We presume a reality D that can be represented
by set-theory including functions and we presume some knowledge of basic set
theory as e.g. summarized in 'Naive Set Theory' by Paul Halmos.
With set-theory developed to the extent it has
functions, one has the means to state how one can measure a domain one can
represent in set-theory by a set T of terms and statements of a language L, and to
define what it is to be true in the domain for a statement of the language.
Domains
D* = {Dj : Dj
a D}
-Di = D-Di
Ø = DiO-Di
Or in more general terms: We have the standard Algebra of Sets on the subsets of
D, and D* = the set of all subsets of D (also known as the powerset of
D); -Di is the complement of Di and
Ø is
the void set.
Functions
e : T |-> D*
# : D* |-> N
T are the well-formed statements and terms (Note
1) in language
L. The function e - say: extension of - maps the statements and terms to
the subsets of D. And the function # - say: number of - maps the subsets of
D to
the natural numbers including zero.
Note on notation for e and #:
In what follows I will suppress the usual
brackets in functional notation for e and #. Thus, 'eTi' = 'e(Ti)',
'#DiODj'
= #(DiODj)
etc. Also, as in these examples, suffixes of introduced terms refer to a term of
that kind: 'Ti' refers to term i and 'Dj' refers to
domain subset j.
Extensions
D = eTi U e~Ti
eTi = eTi&Tj U eTii&~Tj
The union of the extension of a term and the
extension of its complement equals the domain.
The union of the extension of a term conjuncted to any arbitrary term and the
extension of the term conjucted to the negation of the arbitrary term equals
the extension of the term.
These can be seen as properties of
e or as
axioms for e or as assumptions about e or as explanations of the meaning of
e.
Or they can be seen as definitions of D and the extension of arbitrary terms as
certain unions of their - mutually exhaustive and exclusive - subsets. (Note
2)
The yield is that denials and conjunctions map to the subsets of the domain in
the way one expects that they do: Denial maps to complement, conjunction to
intersection and disjunction to union.
Number
#Di = #Dj IFF (Ef)(f : Di 1-1 Dj)
The number of a subset equals the number of another subset precisely if there is
some 1-1 function between them.
This is fairly called Hume's definition of what it is for sets to have the same
number.
#D = #Di + #-Di
#Di = #DiODj
+ #DiO-Dj
The sum of the number of a subset and the number of its complement equals the
number of the domain.
The sum of the number of a subset intersecting with any arbitrary subset and the
number of the subset intersecting with the complement of the arbitrary subset
equals the number of the subset.
These can be seen as properties of # or as
axioms for # or as assumptions about # or as explanations of the meaning of
#.
Or they can be seen as definitions of equality of numbers, number of domain and
number of arbitary subsets as certain sums the numers of their - mutually
exhaustive and exclusive - subsets.
The yield is that complements and conjunctions map to sums in the way one
expects that they do:
The number of a complement of a subset follows by subtraction from that of the number of the
domain and the number of the subset and sets are the sums of the numbers of arbitrary exhaustive
and exclusive subsets of them.
Proportion
(Di) = #Di : #D
(Di|Dj) = #DiODj : #Dj
The proportion of Di equals the number of
Di
divided by the number of the domain.
The proportion of Di in Dj equals the number of the intersection of
Di and Dj
divided by the number of Dj.
These can be seen as properties of | or as
axioms for | or as assumptions about | or as explanations of the meaning of
|.
Or they can be seen as definitions of proportion of a subset to the domain and
of proportion of a subset to a set.
The yield is that | has the expected properties of proportion and provides a
basis for probability, whereas it follows from the numbers subsets have. (See
Classical Probability Theory and Learning from
Experience (Note
3).)
Inference
Qi|=Qj IFF eQi&~Qj=Ø
IFF (Qj|Qi)=1
V (Qi)=0
Qi entails Qj iff the extension of the
conjunction of Qi and ~Qj is void, which is provably equivalent to:
Either the probability
of Qj on Qi is 1 or the probability of Qi is 0.
Again, these can be seen as properties of |= or as axioms for |= or as
assumptions about |= or as explanations of the meaning of |=.
The yield is that entailment has the standard
properties and is definable in terms of proportion (and so in terms of
probability).
Fundamental theorems
Here it makes sense to insert some statements
of simple theorems with sketches of proofs that hold given the above
stipulations concerning e and #. In fact all proofs are quite trivial.
T1: #Ø = 0
Pr: By Ø = DiO-Di
and #DiOD=#Di.
T2: eTi=Ø IFF #Ti=0 IFF (Ti)=0.
Pr: Previous theorem and defs # and ().
T3: Ti IFF Tj --> eTi=eTj (This concerns
statements.)
Pr: By def |= (which is like inclusion).
T4: ti=ti --> eti=etj
(This concerns
terms.)
Pr: By standard properties of =.
T5: Di=Dj --> #Di=#Dj
Pr: By Di a
Dj --> #Di<=#Dj
T6: (Di|DJ) = (DiODj) : (Dj)
Pr: (Di|Dj) = #(DiODj):#(Dj)
= (#(DiODj):#(D)):((Dj):#(D))
= (DiODj):(Dj)
T7: (Di) = (Di|D)
Pr: (Di) = #Di : #D = #DiOD
: #D = (Di|D)
Numbers
For N:
+ : N.N |-> N.N & x+y=y+x & (x+y)+z=x+(y+z) (commutation and association
of +)
* : N.N |-> N.N & x*y=y*x & (x*y)*z=x*(y*z) (commutation and association
of *)
+ : N.N |-> N & x+0=x (identy-preservation of +0)
* : N.N |-> N & x*1=x (identy-preservation
of *1)
x*(y+z) = x*y + x*z (distribution of * over +)
> : N.N |-> {0,1} & 1>0 & x>y IFF x+1>y+1 (Greater
than as truth-function)
Truth-values
There are several possible schemes of mapping to {0,1} i.e. of binary truth-valuation,
that all involve the following:
v : T |-> {0,1}
In the present approach, v must be related somehow to the properties of
e or
#.
Given what we have it is not difficult to see that there are in fact three
fundamental possibilities using e introduced above, that together are mutually
exhaustive and exclusive:
eTi≠Ø & e~Ti≠Ø
eTi≠Ø & e~Tii=Ø
e~Ti≠Ø & eTi≠Ø
e~Ti≠Ø & eTi=Ø
for the first and third are the same (if ~~Ti=Ti),
which suggests
v(+Ti)=1 IFF eTi≠Ø & e~Ti=Ø IFF #Ti>0
& #~Ti=0 IFF (Ti)>0 & (~Ti)=0
v(-Ti)=1 IFF e~Ti≠Ø & eTi=Ø IFF #~Ti>0 & #Ti=0 IFF (~Ti)>0 &
(Ti)=0
v(?Ti)=1 IFF e~Ti≠Ø & eTi≠Ø IFF #Ti>0 & #~Ti>0 IFF (Ti)>0 &
(~Ti)>0
Or just for classical values:
v(Ti) = 1 IFF eTi≠Ø IFF v(+Ti)=1
v(~Ti) = 1 IFF e~Ti≠Ø IFF v(-Ti)=1 V v(?Ti)=1
One can take both e as relating terms to
meanings/ideas and # as relating meanings/ideas to some supposed reality,
where v(Ti)
= 1 amounts to: Ti has at least one instance in the supposed reality.
(Note 4)
So the above rules for valuation are both a
set-theoretical foundation and justification of my Extended Logic.
(Note 5) Furthermore, there is:
T8: The properties of proportion entail
Kolmogorov's axioms for probability.
Pr: By the above, v(Ti)=1 --> (Ti)=1 and v(Ti --> Tj)=1 --> (Ti) <= (Tj) while
also (Ti)=(Ti&Tj)+(Tii&~Tj).
The first two are direct consequences of the assumptions for v(.), and the last
follows by T6 and T7, which entail that if Di=e(Ti)
and Dj=e(Dj) then (~Dj|Di)=1-(Dj|Di),
whence (Di)=(Dj|Ti)(Ti)+(~Dj|Di)(Di),
whence (Ti)=(T&Tj)+(Tii&~Tj).
This is sufficient to derive Kolmogorov's axioms for
probability. (See:
Classical Probability Theory and Learning from
Experience).
Next, it is worthwile to combine the above with my earlier definitions of
representing, say into representing symbolically and numerically,
abbreviated rsn:
rsn(L,D) IFF L e Language & D is a set &
(Ee)(E#)
( e : T |-> D* &
# : D* |-> N &
D = eTi U e~Ti &
eTi = eTi&Tj U eTi&~Tj &
#D = #Di + #-Di &
#Di = #DiODj
+ #DiO-Dj
&
#Di = #Dj IFF (Ef)(f : Di 1-1 Dj) )
Note e preserves denials and conjunctions under
unions, and # preserves complements and intersections under sums.
Also, in the end I should add some considerations about diverse
kinds of
probability - and note that the present proposed proportional foundation
is new, and derives from the actual numbers of the real subsets of real domains
(somehow measured).
Note this approach to probability has another interesting consequence: There simply are
proportional non-extreme probabilities wherever both terms of
#Di = #DiODj + #DiiO-Dj
are non-zero, for whatever reason.
What may be the reason for this is often not so
important as to know that it is so, and what are the approximate frequencies.
(Indeed, one general kind of reason for non-extreme probabilities is this: The
alternatives neither logically exclude nor logically imply each other. This may
not be sufficient (or else there would be more mermaids, for example, supposing
these to be logically possible), but it goes some way, as it is at least
necessary for non-extreme probabilities).
Finally, the above should be combined with ideas and attitudes of persons. This can be
done most simply using the present set-up by taking D to be a set of ideas,
represented by terms and statements.
Maarten Maartensz
maartens@xs4all.nl
Note 1:
In this paper I take a certain amount of standard predicate logic and
set theory for granted. The terms and statements e and # work for are
those expressions - 'mermaid', 'elephant', 'Paris is the capital of
France' - that represent, but do not work for so-called syncategorematic
terms like 'of' and 'by' that only represent something when combined
grammatically with a representing term· Back.
Note 2:
I use the phrase 'These can be seen as ...' etc. repeatedly in this
paper because I want to avoid problems of interpretation. My own view is
that what I propose are axiomatic properties of the functions e
(extension of) and # (number of) that explain when assumed how terms and
ideas and facts and things are related to each other, and thus how terms
and statements can help us understand and describe reality.
Back.
Note 3:
The referred
Classical Probability Theory and Learning from
Experience can be seen as a
sequel to the present paper. See also T8 of the present paper.
Back.
Note 4:
The proposed three-some 'language - ideas - reality' for what is
represented by resp. terms, sets and numbers is not canonical though it
is basic, since the three-some is present whenever and wherever men
think with the help of language about some (presumed) reality.
Back.
Note 5:
As my equations show, there is a fundamental ambiguity in the standard
treatment of negation, for one may cogently mean by 'it is not
true that q' either that q is false or that q is
undetermined i.e. q is neither true nor false. (The last case is quite
common, both in case of socalled 'futura contingentia' statements, like
Aristotle's 'There will be a sea-battle tomorrow' and in case of many
other statements where one just doesn't know whether a statement or its
denial is definitely true.)
This feature of negation
has been discovered and rediscovered from Aristotle to Lukasiewicz, but
to my knowledge the first person to propose the present bivalent
analysis (that mirrors the bivalent analysis of three-somes like
'small', 'tall', 'neither small nor tall' and many more similar
examples) was the Russian logician and philosopher A.A. Zinoviev.
See e.g. his 'Logische Sprachregeln' - and it is an interesting
aside that to cope with similar problems with negation Lukasiewicz
introduced three or more truth-values and Brouwer denied the
validity of the excluded third and founded intuitionist logic.
The present settheoretical
semantics is original (and Zinoviev did not like standard set theory, at
least not for the purpose of analysing logical notions). In 'Logische
Sprachregeln' there are proposed many bi-valent logical systems
involving operators like '+' say 'it is verified that', '-' say 'it is
falsified that' and '?' say 'it is undetermined'. An approach intermediate
between Zinoviev's approach and the present one was part of my
M.A.-thesis, that was concerned with extended propositional logic and
propositional attitudes. Back.