p_a=1              IFF  aB(pVq) & ~aBq   |- aBp 
pV_aq=1 & -p_a>0  |- q_a=1              IFF  aB(pVq) & ~aBp   |- aBq 

pV_aq=1 & -q_a>0  |- p_a>0              IFF  aB(pVq) & ~aBq   |- aBp
pV_aq=1 & -q_a>0  |- p_a>0            IFF  aB(pVq) & ~aBp   |- aBq

Here is a first table that embodies several simplifications to be explained below:

The simplifications are that
(1) in the table the personal suffix is left out, and may be put in as desired (if the same for both p and q)
(2) the cases with the ?! have been left out.

The second point arises from the fact that with propositional atitudes one has, in the proposed notations

q_a V -q_a  V ?q_a V ?!q_a        IFF aBq V aB-q V aB?q V ?!aBq
?!q_a = ~q_a & ~-q_a & ~?q_a   IFF ~aBq & ~aB-q & ~aB?q  

where "?!q_a" is (effectively) "a is not aware of, not acquainted with q".

Normally one will have an explicit or tacit premiss to the effect that ~?!aBq i.e. a is acquainted with q, when considering what a believes about q, but for those who want to face all logical possibilities here is a table:

which may be abbreviated again like this, simply eliminating the cases in ?!:

In almost any case then there will be at least 9 distinct relevant possibilities for logical compounds when considering LPA.

The extended bi-valent propositional logic results from taking all probabilities as being 1 or being 0 or being between 1 and 0, as in

 p_a=1  IFF -p_a=0
-p_a=1  IFF p_a=0
?p_a=1  IFF 0 < p_a < 1

As an aside of the probabilistic notation adopted, another benefit is that it enables an easy brief addition of values, along the same lines:

"[q_a]" for "a's value for q"
"[-q_a]" for "a's value for -q"
"[?q_a]" for "a's value for ?q"

and so on, that also admits - after admitting or when using - non-extreme personal probabilities enable defining concepts like personal expectation of a for q:

exp(a,q)=[q_a].q_a

that is, the product of a's probability and value for q. (It makes sense, incidentally, to use standardized values, that is values converted to a scale between +100 and -100 or +1000 and -1000, or any convenient lower and upper boundaries. Also, clearly and intuitively, negative values of a are undesirable for a, and positive values of a are desirable for a. And one can introduce similar concepts for the society $ - say - that a belongs to, including the possibility that [q_a]≠[q_$] i.e. a's value for q is not the same as a's society $'s value for it, as may happen in fact.)

And it also should be noted that with extended propositional logic as defined, one can do wholly without "-" and "?" by using the theorem that

|-   ~q_a=1 IFF -q_a=1 V ?q_a=1

and therefore likewise one can do wholly without "~" if one desires.

Finally, it is worth noting that whenever -?q_a (which is equivalent with ~?q_a and also with ~?-q_a), the weak and strong logical operators are all equivalent, because in that case ~q_a=-q_a (and effectively and provably everything turns to - equivalents of - classical propositional logic).

But one can admit the case ?q_a i.e. the case that for a some propositions are undecided or uncertain, and give a probabilistic semantics for this that can be given a bi-valent form, as illustrated above.

And here is a sketch of three systems of propositional logic, where I suppose 

p =df q |- p|q.s & q|p.s
p=1     |- p

i.e. if p is q by definition, then p can be substituted for q and q for p in any arbitrary proposition s, and if p has the value 1, then p can be concluded.

For CPL the propositional rules of BL may be presumed. The other e-rules are above, but are all provable, and not listed in the following:

CPL: q=1 or q=0

       ~q=1 =df q=0

       p&q=1 =df p=1 & q=1
       pVq=1 =df p=1 V q=1

       p iff q =df p&q V ~p&~q

EPL: q=0 or q=1 or 0 < q < 1

       ~q      =df    -q V ?q
       -q       =df   ~q&~?q
       ?q        =df   ~q & ~-q

        p=1    =df     p=1 & ?p=0
       -p=1    =df    -p=1 & ?p=0
       ?p=1    =df     p<1 & -p<1

       p&q=1   =df   p=1 & p=1
       p&q=1   =df   p>0  & q>0
       pVq=1  =df    p=1&q>0 V q=1&p>0                  
       pVq=1  =df    p>0 & ?q=0 V q>0 & ?p=0              

       p IFF q =df    p&q V -p&-q V ?p&?q

LPA: q_a=1 or q_a=0 or 0 < q_a < 1 or ?!q_a       IFF   aBq V aB-q V aB?q V ?!aBq |
       ~q_a =df -q_a  V ?q_a V ?!q_a                    IFF   ~aBq
       ?!q_a  =df ~q_a & ~-q_a & ~?q_a               IFF   ?!aBq

        p_a=1    =df   -p_a=0 & ?p_a=0 & ?!p_a=0  IFF    aBp   IFF ~aB-p & ~aB?p & ~?!aBp
       -p_a=1    =df    p_a=0 & ?p_a=0 & ?!p_a=0   IFF    aB-p  IFF ~aBp & ~aB?p & ~?!aBp
       ?p_a=1    =df    p_a>0 & -p_a>0 & ?!p_a=0  IFF    aB?p  IFF ~aBp & ~aB-p & ~?!aBp
       ?!p_a=1   =df    p_a>0 & -p_a>0 & ?p_a<1   IFF    ?!aBp IFF ~aBp & ~aB-p & ~aB?p

       p&_aq=1  =df    p_a=1 & p_a=1                IFF    aBp&q IFF aBp & aBq
       p&_aq=1  =df    p_a>0  & q_a>0               IFF    aBp&q IFF -aB-p & -aB-q

       pV_aq=1  =df    p_a=1&?!q_a=0 V q_a=1&?!p_a=0 
IFF   aBpVq    IFF    aBp&-?!aBq  V aBq&-?!aB-p
       pV_aq=1  =df     p_a>0&?!q_a=0 V q_a>0&?!p_a=0     
IFF   aBpVq   IFF   -aB-p&-?!aBq V -aB-q&-?!aBp  

This works out in terms of tables for strong and weak conjunction as follows (and I adopt the longer more perspicuous notation):

And this works out in terms of tables for strong and weak disjunction as follows:

The tables show the interesting property of there being no weak uncertain binary operators, and indeed also [-aB¬±q]=[aB±q]=[-aB-±q].

There are quite a few details that have to be straightened out. Thus, for substitution of logical equivalents i.e.

       p |- q  & ~?!aBq & ~?!aBq |- aB(p|-q)

there is this:

aB(p |- q) IFF (p |- q) & -?!aBp & -?!aBq   
               IFF p |-_a q IFF p |- q & ~?!p_a & ~?!q_a

c.q.

aB(p |- q) --> aK(p |- q)          IFF p |-_a q --> p |- q

all of which can be summarized by saying that a is a logical reasoner: if a believes a consequence relation, then a's belief is true. And this implies the substitution of logical equivalents of a-acquainted propositions.

This can be refined in various ways, for various kinds of reasoners.

Also, it should be noted all of the above is bi-valent: When this restriction is removed we leave propositional logic and enter probability theory.

And in fact, the above consists in adding a few unary operators to CPL, together with axioms and rules for them, while the main refinement is that of analysing "~q" as "-q V ?q". This also gives rise to ?!q_a for LPA.

Another thing to note is that starting in EPL the analysis of conjunction and especially disjunction is more subtle than in CPL, just as is the analysis of ~.

See also: Algebraic semantics for EPLA

P.S. In so far as this is unclear because it is densely written:
It is a summary of much more research in this field. Also, it is an intermediate and dated report.