Extended
Propositional Logic:
A standard bivalent propositional logic that is able to
distinguish between "it is not true that" and "it is false that", by considering
the former as weaker than the latter, which thereby also has the ability
to distinguish statements that are undecided (undetermined, uncertain), by
defining them as the statements that are neither true nor false. The
interesting fact is that indeed this can be achieved with a
bivalent propositional truthfunctional logic that is a simple extension of standard classical
propositional logic.
This is shown in what follows, using ordinary algebra, and building on the
work done for Classical Propositional Logic.
1. Introduction:
That
"it is not true that" and "it is false that" may be motivated in many different ways. Here I shall
do it by considering the following two arguments, one relating intuitively to
space, the other to time.
First, there is e.g. "it's not true there's
extraterrestial intelligence and it's not false either, because we have
neither found it nor searched exhaustively through all the known universe".
The point is not whether or not there is extraterrestial intelligence (or
what this would mean), but that one may in many cases plausibly deny that a
proposition is true, on the ground it has not been established as true and
deny that the same proposition is false, on the ground that it has also not
been established as false.
Second, there is Aristotle's example, who knew that two
fleets were encircling one another preparing for battle or escape: In such a
case "it is not true that tomorrow there is a seabattle and it is not
false either, because what will happen is not yet decided in this and many
similar cases", of socalled "future contingents".
The conclusion that is usually drawn from arguments like
this is that to do justice to such cases one must reject the postulate of
bivalence, and introduce a third truthvalue, and perhaps a fourth, a fifth
a.s.o.
That conclusion is not drawn here. Instead, we maintain
the postulate of bivalence but introduce a new negation, so that now we have
the possibility of saying "it is false that p" i.e. "p"
and "it is not true that p" i.e. "~p", and also have the
possibility of saying "it is neither true nor false that p" i.e.
"~p&~p" i.e. in brief "?p", which may be read as
"p is undecided" or "p is uncertain". (I tend to use
"undecided" for cases of a spatial nature, and "uncertain"
in temporal cases.)
And the basic reason for distinguishing between it is not
true that p and it is false that p, and therewith the definable notion of p
is undecided or p is uncertain is that there intuitively are possibilities
that are neither true nor false but uncertain, in the sense that there are
many domains that are not fully known, not fully decided, not fully certain
for some reason.
Two important domains of this kind where this distinction applies are
the past and the future, at least if one assumes that what's past is done and
settled, and definitely so or definitely not so (whether or not that is known)
while part of the future is neither done nor settled and neither definitely so
nor definitely not so (and that is what we seem to know about the future).
However, once the basic distinction is drawn
between the two negations, there are
as many possible applications as there are appropriate domains one's
statements are about where a distinction of the given form applies.
There are many possible ways of providing a semantics for
these ideas, but it seems the clearest way is to follow the ideas used above
to give a semantical foundation for CPL, and add to CPL the following
postulates:
2. Syntactics of EPL
As for
CPL plus:
"" and "?" are unary operators.

EPL builds on CPL, and therefore little is
needed in the way of added syntax.
3. Semantics for EPL:
For the semantics for EPL I choose here to do it algebraically, and in
terms of four axioms:
As for CPL plus:
E+ [+p]+[p]+[?p]
= 1
E [p]=[+p]
E? [?p]=[+pVp]
E& [(+p&+q)]=[+pV+q]

Each of these axioms can be motivated intuitively:
E+ parallels C+ by simply refining [~p] as [p]+[?p];
E parallels the theorem [~~p]=[p];
E? says what's the case if it is false that p is undecided/uncertain:
then p is either true or false; and
E& gives the De Morgan law for strong denial, which we shall call
negation.
Note that EPL is an extension of
CPL: Semantically and
syntactically, EPL = CPL + the above four postulates, apart from the
necessary syntaxis for the new operators, that also conforms to CPL. We shall
inquire below into the precise relation between EPL and CPL, and find that
the present claim is correct and can be precisified.
As with CPL, one basic aim will be to derive truthtable
definitions. To do so, here are some theorems with comments.
4. Theorems of EPL in 1 variable:
The first theorem is used to establish later theorems. It
also incorporates several statements concerning consistency: [p&p]=0
etc. that may be summarized as: a proposition and its negation or uncertainty
are never both true. (The same holds for a proposition and its denial, both
in CPL and EPL.)
T1 
[p][p]=[p][?p]=[p][?p]=0
[p&p]=[p&?p]=[p&?p]=0 
Contradictions 
(1) 
[p]+[p]+[?p]=1

E+ 
(2) 
[p][p]+[p][p]+[p][?p]=[p] 
Algebra, (1) 
(3) 
[p]+[p][p]+[p][?p]=[p] 
Algebra and BV 
(4) 
[p][p]+[p][?p]=0 
Algebra, (3) 
(5) 
[p][p]=[p][?p]=0

BV, Algebra. (4) 
The last line follows from the line before it because if a
sum of two quantities is 0 and both quantities are either 0 or 1 both must be
0 by BV (of the four possibilities 1+1, 0+1, 1+0, 0+0 only the last sums to
0).
The second equation on the first line is proved likewise,
and the other equations follow by P&.
T2 establishes a fundamental fact about the negation of
an uncertain proposition: it has the same value as the denial of the same
uncertain proposition:
T2 
[?p]=[~?p]

Denial of uncertainty is
negation of uncertainty

(1) 
[?p]=[pVp] 
E? 
(2) 
=[p]+[p][p][p]

PV, (1) 
(3) 
=[p]+[p]

T1, (2) 
(4) 
=1[?p]

E+, (3) 
(5) 
=[~?p] 
P~, (4) 
T3 establishes that the assertion that a proposition is
uncertain has the same value as the assertion that the negation of the
proposition is uncertain:
T3 
[?p]=[?p]

Uncertainty of
truth is equivalent to uncertainty of falsity 
(1) 
[~?p]=~[pVp] 
E? 
(2) 
= 1([p]+[p][p][p])

PV, (1) (CPL keeps applying) 
(3) 
= 1([p]+[p][p][p]) 
E, (2) 
(4) 
= 1[pVp] 
PV, T1, (3) 
(5) 
= [~?p] 
E+, (4) 
T4 establishes that the assertion that an uncertain proposition is itself
uncertain is always false. (Of course, T4 can be restated as
[??p]=[~??p]=1.)
T4 
[??p]=0

There are no true uncertain
uncertain propositions 
(1) 
[?p]+[?p]+[??p]=1 
E+ 
(2) 
[??p]=1([?p]+[?p]) 
Algebra, (1) 
(3) 
=1([?p]+[~?p]) 
T2, (2) 
(4) 
=11=0 
P~, (3) 
T5 establishes that the value of any proposition equals
the conjunction of the denials of the same proposition when negated and when
uncertain:
T5 
[p]
=[~p][~?p]=[~p&~?p]
[p]=[~p][~?p]
=[~p&~?p]
[?p]=[~p][~p]
=[~p&~p]

Fundamental EPL
equivalences 
(1) 
[~p][~?p]=(1[p])(1[?p]) 
P~ 
(2) 
=1[p][?p]+[p][?p] 
Algebra, (1) 
(3) 
=1[p][?p] 
T1, (2) 
(4) 
=[p] 
E+, (3) 
This proves the first equation. The other two are proved
likewise. Qed.
T6 shows denial can be wholly avoided and defined in
terms of ,? and V. Incidentally, this shows that the classical denial used
in CPL indeed is systematically ambiguous, at least from the point of view of
EPL: saying something is not so generally is ambiguous between claiming it is
false or it is uncertain.
T6 
[~p]=[pV?p]
[~p]=[pV?p]
[~?p]=[pVp]

Denials eliminable 
(1) 
[~p]=1[p] 
P~

(2) 
=[p]+[?p] 
E+ 
(3) 
=[p]+[?p][p][?p] 
T1 
(4) 
=[pV?p] 
PV 
The first equation is proved as indicated, and the second
likewise. The third already follows by T2 and E?.
T7 shows that a negated denial is like a double denial or
double negation:
T7 
[~p]=[p] 
Negation of denial
is affirmation 
(1) 
[~p]=[(pV?p)] 
T6 
(2) 
=[p&?p] 
E, (1) 
(3) 
=[p&?p] 
E, (2) 
(4) 
=[p][?p] 
P&, (3) 
(5) 
=[p](1[?p]) 
T2, (4) 
(6) 
=[p][p][?p] 
Algebra, (5) 
(7) 
=[p] 
T1, (6) 
So now we can write a truthtable for propositions with
one variable. Note that it is considerably larger than for CPL, since there
are more distinct cases to include. Also  and different in principle from the
standard truthtables for standard CPL  the leftmost column gives the
fundamental possibilities for p (thus avoiding the need for three
truthvalues) while the upper row gives the diverse propositions with
"p" that are evaluated for each fundamental possibility.
Fundamental truthtable for EPL

Pos

"p"

"p"

"?p"

"~p"

"~p"

"~?p"

"?p"

"?p"

"??p"

"p"

"~~p"

"~p"

'p'

1

0

0

0

1

1

1

0

0

1

1

1

'p'

0

1

0

1

0

1

1

0

0

0

0

0

'?p'

0

0

1

1

1

0

0

1

0

0

0

0

The point of the two styles of quotationmarks in the above table is to
distinguish between the propositions that are evaluated, listed in the top
row between double quotes, so that "p" may conveniently be read as
"the proposition that p", and the possible facts that would make
these propositions true or false, so that 'p' may be conveniently read as
"the fact that p".
Note that both "p" and 'p' are linguistic
notation, the first for the linguistic expression the double quotes embrace,
the second for the possible fact represented by the linguistic expression the
single quotes embrace. Also, the map that is encoded by the above table may
be read in two ways: from singlequoted formulas to double quoted formulas,
or from possible facts represented by single quoted formulas to ideas
represented by double quoted formulas.
In the tables that follow both kinds of quotation
marks will be
left out as understood for reasons of space and clarity. (And the reader who
is familiar with logic should realize that the present approach to
truthtables and semantics of propositional logic is in some ways subtly
different from the received way, even though the resultant tables look much
the same.)
5. Theorems of EPL in 2 variable:
T8 establishes a kind of general expansion law,
corresponding to the PL version [p]=[p&q]+[p&~q], which may be
proved in a similar way:
T8 
[p] =[ p&q]+[
p&q]+[ p&?q]
[p]=[p&q]+[p&q]+[p&?q]
[?p]=[?p&q]+[?p&q]+[?p&?q]

EPLExpansion 
(1) 
[p]=[p]([q]+[q]+[?q]) 
E+

(2) 
=[p][q]+[p][q]+[p][?q] 
Algebra, (1) 
(3) 
=[p&q]+[p&q]+[p&?q] 
T1, (2) 
The first equation is proved as indicated.The other two
are proved likewise. Qed.
T9 uses T8 to establish the EPLcounterpart of
[p&q]+[p&~q]+[~p&q]+[~p&~q] =1, that may be proved in the
same manner:
T9 
[p&q]+[p&q]+[p&?q]+
[p&q]+[p&q]+[p&?q]+
[?p&q]+[?p&q]+[?p&?q]=1

EPLTautology 
The proof uses T8 and E+. Qed.
Note that for CPL there are four distinct possibilities
for any two distinct propositions, whereas for EPL there are nine distinct
fundamental possibilities for any two distinct propositions (which also shows
EPL cannot coincide with a 3valued logic, since that distinguishes instead
2^3=8 distinct fundamental possibilities: EPL just involves a somewhat finer
grid of logical distinctions, so to speak.)
Note also that by Bivalence and Algebra it follows as in
CPL that precisely one of these possiblities will hold in any case any of
these may hold or else none holds, if the proposition is a contradiction.
T10 shows that an uncertain conjunction has the same
value as either conjunct being true and the other uncertain or both being
uncertain. It also shows that ? in front of a conjunction can be worked
inwards:
T10 
[?(p&q)]=[p&?q]+[?p&q]+[?p&?q]

Uncertainty conjunction 
(1) 
[?(p&q)]
=1[p&q][(p&q)] 
E+

(2) 
=1[p&q][pVq)] 
E&, (1) 
(3) 
=1[p&q]([p]+[q][p][q]) 
PV, (2) 
(4) 
=1[p&q][p&q][p&q]
[p&?q][q&p][q&p]
[q&?p]+[p][q]

T8, (3) 
(5) 
=1[p&q][p&q][p&q]
[p&?q][q&p][q&p]
[q&?p]+[p&q] 
P&, (4) 
(6) 
=1[p&q][p&q][p&?q]
[p&q][p&q][q&?p] 
Algebra 
(7) 
=[p&?q]+[?p&q]+[?p&?q] 
T9, (6) 
T11 expands E to a more precise statement. Note that after
this point the truthtable for conjuctions can be written:
T11 
[(p&q)]=[p&q]+[p&q]+[p&q]+[p&?q]+[?p&q]


The proof is by T9, T10, P&, E and E+.
Here is the truthtable for the basic conjunctions, which
one may compare with one's intuitions, noting especially (1) that a
conjunction is uncertain iff one conjunct is true and another uncertain or
both conjuncts are uncertain and (2) that in fact the frontal ""
and "?" can be worked inside i.e. detached from the compound
formula and attached to (a disjunction of conjunction of) its components like
T10 and T11 show.

(p&q)

(p&q)

?(p&q)

p q

1



p –q


1


p ?q



1

p q


1


p q


1


p ?q


1


?p q



1

?p q


1


?p ?q



1

T12 parallels T10: An disjunction is uncertain iff either
one disjunct is false and the other uncertain or both disjuncts uncertain:
T12 
[?(pVq)]=[p&?q]+[?p&q]+[?p&?q]

Uncertainty disjunction 
(1) 
?(pVq)
=~(pVq)&~(pVq)

T5

(2) 
=~p&~q&(~pV~q) 
PL 
(3) 
=~p&~q&~p
V ~p&~q&~q 
PL 
(4) 
=?p&q V
?p&?q V ?q&p 
T5 
T13 establishes the counterpart of EM, i.e. the dual of De
Morgan's Law. After this point we also can write the EPLtruthtable for V.
T13 
[(pVq)]=[p&q]

De Morgan for negation 
(1) 
[(pVq)]=1[pVq][?(pVq)] 
E+

(2) 
=1[p][q]+[p&q]?(pVq) 
PV, (1) 
(3) 
=1[p&q][p&q][p&?q][q&p]
[q&p][q&?p]?[pVq] 
T5, (2) 
(4) 
=[p&q]+[p&?q]+
[?p&q]+[?p&?q]?[pVq] 
T9, (3) 
(5) 
=[p&q]+[p&?q]+
[?p&q]+[?p&?q]
[?p&q][?p&?q][?q&p] 
T11, (4) 
(6) 
=[p&q] 
Algebra, (5) 
Here is the truthtable for the basic disjunctions:

(pVq)

(pVq)

?(pVq)

p q

1



p q

1



p ?q

1



p q

1



p q


1


p ?q



1

?p q

1



?p q



1

?p ?q



1

6. Theorems about basic uncertainties
In a sense, the semantical basis of EPL has been
completed, for as in the case of CPL any formula may be replaced by an equivalent
DNF (=Disjunctive Normal Form: A disjunction of mutually exclusive
conjunctions). However, we are interested in more than the minimally adequate
semantical basis, and want to know something about the behaviour of ?; how we
may introduce implication and equivalence; and what we can say in the context
of EPL about the problems we have raised about CPL.
So let's first consider ?.
First, let's prove two theorems that parallel T3:
[?p]=[?p]. These have the same import: For simple propositions, conjunctions
and disjunctions, the uncertain true cases have the same values as the
uncertain false cases:
T14 
[?(p&q)] =[?(p&q)]


(1) 
[?(p&q)]
=
[p&?q]+[?p&q]+[?p&?q] 
T10

(2) 
[?(p&q)]
=
[?(pVq)] 
E 
(3) 
= [p&?q]+[?p&q]+[?p&?q]

E, T12 
And
T15 
[?(pVq)]=[?(pVq)]


(1) 
[?(pVq)] =[p&?q]+[?p&q]+[?p&?q] 
T12

(2) 
[?(pVq)]=[?(p&q)] 
T13 
(3) 
=[p&?q]+[?p&q]+[?p&?q]

E, T10 
Next, while we have that [?p]=[?p],
we also have that denials when fronted with ? are always false:
T16 
[?(~p)]=0


(1) 
[?(~p)]=[?(pV?p)] 
T6

(2) 
=[p&??p]+[?p&?p]+[?p&??p]

T12 
(3) 
=0 
T1, T4 
We can restate T16 thus, stressing
several tautologies in only one proposition. Indeed, in T17 the basic
tautology of CPL is restated equivalently in several ways not available in
CPL:
T17 
[~?(~p)]=[?(~p)]=1

EPLtautologies in
one proposition 
(1) 
=[~pV~p] 

(2) 
=[~pVp] 

(3) 
=[~(p&p)] 

(4) 
=[~(p&~p)] 

Next, we can use T10 to derive
the various cases of conjunctions of uncertain propositions. The first two
are nearly immediate from T10:
T18 
[?(p&q)]=[p&?q]+[?p&q]+[?p&?q]

T10 
And
T19 
[?(p&q)]=[p&?q]+[?p&q]+[?p&?q]

T10 
The next two can also be read off from earlier results,
but I'll give the proof of the first, showing the interplay between denial,
negation, and uncertainty:
T20 
?(p&?q)]=[?p&q]+[?p&q]


(1) 
[?(p&?q)] =
[~(p&?q) & ~(p&?q)]

E?

(2) 
= [(~pV~q) & ~(p V ?q)] 
E~, E, (1) 
(3) 
= [(~pV~q) & ~p & ?q)] 
E~, (2) 
(4) 
= [?p&q V ?p&q] 
E?, (3) 
(5) 
= [?p&q]+[?p&q] 
E+, (4) 
And
T21 
[?(p&?q)]=[?p&q]+[?p&q]

T20 
The last two cases involving uncertain conjunctions
involving uncertain propositons conform to T4:
T22 
[?(?p&?q)]=1 

(1) 
[?(?p&?q)]=[~?(?p&?q)] 
T4 
(2) 
=[~(~(?p&?q) & ~(?p&?q)] 
T5, (1) 
(3) 
=[~((?pV?q) & (?p&?q)] 
T4, (2) 
(4) 
=[(?p&?q) V ~(?p&?q)] 
T4, (3) 
(5) 
=1 
P+, (4) 
And so
T23 
1[?(?p&?q)]=1

T22, T4 
Note that there are three possibilities for a conjunction
of two certain propositons to be uncertain; two possibilities for a
conjunction of a certain and an uncertain proposition to be uncertain; while no
conjunction of two uncertain propositions can ever be uncertain. This
also may be stated as follows:
T24 
[+(?p&?q) V (?p&?q)]=1

A conjunction of two
uncertainties is true or false 
T12 can likewise be used to derive the various cases of disjunctions
of uncertain propositions that mirrors the case for uncertain conjunctions in
a way similar to De Morgan's Laws. The proofs are similar to the case of
conjunction:
T25 
[?(pvq)]=[p&?q]+[?p&q]+[?p&?q]

T12 
T26 
[?(pvq)]=[p&?q]+[?p&q]+[?p&?q] 
T12 
T27 
[?(pV?q)]=[?p&q]+[?p&q] 
T12 
T28 
[?(pV?q)]=[?p&q]+[?p&q] 
T12 
Here is a set of noteworthy consequences of T20, T21, T25
en T26:
T30 
[?(pV?q)] =[?(pV?q)]


(1) 
=[?(p&?q)] 

(2) 
=[?(p&?q)] 

(3) 
=[?p&q]+[?p&q]=[+(?pV?q)]


This shows that uncertain conjunctions and disjunctions
of a certain and an uncertain proposition amount to the same. An intuitive way
of making sense of this in terms of time is that such disjunctions and
conjunctions refer in one conjunction or disjunction to both the future and
the nonfuture (past or present).
Here are the basic tables for conjunctions and
disjunctions collected in one table:

(p&q)

(p&q)

?(p&q)

(pVq)

(pVq)

?(pVq)

p q

1



1



p q


1


1



p ?q



1

1



p q


1


1



p q


1



1


p ?q


1




1

?p q



1

1



?p q


1




1

?p ?q



1



1

7. Theorems about compounded uncertainties
T31 uses T10 and T11 to state when the possible values of
an EPLform in two variables are classical: Iff both their conjunction and
disjunction is certain.
T31 
[?(p&q) &?(pVq)]=
[p&q]+[p&q]+[p&q]+[p&q] 
EPL to CPLresolution 
(1) 
[?(p&q)&?(pVq)]=[?(p&q)][?(pVq)]

P& 
(2) 
=[~?(p&q)][~?(pVq)] 
T2, (11) 
(3a) 
=[[p&q]+[p&q]+[p&q]+
[p&q]+[p&?q]+[?p&q]]

T10, (2) 
(3b) 
*
[[p&q]+[p&q]+[p&?q]+
[p&q]+[p&q]+[?p&q]]

T12, (2) 
(4) 
=[p&q]+[p&q]+[p&q]+[p&q] 
T1, (3a), (3b) 
Note this does give a fairly neat criterion and
distinction: two propositions are both either true or false precisely if both
their conjunction and their disjunction is either true or false. Of course, this
also runs the other way: Of two propositions at least one is uncertain iff
their conjunction or disjunction is uncertain.
In the proof of T31 the third and fourth line are one
long multiplication printed over two lines for readability. This will be done
below also without clarification. Also T1 is applied to quickly dispose of
inconsistent conjunctions.
It helps intuitively to consider the other three similar
cases.
I start with an uncertain conjunction but a decided
disjunction: This turns out as equivalent with one of the two propositions
true and the other uncertain
T32 
[?(p&q) &
?(pVq)]=[p&?q]+[?p&q]


(1a) 
[?(p&q) & ?(pVq)]=
[[p&?q]+[?p&q]+[?p&?q]]* 
T9, T10, T12

(1b) 
[[p&q]+[p&q]+[p&?q]+[p&q]+
[p&q]+[?p&q]] 

(2) 
=[p&?q]+[?p&q] 
T1, (1a), (1b) 
Next, a decided conjunction but an uncertain disjunction
of two propositions is equivalent with the falsity of one proposition and the
uncertainty of the other:
T33 
[?(p&q) &
?(pVq)]=[p&?q]+[?p&q]


(1a) 
[?(p&q) &
?(pVq)]=
[[p&q]+[p&q]+[p&q]+
[p&q]+[p&?q]+[?p&q]]* 

(1b) 
[[p&?q]+[?p&q]+[?p&?q]] 

(2) 
=[p&?q]+[?p&q] 

Finally, when both the conjunction and disjunction of two
propositions are uncertain, both propositions are uncertain, and conversely:
T34 
[?(p&q) &
?(pVq)]=[?p&?q]


(1a) 
[?(p&q) &
?(pVq)]=[[p&?q]+[?p&q]+[?p&?q]]*


(1b) 
[[p&?q]+[?p&q]+[?p&?q]] 

(2) 
=[?p&?q] 

Here is a table summarizing T30T32, showing some neat
symmetries, that also show that combinations of conjunctions and disjunctions
can neatly isolate the possibilities with no uncertainties, one uncertainty
and two uncertainties.
It is also interesting to note  in the middle two
columns  that in the most interesting case of one uncertainty, it does not
follow which of the two propositions is uncertain:

?(p&q)&?(pVq)

?(p&q)&?(pVq)

?(p&q)&?(pVq)

?(p&q)&?(pVq)

+p+q

1




+p q

1




+p ?q


1



p +q

1




p q

1




p ?q



1


?p +q


1



?p q



1


?p ?q




1

8. Theorems about implication and equivalence
Now let's briefly consider implication and equivalence in
EPL, since implication and equivalence are the basic principles of inference
(when inference is construed intuitively as did Leibniz: as concerned with
inferring necessary consequences from given assumptions and substituting
definitions (of equivalents) in known theorems).
Let's start with a simple and basic point about
implication.
It is obvious that in EPL several styles of implication
can be defined, notably the following two, where I shall write "I"
for "implies": (1) [pIq]=[~pVq] or else (2) [pIq]=[pVq]. So the
question rises which of these two (and related other ones) is the intuitively
correct rendering of implication, where "intuitively correct" means
that the chosen definition should conform to one's intuitions about valid
inferences.
Put in these terms, it is obvious that [pIq]=[~pVq] is to
be preferred over [pIq]=[pVq], because the latter if adopted would validate
inferences of the form "pIq & ?p therefore q" (as ?p and p
exclude each other). It makes no sense to infer conclusions from uncertainties
plus implications in this way, and so the basic implication is the one used
in CPL.
The first truthtable with relevant information is this,
with "T" written for "1" in some of the lines for
clarity's sake:

(~pVq)

(~qVp)

(pBq)

+p >+q &
p > q & ?p > ?q

+q >+p &
q > p &
?q > ?p

(pEq)

p q

1

1

1

1

1

1

p q


1





p ?q


1





p q

1






p q

1

1

1

1

1

1

p ?q

1

1

1




?p q

1






?p q

1

1

1




?p ?q

1

1

1

1

1

1

Columns 1 and 2 provide the definitions corresponding to the
[~pVq]=[p>q] and [~qVp]=[q>p]. If we call the lines on which a ? is
occurring a nonclassical line, and the other ones, i.e. 1,2,4 and 5
classical it will be seen that EPL adds 5 nonclassical cases to the 4
classical ones, in all of which but one, namely [p&?q], [p>q] holds,
all simply by definition.
Now if we follow the definition of equivalence as in CPL
we get [p iff q] = [(p>q)&(q>p)]. This leads to the cases
[(p&?q) > (p iff q)]=1 and [(?p&q) > (p iff q)]=1, as
column 3 shows, with "B" ("biimplication") for
"iff" for brevity.
This is not intuitive: although in both cases neither p
nor q is true, it does not follow intuitively that in these cases therefore p
and q are to come out as equivalent.
So here we reap a consequence of refining ~p to pV?p: In
CPL  and ? collapse to ~, and so both (p&?q) and (?p&q) only can
come out as (~p&~q), which does imply (p iff q) intuitively  in CPL.
But in EPL matters are more refined and not quite as
simple.
Columns 4, 5 and 6 provide the outlines for a solution,
which resides in the fact that
[p>q&p>q&?p>?q]=[q>p&q>p&?q>p]=[p&q]+[p&q]+
[?p&?q]=1, abbreviated to "pEq" in column 6. Note that each of
the columns 4, 5 and 6 amount to the same, i.e. that each of these amounts to
pEq i.e. [p&q]+ [p&q]+[?p&?q]. Put as a theorem:
T35 
[pEq] =
[p&q]+[p&q]+[?p&?q] 

(1) 
= [p>q & p>q & ?p>?q]


(2) 
= [q>p & q>p & ?q>?p] 

Note there is a parallel in CPL: In CPL [pEq]=[p>q
& ~p>~q]. As before, EPL merely refines the analysis of denials.
Further relevant facts concerning pEq can be seen from
the next table:

1

2

3

4

5

6


piffq

piffq

?piff?q

(pEq&pEq) E
(pEq&?qE?p)

(pEq&?pE?q) E
(pEq&pEq)

pEq

+p +q

1

1

1

1

1

1

+p q







+p ?q


1





p +q

1


1




p q

1

1

1

1

1

1

p ?q







?p +q


1





?p q

1






?p ?q

1

1

1

1

1

1

The theorem that can be read from this is to the effect
that pEq is true precisely if p and q have the same signs. In other words: p and
q are equivalent iff p and q are equivalent iff ?p and ?p are equivalent 
as fits one's linguistic intuitions that equivalent propositions must have the
same basic unitary operators (to which "~" does not belong):
T36 
(pEq&pEq)=(pEq&?qE?p)


(1) 
=(pEq&?pE?q) 

(2) 
=(pEq&pEq) 

(3) 
=pEq 

This in turn suggests the following definitions of
compatible, incompatible and contrary propositions:
Compatible(p,q) 
= def 
= pEq =
[+p&+q]+[p&q]+[?p&?q]

Incompatible(p,q)

= def 
=
[+p&?q]+[p&?q]+[?p&+q]+[?p&q]



=
[?(pVq)] 


=?[(+p&+q)] 


= [?(+p&+q)] 


= ?(pEq) 
Contraries(p,q) 
= def 
=
[+p&q]+[p&+q] 


= (pEq) 
Briefly, compatible propositions are equivalent i.e. have
the same basic operator; incompatible propositions are uncertainly
inequivalent or conjunctively
uncertain in that one is certain and the other uncertain, which is the case
precisely if their conjunction is uncertain; and contrary propositions are
each others negations. Note that it follows from the definitions that, since
the alternatives are exhaustive and exclusive that:
T37 
Any two propositions are
either compatibles or else incompatibles or else contraries. 

Again, one fundamental way of
making sense of incompatible propositions is by noting that intuitively, when
speaking of uncertainties and time, two propositions are incompatible if one
refers to the contingent future and the other doesn't i.e. one is uncertain
and the other certain. It is noteworthy that the nonincompatible pairs of
propositions are precisely the classical ones plus the case both are
uncertain.
Hence another way of dividing
the 9 cases of pairs of propositions is: Both uncertain; both certain and
equivalent or both certain and inequivalent; or mixed i.e. one certain the
other uncertain.
