proposition

A proposition consists of a description and a qualifier.

description

The description inform of circumstances.

qualifier

The qualifier expresses how this description is related to factual circumstances.

Usually expresses, or understands tacitly, a proposition, that its description corresponds to factual circumstances.

proposition

A proposition is equivalent to a question + an answer if they inform of the same.

question

A question is a description without any qualifier.

answer

An answer is a qualifier which can be attached by a complementing description.

If an answer to a question only can be "yes" or "no", the question contains the whole description and the answer only the qualifier. The proposition "It's not raining." expresses exactly the same as the question-answer pair "Is it raining? No!"

truth

Truth is generally the correspondence between a description (statement, image, model, theory) and the corresponding factual circumstances it describes (the real world).

The qualifier to the description specify how the truth / correspondence will be interpreted.
In this generally concept there can be many variants of truth for different applications and with different properties and strength, as e.g. logical truth, factual truth, probability etc.

factual truth

A propositions factual truth consists of the degree of correspondence between the proposition and the temporarily observed factual circumstances it describes.

logical truth

A propositions logical truth consists of the "degree" of necessarity in correspondence between the proposition and the factual circumstances it describes.

Factual truth can has all degrees of correspondence / truth, but logical truth can only be "true" or "false"

partly truth

part observation

An observation can establish a degree of factual truth, partly truth, to a proposition if it can be divided in part observations with known weights related to the full observation.

equal weight

Especially, can an observation establish partly truth if it can be divided in N equal disjunctive parts with equal weight.

resolution

If an observation consists on N equal weighty part observation, its resolution is N.

supports

A part observation supports the truth of a proposition if its qualified description correspond to the observation.

weaken

A part observation weaken the truth of a proposition if its qualified description not correspond to the observation.

satisfied

A proposition is satisfied by a part observation if it supports the factual truth of the proposition.

unsatisfied

A proposition is unsatisfied by a part observation if it weakens the factual truth of the proposition.

relevant

A part observation is relevant for a proposition if it either supports or weakens the factual truth of the proposition.

irrelevant

A part observation is irrelevant for a proposition if it neither supports nor weakens the factual truth of the proposition.

true

A proposition is true if all part observations of an observation satisfies the proposition.

false

A proposition is false if not all part observations of an observation satisfies the proposition.

existent

A proposition is existent if some part observations of an observation satisfies the proposition.

unexistent

A proposition is unexistent if none part observations of an observation satisfies the proposition.

"False" is identical to "not true"
"Unexistent" is identical to "not existent"

polyvalued truth value

truth value

The polyvalued truth value of a proposition p, or shorter, its truth value, defines by the rate:
the number Np of equal-weighty-part-observations satisfying the proposition, divided by the total number N of relevant equal-weighty-part-observations, i.e. the rate: Np / N.

This is the same as a "relative frequency" or "relative occurence".

polyvalued truth value part system

An observation with the resolution N defines a polyvalued truth value part system with N+1 discrete truth values, which are rational numbers in the interval [0,1].

The resolution N depends of what the proposition describes and how this can be observed, but all different truth value part systems belongs to the same value interval [0,1] and are mutually comparable to each other. Every polyvalued truth value part system can be characterized as one many-valued system, i.e. a (N+1)-nary truth value system.

polyvalued truth value space

The polyvalued truth values are able to make every (N+1)-nary truth value part system characterized above. N=1 makes a binary system, N=2 makes a trinary etc. All together they makes a infinite system of many-valued systems, a polyvalued truth value space.

The characterizing as infinite-many-many-valued has been rationalized to the more handy name "polyvalued" (analogue to using of the word 'poly' in chemistry)

complements

Let p represent an arbitrary proposition and not-p its denying. p and not-p describes the same but expresses the opposite and they complement each other, they are complements, in the following sense: If a part observation is relevant for a proposition is it either satisfied or unsatisfied.

If p is satisfied not-p must be unsatisfied.
If not-p is satisfied must p be unsatisfied.
If p is irrelevant not-p must be irrelevant too.
If not-p is irrelevant p must be irrelevant too.

meaning

The meaning of a proposition is determined by two factors: the circumstances when it is defined and when it is satisfied.

definition range

The description part of a proposition informs of the circumstances when the proposition is defined, it informs of its definition range(set).

satisfaction range

The qualifier part of a proposition informs of the circumstances when the proposition is satisfied, it informs of its satisfaction range(set).

The satisfaction range(set) is always a subrange(subset) to the definition range(set). p and not-p has the same description (of circumstances) but different qualifiers. p and not-p has the same definition range but different satisfaction range.

disjunctive meanings

The meanings of p and not-p are disjunctive to each other in the sense they has no overlapping satisfaction ranges.

complementary meanings

The meanings of p and not-p are complementary to each other in the sense they cover together the whole definition range.

It is not possible to introduce a third meaning to a description of p beside or between p and not-p (but each of the two meanings itself can often be split up in smaller specified parts generating more alternatives. All of them must be submeanings belonging to either p or not-p)

Let Np represent the number of part observations of an observation that satisfies p and let N~p represent the number that satisfies not-p. Np + N~p is then representing the number of part observations there the description of p are defined. The polyvalued truth value of p can be calculated by the rate Np / (Np+N~p) and the truth value of not-p by the rate N~p / (Np+N~p). For an arbitrary proposition p this can be generalized to the rate: "the number of part observations there p are satisfied" divided by "the number of part observations there p are defined".

relativ appearance

The rate Np / (Np+N~p) is the relativ appearance of p and will be represented by Rp.

The terms 'polyvalued truth value' and 'relative apperance' will be used as synonyms in the following and are representing the measurement of truth. Terms as 'partly truth', 'factual truth' and 'logical truth' are representing different properties of truth.

value identical =

meaning identical

If two propositions always respond to part observations in the same way (satisfied, unsatisfied or irrelevant) independent of every arbitrary situation, they has necessarily the same polyvalued truth value and are value identical, and they expresses necessarily the same meaning and are meaning identical.

identity

Both value identical and meaning identical are represented by the symbol '' which often short will be referred to as identity.

mathematical equality

Value identity expresses a stronger form of equality - a necessary equality - an equality, valid independent of every arbitrary assignment and situation. Normal mathematical equality expresses only temporary value equality.

linguistic equivalence

Meaning identity expresses a stronger form of equivalence - a necessary equivalence - an equivalence, valid independent of every arbitrary assignment and situation. Normal linguistic equivalence expresses only temporary two-ways conditionality.

To find the factual truth value for a proposition p we must observe it, Assume this observation consists of N equal- weighty-part-observations able to satisfy or unsatisfy the proposition. The degree of knowledge on p increases during this process and after N part observations we know exactly how many times Np our proposition has been satisfied. The rate Np / N gives us the relative occurrence, i.e. the polyvalued truth value.

All part observations satisfies either p or not-p, so Np + N~p N must be true. Dividing both sides by N transforms this to Np/N + N~p/N N/N which also can be written as Rp + R~p 1. Subtracting both sides by Rp transforms this to

R~p 1 - Rp.

This is our first real polyvalued truth function R~p = f(Rp) and this is derived using only the normal mathematical operations. Knowing the polyvalued truth value of Rp the truth value of R~p can immediately be calculated.

Already in the Np + N~p N the logical relation between p and not-p is expressed. The transformations using only mathematical operations preserves this logical relation without using any other logical rules or axioms. This ability is one of the basic characteristics of polyvalued functions based on polyvalued truth.

conjunct pair

Two propositions p and q validated by the same observation, forms a conjunct pair.

In a sequence of part observations, we can record not only how many times p respectively q are satisfied but also how many times they are satisfied together. However, from the information about the polyvalued truth value of p and q it is impossible to calculate the truth value for how many times they are satisfied together. There are lack of information.
In a polyvalued logic it is impossible to calculate the truth value, not only of p&q, but of pvq, p=>q, p<=>q and so on, from any functions of type f(p,q). We need instead functions of type f( p&q , p&~q , ~p&q , ~p&~q ) for two variables, and we need functions of type f( p&q&r , p&~q&r , ~p&q&r , ~p&~q&r , p&q&~r , p&~q&~r , ~p&q&~r , ~p&~q&~r ) for tree variables (conjunct triplets), and so on.

In a minimally polyvalued logic of two propositions, truth value functions must be of type f(a,b,c,d) there a, b, c and d are the four possible alternative conjunct pairs of two variables defined as:

PV0 alternative set

a "p and q are simultaneous satisfied"
b "p and not-q are simultaneous satisfied"
c "not-p and q are simultaneous satisfied"
d "not-p and not-q are simultaneous satisfied"

The number of possible alternatives depends on the number of proposition variables and the number of qualifiers to the description (1.1) to each proposition variable. One proposition p has the set {p,~p} of alternatives, i.e. 2 alternatives. Two propositions p and q has the set {p&q, p&q, ~p&q, ~p&~q} of alternatives. i.e. 2*2 = 4 alternatives. Three propositions has 2*2*2 = 8 alternatives, and so on.

disjunctive alternative set

The set of disjunctive and complementary alternative propositions with the same description shall be referred to as the disjunctive alternative set of a polyvalued logic.

In this case it is the disjunctive alternative set of a minimally polyvalued logic referred to as PV0.

application range

The meaning (description) the disjunctive alternative set expresses, defines which applications the logic can be applied to, or shorter, its application range.

It is possible to introduce other disjunctive alternative sets with other meaning of its elements (propositions). In this case the connective definitions and the axiom sets often must be modified too, and they builds up other logics in the polyvalued truth value space with other applications.
The set a-d above can e.g. be modified to express a "first p satisfied and then q satisfied" ... (temporal and causal logic).

In a minimal system (PV0) used for demonstration of properties in polyvalued logic, we need to define at least the most common used meanings of the logical connectives in our natural language: "not p", "p and q", "p or q", "if p then q" and "if p or q then both" then referred to as "negation", "(exclusive) conjunction", "(inclusive) disjunction", "implication" and "equivalents". Their meanings formalizes (defines) and interprets as:

PV0 connectives

(1) R~p (Nc + Nd) / (Na + Nb + Nc + Nd)
(2) R(p&q) Na / (Na + Nb + Nc + Nd)
(3) R(pvq) (Na + Nb + Nc) / (Na + Nb + Nc + Nd)
(4) R(p=>q) Na / (Na + Nb)
(5) R(p<=>q) Na / (Na + Nb + Nc)

The definition range of "not p", "p and q" and "p or q" can be read as "always defined" or more accurate as "defined every time either a, b, c or d is satisfied".
The definition range of "if p then q" can be read as "defined only if p is satisfied".
The definition range of "if p or q then both" can be read as "defined only if p or q or both are satisfied".
These definition ranges is interpreted from / adapted to the most common intuitive meanings of the linguistic connectives above.

Definitions in polyvalued logic are typically defined at the highest "understandable level" - their meanings shall be immediately understandable and their necessary truth for the application shall be easy to see and agree with.

At least following axioms are needed for a minimal system (PV0):

PV0 axioms

(6) Np + N~p N
(7) Na + Nb + Nc + Nd N
(8) R(p&q) R(q&p)
(9) R(p&~p) 0

Axioms in polyvalued logic are typically defined at the highest "understandable level" as well as for definitions. This is the best guarantee for the validity of the system for the application

The relative occurrence are denoted by the prefix R and the number of satisfying observations by the prefix N. In the definitions (1)-(5) all N-terms in the numerator and the denominator can be divided by N and according to axiom (7) is N the same as (Na + Nb + Nc + Nd). To divide by N transforms all terms to relative occurrences. In (1)-(5) all N can be replaced by R. If all terms has the same prefix this can be omitted. This will transform (1)-(9) to the more handy notation:

(1) ~p (c + d) / (a + b + c + d)
(2) (p&q) a / (a + b + c + d)
(3) (pvq) (a + b + c) / (a + b + c + d)
(4) (p=>q) a / (a + b)
(5) (p<=>q) a / (a + b + c)
(6) p + ~p 1
(7) a + b + c + d 1
(8) (p&q) (q&p)
(9) (p&~p) 0

The formalism of polyvalued logic can be used also for other truth value conceptions than factual truth, as e.g. probability and its variants and mix them with factual truth values. In this case or for other clarifying use, the prefix R is used again to represent factual truth.

unconditional

If a proposition's definition range is equal to the disjunctive alternative set, it is referred to as an unconditional proposition. The denominator in the polyvalued function to an unconditional proposition is always 1 and can be omitted.

conditional

If a proposition's definition range is not equal (but a subset) of the disjunctive alternative set, it is referred to as a conditional proposition.

With the denominator omitted for the unconditional statements the connective definitions (1)-(5) can be written in their most simple form as:

elemental form

(1) ~p c + d
(2) (p&q) a
(3) (pvq) a + b + c
(4) (p=>q) a / (a + b)
(5) (p<=>q) a / (a + b + c)

This form, using the elements in the disjunctive alternative set, will be referred to as the elemental form of polyvalued functions. In elemental form are all connectives expressed by the element propositions of the disjunctive alternative set.

Using the definitions and axioms the elemental form of (1)-(5) can also be transformed to:

connective form

(1) ~p 1 - p
(3) (pvq) (p&q) + (p&~q) + (~p&q)
(4) (p=>q) (p&q) / p
(5) (p<=>q) (p&q) / (pvq)

This form will be referred to as connective form of polyvalued functions.

Of course it is possible to mix elemental and connective form and the connectives form can use alternative formulations. E.g. can (3) also be expressed by the alternatives (pvq) p + (~p&q) and (pvq) q + (p&~q), both in connective form, and (pvq) p + c in mixed form.

Polyvalued connectives can not be described by the use of truth tables, because of true and false are two possibilities of infinite many possible truth values. On the other hand,

observation table

polyvalued connectives can be described by the use of observation tables using only the tree alternatives: 'satisfied', 'unsatisfied' and 'irrelevant', here represented by '1', '0' and '-'.

The polyvalued connectives in PVL0 can be described in table form:

{a,b,c,d}
(1) {0,0,1,1} : ~p
(2) {1,0,0,0} : p&q
(3) {1,1,1,0} : pvq
(4) {1,0,-,-} : p=>q
(5) {1,0,0,-} : p<=>q

These table form definitions (tuples) are complete isomorph to the corresponding function form definitions and can immediately be used as coefficients in these functions, in the following manner:

1. First, using the operator for division, can e.g. the {1,0,-,-} tuple be alternatively formulated as {1/1,0/1,0/0,0/0}
2. Then, the tuple must be separated in value coefficients and definition coefficients. The value coefficients shall be read out as the tuple {1,0,0,0} and the definition coefficients as the tuple {1,1,0,0}.
   
3. Then, the function form is constructed as:

(1*a + 0*b + 0*c + 0*d) / (1*a + 1*b + 0*c + 0*d)
a / (a + b)
(p=>q)

Table form can be used to demonstrate the meaning of a connective. The table form for e.g. (5) can be derived by the sequence:

(p<=>q)
(p&q) / (pvq)
a / (a + b + c)
{1,0,0,0} / ({1,0,0,0} + {0,1,0,0} + {0,0,1,0})
{1,0,0,0} / {1,1,1,0}
{1/1,0/1,0/1,0/0}
{1,0,0,-}

Because table form only is another representation of the function elemental form, it is possible to directly read out from the elemental form then the function is satisfied, unsatisfied and irrelevant. Of that reason the table form can be omitted. On the other hand, table form is a compact and suitable representation in certain applications, e.g. in programming.

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Page generated with items (1.1)-(1.11), (2.1)-(2.7) and (3.1)-(3.10) from postings in the discussion groups phil-logic@bucknell.edu and sci.logic.

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Item (3.2), (3.4) and (3.5) updated.

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Chance to use of DT and DD tags. Item (2.4) updated.

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