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* = {D**_{j} : D_{j} a D}

-D_{i} = D-D_{i}

Ø = D_{i}O**-D**_{i}

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**); **-D**_{i} is the complement of **D**_{i} 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, '**eT**_{i}' = 'e(T_{i})', '**#D**_{i}**O****D**_{j}' = #(D_{i}OD_{j}) etc. Also, as in these examples, suffixes of introduced terms refer to a term of that kind: '**T**_{i}' refers to **term** i and '**D**_{j}' refers to **domain subset** j.

**Extensions**

**D = eT**_{i} U e~T_{i}

eT_{i} = eT_{i}&T_{j} U eTi_{i}&~T_{j}

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**

**#D**_{i} = #D_{j} IFF (Ef)(f : D_{i} 1-1 D_{j})

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 = #D**_{i} + #-D_{i}

#D_{i} = #D_{i}OD_{j} + #D_{i}O**-D**_{j}

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**

**(D**_{i}) = #D_{i} : #D

(D_{i}|D_{j}) = #D_{i}OD_{j} : #D_{j}

The proportion of **D**_{i} equals the number of **D**_{i} divided by the number of the domain.

The proportion of **D**_{i} in **D**_{j} equals the number of the intersection of **D**_{i} and **D**_{j} divided by the number of **D**_{j}.

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**

**Q**_{i}|=Q_{j} IFF eQ_{i}&~Q_{j}=Ø

IFF (Q_{j}|Q_{i})=1 V (Q_{i})=0

**Q**_{i} entails **Q**_{j} iff the extension of the conjunction of **Q**_{i} and **~Q**_{j} is void, which is provably equivalent to: Either the probability of **Q**_{j} on **Q**_{i} is 1 or the probability of **Q**_{i} 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 = D_{i}O-D_{i} and #D_{i}OD=#D_{i}.

**T2**: **eT**_{i}=Ø IFF #T_{i}=0 IFF (T_{i})=0.

Pr: Previous theorem and defs # and ().

**T3**: **T**_{i} IFF T_{j} --> eT_{i}=eT_{j} (This concerns **statements.**)

Pr: By def |= (which is like inclusion).

**T4**: **t**_{i}=t_{i} --> et_{i}=et_{j} (This concerns **terms**.)

Pr: By standard properties of =.

**T5**: **D**_{i}=D_{j} --> #D_{i}=#D_{j}

Pr: By D_{i} a Dj --> #D_{i}<=#D_{j}

**T6**: **(D**_{i}|D_{J}) = (D_{i}**O****D**_{j}) : (D_{j})

Pr: (D_{i}|D_{j}) = #(D_{i}OD_{j}):#(D_{j}) = (#(D_{i}OD_{j}):#(D)):((D_{j}):#(D)) = (D_{i}OD_{j}):(D_{j})

**T7**: **(D**_{i}) = (D_{i}|D)

Pr: (D_{i}) = #D_{i} : #D = #D_{i}OD : #D = (D_{i}|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:

**eT**_{i}≠Ø & e~T_{i}≠Ø

eT_{i}≠Ø & e~Ti_{i}=Ø

e~T_{i}≠Ø & eT_{i}≠Ø

e~T_{i}≠Ø & eT_{i}=Ø

for the first and third are the same (if ~~T_{i}=T_{i}), which suggests

**v(+T**_{i})=1 IFF eT_{i}≠Ø & e~T_{i}=Ø IFF #T_{i}>0 & #~T_{i}=0 IFF (T_{i})>0 & (~T_{i})=0

v(-T_{i})=1 IFF e~T_{i}≠Ø & eT_{i}=Ø IFF #~T_{i}>0 & #T_{i}=0 IFF (~T_{i})>0 & (T_{i})=0

v(?T_{i})=1 IFF e~T_{i}≠Ø & eT_{i}≠Ø IFF #T_{i}>0 & #~T_{i}>0 IFF (T_{i})>0 & (~T_{i})>0

Or just for classical values:

**v(T**_{i}) = 1 IFF eT_{i}≠Ø IFF v(+T_{i})=1

v(~T_{i}) = 1 IFF e~T_{i}≠Ø IFF v(-T_{i})=1 V v(?T_{i})=1

One can take both e as relating terms to **meanings/ideas** and # as relating meanings/ideas to **some supposed reality**, where v(T_{i}) = 1 amounts to: T_{i} 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(T_{i})=1 --> (T_{i})=1 and v(T_{i} --> T_{j})=1 --> (T_{i}) <= (T_{j}) while also (T_{i})=(T_{i}&T_{j})+(Ti_{i}&~T_{j}). The first two are direct consequences of the assumptions for v(.), and the last follows by **T6** and **T7**, which entail that if D_{i}=e(T_{i}) and D_{j}=e(D_{j}) then (~D_{j}|D_{i})=1-(D_{j}|D_{i}), whence (D_{i})=(D_{j}|T_{i})(T_{i})+(~D_{j}|D_{i})(D_{i}), whence (T_{i})=(T&T_{j})+(Ti_{i}&~T_{j}).

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 = eT_{i} U e~T_{i} &

eT_{i} = eT_{i}&T_{j} U eT_{i}&~T_{j} &

#D = #D_{i} + #-D_{i } &

#D_{i} = #D_{i}**OD**_{j} + #D_{i}O-D_{j} &

#D_{i} = #D_{j} IFF (Ef)(f : D_{i} 1-1 D_{j}) )

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 #D_{i} = #D_{i}OD_{j} + #Di_{i}O-D_{j} 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

[email protected]

**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.