|
Extended Propositional Logic - Algebraic semantics:
Non-standard bivalent extension of standard bivalent propositional
logic, to cover also
reasoning with
uncertainties and
propositional attitudes.
This gives a sketch of the foundations of an
Algebraic semantics for EPLA. See
Extended Propostional Logic.
A. Rules, definitions and axioms
The term "PV" abbreviates
"Propositional Variable", and "Prop" abbreviates "Proposition". In what
follows, I shall use "p", "q", "r", "s" as PVs, and allow subscripts
(letters or numbers) to produce as many PVs as one needs. Note that what
follows is an informal formal explanation: It is fairly but not fully
precise and specific, but I leave the full details for later: This is
just a brief sketch of basic ideas.
The original motive for LPA (that follows below) was: To find a logic
for propositional attitudes that is as
simple as possible, and that requires neither modal logic nor more than
two truth-values. My desire for simplicity was motivated both by logical
and mathematical reasons (try to do with as few basic notions as are
adequate to solve the problem), and by the empirical observation that
not only all non-idiotic humans but also quite a few social mammals, at
least, and perhaps also some birds, seem to have something like a theory
of mind, that allows them to somehow reckon with the beliefs and desires
of others.
B. Logical inference rules
CPL= Classical Propositional Logic
Syntax: Propositions
If X is a PV, X is a Prop.
If X is a Prop, then ¬X is a Prop.
If X is a Prop and Y is a Prop, then (X&Y) is a Prop and
(XVY) is a Prop.
Nothing else is a Prop, except by definition.
Note this does not say what are the
Props, but only how to produce them from PVs or other Props.
Axioms: pV¬p
Definitions:
p --> q =d ¬p V q
p iff q =d p-->q & q-->p
Rules: p&q |- p
p&q |- q
p,q |- p&q
p |- pVq
q |- pVq
pVq, ¬pVq |- q
The above is a version of Classical
Propositional Logic. The axiom is most conveniently read as "any
proposition p is true or is not true". It can be proved from the rules,
but is added for clarity.
EPL= Extended Propositional Logic
EPL = CPL +
Syntax:
If X is a Prop, then +X and -X and
?X are Props.
Axioms: +p V -p V ?p
Definitions
+p =d ¬-p & ¬?p = p
-p =d ¬+p & ¬?p
?p =d ¬+p & ¬-p
piffq =d p&q V -p&-q V
?p&?q
p&q =d ¬-p & ¬-q
pVq =d ¬-p V ¬-q
Extended Propositional Logic adds the
distinctions between the weak and strong logical operators
(¬ and -, & and &, and V and V, iff and iff
respectively), and adds the prefixes +, - and ? (respectively true,
false and uncertain, or also verified, falsified and undecided).
One way to motivate the need for EPL is
to point to logically contingent propositions about the future, such as
"Tomorrow I will still be alive", which by many, such as Aristotle and
Ockham, have been believed to be neither true nor false today.
It can be seen from the above and CPL
that "¬p" amounts to "-p V ?p", and that accordingly EPL starts with a
refinement of negation, and proceeds with refinements of the other
logical operators.
EPLA = Extended Propositional Logic
with attitudes = EPL +
Syntax: If X is a Prop, and A is a
term for an attitude and a is a term for person, then aAX is a Prop.
Attitudes are terms like "believes",
"desires", "chooses" and many more. Terms for persons may include terms
for personified entities, such as "the people of France" or "my kitten".
Definitions:
-aAp =d aA¬p
aKp =d aBp&p
The first of these makes a sort of
reduction of EPLA to EPL possible, as will be shown
Axioms: aAp V aA-p V aA?p V ?aAp
¬aAaAp V aAp
¬aB(p |- q) V (p |- q)
The difference between "aA?p" and
"?aAp" will be taken as that between a's believing (etc.) that p is
uncertain and a's not being acquainted with p. Thus, if you are not me,
presumably you are not acquainted with the proposition "The pope had a
child by my grandmother". (This may be made more plausible by reading
"?" as "uncertifiable".
Rules:
aA(p&q) -||- aAp & aAq
aA(pVq) -||- (aAp & -?aAq) V (aAq & -?aAp)
aA(p&q) -||- -aA-p & -aA-q
aA(pVq) -||- (-aA-p & -?aAq) V (-aA-q & -?aAp)
aB(p |- q) -||- aK(p |- q)
Extended Propositional Logic with
Attitudes adds attitudes to EPL and some axioms for these.
The last rule is equivalent to the
assumption that a is fully correct in his beliefs about logical
consequence. This enables substitutions of equivalents, when a is
acquainted with the involved propositions, and entails a knows all valid
formulas involving only those propositions that a is acquainted with.
One may conditionalize or weaken or
relativize this in various ways. Also, the relativization may take the
general form K(a) |- ... with K(a) the knowledge a possesses or
presumes, or a (certain) subset thereof.
As will be seen in the following
section, apart from the axioms and what follows from these, and apart
from the uncertifiable propositions for a, which a just doesn't know of,
though others may, EPLA is much like a notational variant of EPL.
C. Logical semantical rules
The above can be also given a simple
semantical and algebraic formulation. For ease of writing and reading I
use "[p]" = "the value of p" (rather than the more prolix "v[p]" or
"v(p)"), but of course set-theoretically this is just a function from
propositions to the set {0,1}. The values are numbers and can be dealt
with as in ordinary algebra, it is assumed.
Extended Valuations = EPLA
+
If X is a Prop and y=1 or y=0, [X]=y
is a valuation.
If V is a valuation, V is a Prop.
This makes it possible to aggregate the
semantics of EPL and its logic.
CPL Valuations
VB: [p]=1 V [p]=0
V[]: [[p=1]=1]=[p=1]
V¬: [¬p]=1-[p]
V&: [p&q]=[p]*[q]
VV: [pVq]=[p]+[q]-[p]*[q]
Viff: [piffq]=[p&q]+[¬p&¬q]
V|-: [p|-q]=[¬(p&¬q)]
Apart from V[], that allows the
collapsing of iterated truth-values to single ones, this is all
standard. VB is the rule of bi-valence.
EPL Valuations = CPL Valuations plus
VE0: [+p]=[p]
VE+: [+p]=[¬-p]*[¬?p]
VE-: [-p]=[¬+p]*[¬?p]
VE?: [?p]=[¬+p]*[¬-p]
VE-+: [-+p] = [-p]
VE--: [--p] = [+p]
VE-: [-?p] =
[+p V -p]
VE&: [+(p & q)] = [+p & +q]
[-(p & q)]
= [-p V -q]
[?(p & q)]
= [p&?q V q&?p V ?p&?q]
VEV: [+(p V q)] = [+p V +q]
[-(p V q)]
= [-p & -q]
[?(p V q)]
= [-p&?q V -q&?p V ?p&?q]
VE&: [+(p & q)]
= [~-p & ~-q]
[-(p & q)]
= [-p V -q]
VEV: [+(p V q)] = [~-p V ~-q]
[-(p
V q)] = [-p & -q]
VEIFF: [piffq]=[p&q]+[-p&-q]+[?p&?q]
Note that the weak operators are all
implied by the strong operators of the same kind, but that the converse
is not true and that e.g. V|- gets more sophisticated than is possible
in CPL, and amounts to [p|-q]=[-pV?p V +q)]
LPA Valuations = EPL Valuations plus
VA-: [-aAp]=[aA¬p]
VA&: [aA(p&q)]=[aAp]*[aAq]
VAV: [aA(pVq)]=[aAp]*[-?aAq]+[aAq]*[-?aAp]-[aAp]*[aAq]
VA&: [aA(p&q)]=[-aA-p]*[-aA-q]
VAV: [aA(pVq)]=[-aA-p]*[-?aAq]+[-aA-q]*[-?aAp]-[-aA-p]*[-aA-q]
VASb: [aB(p |- q) & ¬(p |- q)]=0
VAA: [aAaAp & ¬aAp]=0
Here are tables that correspond to the
above, that may be derived from the stated and similar theorems:
Tables EPL:
Tables EPL in 1 variable:
|
|
p |
-p |
?p |
~p |
~-p |
~?p |
--p |
-?p |
?-p |
| 1 |
p |
1 |
|
|
|
1 |
1 |
1 |
1 |
|
| 2 |
-p |
|
1 |
|
1 |
|
1 |
|
1 |
|
| 3 |
?p |
|
|
1 |
1 |
1 |
|
|
|
1 |
Note this contains an innovation as
regards truth-tables: To preserve bi-valence, the first column lists all
the basic possibilities for all the propositions involved, as also shown
in the following tables.
Accordingly, the full truth-tables for n propositions will have 2n
lines in CPL, but 3n lines in EPL, and 4n
in EPLA, to list all possible basic conjunctions for these n
propositions.
Also, in EPL "+p" and "p" are
equivalent, and the frontal "+" is usually left out.
Tables EPL in 2 variables:
| |
|
(p&q) |
(p&q) |
(pVq) |
(pVq) |
p |
¬-p |
| |
|
¬(¬pV¬q) |
¬(-pV-q) |
¬(¬p&¬q) |
¬(-p&-q) |
¬(-pV?p) |
(pV?p) |
| 1 |
p q |
1 |
1 |
1 |
1 |
1 |
1 |
| 2 |
p -q |
|
|
1 |
1 |
1 |
1 |
| 3 |
p ?q |
|
1 |
1 |
1 |
1 |
1 |
| 4 |
-p q |
|
|
1 |
1 |
|
|
| 5 |
-p -q |
|
|
|
|
|
|
| 6 |
-p ?q |
|
|
|
1 |
|
|
| 7 |
?p q |
|
1 |
1 |
1 |
|
1 |
| 8 |
?p -q |
|
|
|
1 |
|
1 |
| 9 |
?p ?q |
|
1 |
|
1 |
|
1 |
These are the basic connectives. Note
that all lines in the above and the following truth-tables follow from
the semantical rules and that truth-tables are a notational device.
Tables for strong and weak
conjunction in EPL:
| |
|
(p&q) |
-(p&q) |
?(p&q) |
(p&q) |
-(p&q) |
?(p&q) |
| |
|
¬(¬pV¬q) |
(-pV-q) |
|
¬-p&¬-q |
(-pV-q) |
|
| 1 |
p q |
1 |
|
|
1 |
|
|
| 2 |
p -q |
|
1 |
|
|
1 |
|
| 3 |
p ?q |
|
|
1 |
1 |
|
|
| 4 |
-p q |
|
1 |
|
|
1 |
|
| 5 |
-p -q |
|
1 |
|
|
1 |
|
| 6 |
-p ?q |
|
1 |
|
|
1 |
|
| 7 |
?p q |
|
|
1 |
1 |
|
|
| 8 |
?p -q |
|
1 |
|
|
1 |
|
| 9 |
?p ?q |
|
|
1 |
1 |
|
|
Accordingly, a strong conjunction has
all conjuncts are true to be true itself, and a weak conjunction has all
conjuncts not false to be true itself.
Tables for strong and weak disjunction in EPL:
| |
|
(pVq) |
-(pVq) |
?(pVq) |
(pVq) |
-(pVq) |
?(pVq) |
| |
|
-(-p&-q) |
(-p&-q) |
|
¬-p&¬-q) |
(-p&-q) |
|
| 1 |
p q |
1 |
|
|
1 |
|
|
| 2 |
p -q |
1 |
|
|
1 |
|
|
| 3 |
p ?q |
1 |
|
|
1 |
|
|
| 4 |
-p q |
1 |
|
|
1 |
|
|
| 5 |
-p -q |
|
1 |
|
|
1 |
|
| 6 |
-p ?q |
|
|
1 |
1 |
|
|
| 7 |
?p q |
1 |
|
|
1 |
|
|
| 8 |
?p -q |
|
|
1 |
1 |
|
|
| 9 |
?p ?q |
|
|
1 |
1 |
|
|
Accordingly, a strong disjunction has at least one disjunct true to
be true itself, and a weak disjunction has at least one disjunct that is
not false to be true itself.
Tables for strong and weak equivalence in EPL:
| |
|
(piffq) |
-(piffq) |
?(piffq) |
(piffq) |
-(piffq) |
?(piffq) |
| |
|
(p&q)V(¬p&¬q) |
|
|
(p&q)V(-p&-q)V(?p&?q) |
|
|
| 1 |
p q |
1 |
|
|
1 |
|
|
| 2 |
p -q |
|
1 |
|
|
1 |
|
| 3 |
p ?q |
|
|
1 |
|
1 |
|
| 4 |
-p q |
|
1 |
|
|
1 |
|
| 5 |
-p -q |
1 |
|
|
1 |
|
|
| 6 |
-p ?q |
1 |
|
|
|
1 |
|
| 7 |
?p q |
|
|
1 |
|
1 |
|
| 8 |
?p -q |
1 |
|
|
|
1 |
|
| 9 |
?p ?q |
1 |
|
|
1 |
|
|
A weak equivalence follows CPL, but confuses non-truth and falsehood
in EPL-terms. The strong equivalence has the intuitively nice feature of
being true precisely if the propositions involved have the same prefix:
both true, both false, or both uncertain.
D. Extending EPL to EPLA
To start with when considering EPLA, it is noteworthy that the given
tables for EPLA and for EPL are isomorphic - but it must be kept in mind
that in EPLA there are the additional lines when a is not acquainted
with some involved proposition.
This also explains the additions in the
the EPLA valuation rules for disjunction: To believe a disjunction,
whether weak or strong, one must know of all the involved propositions.
Thence
VAV:
[aA(pVq)]=[aAp]*[-?aAq]+[aAq]*[-?aA?p]-[aAp]*[aAq]
VAV: [aA(pVq)]=[-aA-p]*[-?aAq]+[-aA-q]*[-?aA?p]-[-aA-p]*[-aA-q]
But apart from that, the basics for EPL
and EPLA are just the same, and it may also be noted about disjunctions
and distribution that:
Strong or has the strong distribution property that one of the
alternatives is true.
Weak or has the weak distribution property that one of the
alternatives is not false.
Here are tables:
Tables ELPA full:
| |
|
aB(p&q) |
aB(p&q) |
aB(pVq) |
aB(pVq) |
aBp |
-aB-p |
| |
|
-aB(~pV~q) |
-aB(-pV-q) |
-aB(~p&~q) |
-aB(-p&-q) |
-aB(-pV?p) |
aB(pV?p) |
| 1 |
aBp aBq |
1 |
1 |
1 |
1 |
1 |
1 |
| 2 |
aBp aB-q |
|
|
1 |
1 |
1 |
1 |
| 3 |
aBp aB?q |
|
1 |
|
1 |
1 |
1 |
| 4 |
aBp ?aBq |
|
|
|
1 |
|
1 |
| 5 |
aB-p aBq |
|
|
1 |
1 |
|
|
| 6 |
aB-p aB-q |
|
|
|
|
|
|
| 7 |
aB-p aB?q |
|
|
|
|
|
|
| 8 |
aB-p ?aBq |
|
|
|
|
|
|
| 9 |
aB?p aBq |
|
1 |
1 |
1 |
|
1 |
| 10 |
aB?p aB-q |
|
|
|
|
|
1 |
| 11 |
aB?p aB?q |
|
1 |
|
1 |
|
1 |
| 12 |
aB?p ?aBq |
|
|
|
|
|
1 |
| 13 |
?aBp aBq |
|
|
|
|
|
|
| 14 |
?aBp aB-q |
|
|
|
|
|
|
| 15 |
?aBp aB?q |
|
|
|
|
|
|
| 16 |
?aBp ?aBq |
|
|
|
|
|
|
Tables ELPA small: The above may
be abbreviated again like this, simply eliminating the cases with
frontal ? (that intuitively means for e.g. ?aB(pVq), and any other
binary operator, that a does not know of p or of q, which is not at all
the same as aB?(pVq), that says a believes that the disjunction p or q
is uncertain, for which a does need to know of both, in order to
evaluate them.
Note this possible frontal ? must be
accounted for in the rules of inference, but since it clutters up tables
the above may be simplified to:
| |
|
aB(p&q) |
aB(p&q) |
aB(pVq) |
aB(pVq) |
aBp |
-aB-p |
| |
|
-aB(~pV~q) |
-aB(-pV-q) |
-aB(~p&~q) |
-aB(-p&-q) |
-aB(-pV?p) |
aB(pV?p) |
| 1 |
aBp aBq |
1 |
1 |
1 |
1 |
1 |
1 |
| 2 |
aBp aB-q |
|
|
1 |
1 |
1 |
1 |
| 3 |
aBp aB?q |
|
1 |
|
1 |
1 |
1 |
| 4 |
aB-p aBq |
|
|
|
1 |
|
|
| 5 |
aB-p aB-q |
|
|
1 |
|
|
|
| 6 |
aB-p aB?q |
|
|
|
|
|
|
| 7 |
aB?p aBq |
|
1 |
1 |
1 |
|
1 |
| 8 |
aB?p aB-q |
|
|
|
|
|
1 |
| 9 |
aB?p aB?q |
|
1 |
|
1 |
|
1 |
Tables for strong and weak
conjunction in EPLA:
| |
|
aB(p&q) |
aB-(p&q) |
aB?(p&q) |
aB(p&q) |
aB-(p&q) |
aB?(p&q) |
| |
|
-aB(~pV~q) |
aB(-pV-q) |
|
aB~-p&~-q |
aB(-pV-q) |
|
| 1 |
aBp aBq |
1 |
|
|
1 |
|
|
| 2 |
aBp aB-q |
|
1 |
|
|
1 |
|
| 3 |
aBp aB?q |
|
|
1 |
1 |
|
|
| 4 |
aB-p aBq |
|
1 |
|
|
1 |
|
| 5 |
aB-p aB-q |
|
1 |
|
|
1 |
|
| 6 |
aB-p aB?q |
|
1 |
|
|
1 |
|
| 7 |
aB?p aBq |
|
|
1 |
1 |
|
|
| 8 |
aB?p aB-q |
|
1 |
|
|
1 |
|
| 9 |
aB?p aB?q |
|
|
1 |
1 |
|
|
Tables for strong and weak
disjunction in EPLA:
| |
|
aB(pVq) |
aB-(pVq) |
aB?(pVq) |
aB(pVq) |
aB-(pVq) |
aB?(pVq) |
| |
|
-aB(~p&~q) |
-aB~(-p&-q) |
|
-aB(-p&-q) |
aB(-p&-q) |
|
| 1 |
aBp aBq |
1 |
|
|
1 |
|
|
| 2 |
aBp aB-q |
1 |
|
|
1 |
|
|
| 3 |
aBp aB?q |
1 |
|
|
1 |
|
|
| 4 |
aB-p aBq |
|
|
|
1 |
|
|
| 5 |
aB-p aB-q |
|
1 |
|
|
1 |
|
| 6 |
aB-p aB?q |
|
|
1 |
1 |
|
|
| 7 |
aB?p aBq |
1 |
|
|
1 |
|
|
| 8 |
aB?p aB-q |
|
|
1 |
1 |
|
|
| 9 |
aB?p aB?q |
|
|
1 |
1 |
|
|
The tables show the interesting
property of there being no true weak uncertain binary
operators, that also holds for EPL.
It should also be remarked that many of
the attitudes people do have are weak rather than strong -
which seems to be something most people, whether logicians or not, seem
to have overlooked.
Normally when saying e.g. "I believe I
will be going on holiday next week to Greece or to France" one means no
more than that it is false one does believe one will be going on holiday
next week to neither country. ("If I don't go to Greece nor to France
next week, I won't be going on holiday at all, but I don't know yet
where I will go.")
It should also be mentioned that the following theorems can be
proved. First, there is
T11. ¬?aBp |- aB(pV-p)
which says that one believes weakly
that p is true or false if one knows of p (but usually one does not
believe so strongly, since this involves believing that the one or the
other is true, which one often does not believe until one has more
evidence). And it may be noted also that aB(pV-p) -||- -aB(p&-p)
-||- aB¬(p&-p) -||- aB¬pV-¬p) -||- aB(+pV-pV?p).
And the parallelism between EPL and
EPLA is mostly due to
T12. -aAp -||- aA¬p
which is to say that it is false that a
has an attitude to p iff it is true that a has that attitude to not p.
Next the VASb leads to (and in fact is
equivalent with, given the other rules)
T13. aB(p |- q) |- (p |- q)
which is to say that a is a logical
reasoner as regards logical consequence: a is never mistaken
in his beliefs about these. This allows substitutions of equivalent
propositions, provided a knows of them. (Here the point of (p |- q) is
that (|- (¬p V q)) is meant, i.e. a semantically valid disjunction of
that form, and not merely (¬p V q) which may be true by accident.)
Something like T13 is necessary
for substitutions, though one may restrict this in various ways (e.g. a
may be quite logical, but may have forgotten all but the last 7
propositions he thought of).
Finally the VAA leads to
T14. aAaAp |- aAp
that is if a a believes that a believes
p (or desires that a desires, or chooses that a chooses etc.) then
indeed a believes (or desires or chooses etc.) p, or in other words,
what a is conscious of as regards a's attitudes is true (though a may
believe - desire, (try to) cause, know, experience - many things a is
not conscious of.
Also, it gives the basics for making one's own attitudes: In the case
of belief, by believing one believes p (or believes not p or believes p
is uncertain).
This is involved in both
consciousness - believing one believes, experiencing one experiences
etc. - and in
choice, in as much as that by the above to make one believe
something all one needs to do is believe one believes it, and the same
for other attitudes.
Note this is just a sketch and a
prepublication.
It also refines and restates parts of
my M.A.-thesis. |
