Contents:

1. First Order Logic
2. Intuitionist and modal Logic
3. Functions of a trivalent logic
Conclusions:

First Order Logic:

First order logic is defined as that logic which considers only boolean propositions - that is propositions which must be either true or false. It is generally used as a synonym for formal logic, which itself is generally synonymous with Aristotle and his medieval scholastic successors.

Aristotle does briefly considers statements which are neither true nor false - a prayer, for example - but then dismisses the category of statements which are neither true nor false. The formal syllogistic proposed by Aristotle is founded upon the presupposition that all statements are either true or false.

First order logic is of course not the only formal system for the representation of thought. Indeed the limits of first order logic ar already implicit in many paradoxes. The paradox of Epimenides (all Cretans are liars... I am Cretan) shows the problem. Not all statements can be evaluated as either true or false.

If first order logic is limited by its binary nature any escape from its limitations would arive through a consideration of multivariate logics. Digital logics are based on a presumption of discontinuity, and currently dominate machine intelligence. However prior to the invention of the vacuum tube, analog logics were dominant. Analog systems are founded upon a presumption of continuity. The difference is not purely technical: calculus is founded upon the question of whether entities are continuous or discontinuous. So also is the question of ancient atomic theory.

In determining whether quantities are discontinous or not we may look in fact to certain paradoxes. Xeno's paradox (the hare which runs faster than Achilles) in fact illustrates one of the problems of presuming that entities are infinitely divisible - and thus appears to be an argument for an atomist position that entities are wholes. Yet the fact that we cannot express the ratio of a circle to its circumference or the square root of two seem to be evidence that entities are in fact infinitely divisible. The question in any event seems resolved from the perspective of physics: if matter and energy are two aspects of the same phenomenon, then any material dimension of an object is variable. The question of continuity apparently resolves itself in fact as a false dichotomy based on the erroneous proposition that matter is static.

In terms of logical theory however we are still left with two models: analog (continuous) and digital (discontinuous). Computational devices can of course be built (and have been built) upon either model. So an alternative multivariate logic could be either analog or digital.

Probabalistic reasoning (if x occurs then it is 37% likely that y will occur) is one example of a form of analogical reasoning. In probabalistic reasoning N trials over D units yield R result. For example, in a bag there are 10 white beans and 90 red beans. It is thus 10% likely that in 1 attempt I will draw a white bean. The reason that I say that probabilistic reasoning in practice is continuous is that we may have an infinite number of trials working with an unknown domain. A finite number of trials and a finite domain of data can be analyzed as discontinuous. But if the denominator (the domain being tested) is infinite then the analysis would have to be indeterminate or calculated using Newton's calculus.

*Suppose for example that our bag of beans had 100 beans, 10 white, 90 red, but that with each bean removed a new bean is introduced which may be either color. In such a case the probability of drawing any bean after the first is indeterminate.

While analog models of reasoning are admissible (e.g., it is n% probable that x is true) our consideration of alternatives to first order logic will focus primarily on trivalent logic. This perspective is chosen because for several reasons. First, while limiting ourselves to only three values is a simplification, this very simplicity also limits the field of analysis to a definite number of functions. Second, the vast majority of todays computing devices are digital. Barring a breakthrough in biological computing (using living organisms as computing devices) or in fuzzy set theory, or in crystalography computing devices will likely remain digital. And finally, there exist already three valued logics proposed as alternatives to first order logic.

2. Intuitionist and modal Logic

We also wish to note that modal logic, the logic of what is possible and necessary, adresses the question of sets. Since these sets can be universal, partial, or empty there is a numerical link between modal and trivalent logic which either permits intuition - or confusion. The empty and universal sets would respectively refer to statements which are known as true or false, while the partial set would be analogic to the indeterminate value. Such analogy, while possible, may like any analogy be true or false depending upon the context in which it is used. Modal logic does however raise an interesting paradox: In practice, no universal statement is true. Which is of course a universal statement. We can of course escape from the paradox simply by shifting from practical reasoning (here, the object language) to theoretical reasoning (here, the meta language). The practical counsel can also be rephrased to avoid paradox: it is imprudent to state case as universally true or false. In fact latent in modal logic is the possibility of probabalistic logic. For it is not so outlandish to add terms such as 'more often than not' and 'less often than not'. And from there, it is a short step to a statistical analysis.

3. Functions in trivalent logic

It seems to this author, in practical terms, that human reasoning is based upon two propositions:

1) A known antecedent may imply a knowable consequence

2) From an unknown antecedent no consequence can be derived

We intend to consider these propositions as 'primitives' in our system of thought.

Our consideration of trivalent logic will commence with unary operations on 1 variable. Given a variable 'a' which may have any of three values true=1, false=0, or unknown ¿, there are 27 possible functions to determine the value of 'a' numbered below as f1 through f27:
 

  fffffffffffffffffffffffffff
  123456789111111111122222222
           012345678901234567
-----------------------------
0 000000000111111111¿¿¿¿¿¿¿¿¿
1 000111¿¿¿000111¿¿¿000111¿¿¿
¿ 01¿01¿01¿01¿01¿01¿01¿01¿01¿

Expressed in english, f1(a) states: Whether a is true, false or unknown is irrelevant to the consequence of a which is false. We could calls this “the indifference functor” – except f13 and f27 are also indifferent.

If we presume that fn(a) -> b we may then state that function f1 in English as "Whether 'a' is true or false is indifferent: It is not the case that'b'".

Function f2 would then state: "If 'a' is known then it is not the case that 'b'. But if 'a' is known then it is the case that 'b'."

In a similar fashion,
f3(b): "If 'a' is the case then it is true that 'b'; otherwise 'b' is not so.

Each of these functions can be translated into ordinary English, though that would be rather tedious. However while tedious these function are neither trivial nor absurd. The reason that we delineate them is twofold:

1) First order logic tends to ignore that its representation of the functor of implication is in fact not realistic.

Consider the English statemen 'If p (it is raining) then q (it is wet)'. The difficulty which arises is what value to assign this relation of where not q (it is not raining). Because of this ambiguity, the English statement described above could be expressed as the four following functions:
 

pq 
00 0011
01 0101
10 0000
11 1111

Potential sources of confusion arise first in that the function is binary; that is we have two functors and a relation. To avoid the potential confusion between the value of p and q and the relation of p and q we consider first only unary operations. (This consideration is also justified by the fact that the possible number of functors in a binary trivalent system is 222222222 (trinary) or 27*27. By limiting our consideration to 9 unary functions we expose the foundation of this system, hopefully both elegantly and simply.

Returning to our original source of consternation, what does it mean when we say 'If p (it is raining) then q (it is wet)?' With 3 values we can say:

p (It is raining)
not p (It is not raining)
p unknown (We do not know whether it is raining).

If we return to our two 'laws'

1) A known antecedent may imply a knowable consequence
2) From an unknown antecedent know consequence can be derived

Then we know that in our system only the following functions may be valid:

f3, f6, f9, f12, f15, f18, f21, f24, f27

for the other functors derive known conclusions from unknown antecedents.
 

  fffffffffffffffffffffffffff
  123456789111111111122222222
           012345678901234567
0 000000000111111111¿¿¿¿¿¿¿¿¿
1 000111¿¿¿000111¿¿¿000111¿¿¿
¿ 01¿01¿01¿01¿01¿01¿01¿01¿01¿

And the relation expressed by if p (it is raining) then q (it is wet) is f24

For it is in fact impossible to determine the consequences of an implication wherein the precedent is false. This pragmatic application of Occam's razor may be inelegant but is in terms of praxis irrefutable.

Conclusions:

To conclude we should note that the reader or another logician can develop functions arbitrarily. Thus if we wish we could define a function funct(p) : p = 3*p. One of the theses of intuitionist logic is that propositions are constructed arbitarily. Nth order logics can be developed - with an exponential expansion of the number of functors considered. It is however our position that a purely binary system of logic is too limited to consider certain paradoxes which arise out of the 'law' (p or not p). The existence of paradoxes and probabalistic reasoning tend to show that first order systems of logic are inadequate. First order logic cannot in any case analyze the paradoxes they generate. Logics of n-valence converge from digital to analog logic as n approaches infinity, and are thus the easiest method of representation and transition from binary to multivariate and probabilistic logics.

Note:
André Delessert, Introduction à la logique, Presses Polytechniques Romandes, p. 172.