QUATERNARY LOGICAL SYSTEM
Dr. Viraj K. Hewage
1.
Binary System
In logic, the principle of bivalence states that
every proposition takes exactly one of two truth values (e.g. truth or falsehood).
The laws of bivalence, excluded middle, and non-contradiction are related, but
not the same. The law of bivalence is compatible with classical logic, but not
intuitionistic logic, linear logic, or multi-valued logic.
For any proposition P, at a given time, in a given respect, there are three
related laws:
·
Law of bivalence:
For any proposition P, P is either true or false.
·
Law of the excluded middle:
For any proposition P, P is true or 'not-P' is true.
·
Law of non-contradiction:
For any proposition P, it is not the case that both P is true and 'not-P'
is true.
Through the use of propositional variables, it is possible to formulate
analogues of the laws of non-contradiction and the excluded middle in the
formal manner of the traditional propositional logic:
·
Excluded middle
·
Non-contradiction
In second-order logic, second-order quantifers are
available to bind the propositional variables, allowing one to formulate closer
analogues:
·
Excluded middle
·
Non-contradiction
Note that both the aforementioned logics assume the law
of bivalence. The law of bivalence itself has no analogue in either of these
logics: on pain of contradiction, it can be stated only in the metalanguage
used to study the aforementioned formal logics.
Analogues of excluded middle are not valid in
intuitionistic logic; this rejection is founded in intuitionists'
constructivist as opposed to Platonist conception of truth and falsity. On the
other hand, in linear logic, analogues of both excluded middle and
non-contradiction are valid,[1]
and yet it is not a two-valued (i.e., bivalent) logic.
These different principles are closely related, but there
are certain cases where we might wish to affirm that they do not all go
together. Specifically, the link between bivalence and the law of excluded
middle is sometimes challenged.
A famous example is the contingent sea battle case
found in Aristotle's work, De Interpretatione, chapter 9:
Imagine P refers to the statement "There will be a sea battle
tomorrow."
The law of the excluded middle clearly holds:
There will be a sea battle tomorrow, or there won't be.
However, some philosophers wish to claim that P is
neither true nor false today, since the matter has not been decided yet.
So, they would say that the principle of bivalence does not hold in such a
case: P is neither true nor false. (But that does not necessarily mean that it
has some other truth-value, e.g. indeterminate, as it may be truth-valueless).
This view is controversial, however.
This logic system
shows four possible out comes for the event .It does not consider events on
dichotomously. The quaternary logic system considers another two out comes
which are neglected by the binary logic. The quaternary logic2 provides four
logical alternatives concerning any event as follows: A, not-A, A and not-A,
neither A nor not-A. It will be argued that these
reflect a progressive complexification of
understanding to whatever event they are applied.
Applied to any
exercise, it is proposed to: start with the constraints of the text (A); to
ignore those constraints, treating the text metaphorically, in search of a
patterning principle (not-A); to endeavor to fit the pattern to the text with
any necessary adjustments (A and not-A); and then to explore the insights
implied by that result, reaffirming the pattern, but unconstrained by the
limitations of the particular representation (neither A nor not-A).
7.
Aristotelian Logic Vs Nirvanic views
As noted above a
number of 4-fold sub-sets (whether or not part of a larger sub-set) are
explicitly structured in terms of: A, not-A, A and not-A, neither A nor not-A.
As noted, these suggest a progressive complexification
beyond the constraints of Aristotelian logic, which remain, however, a sub-set
of the sequence. This implies a progressively increasing challenge to
comprehension, suggesting that the more "nirvanic"
views might be based on elements of the pattern governed by "neither A nor
not-A" (as implied by the phrase "not this, not that",
traditionally associated with that state). Correspondingly, the least nirvanic views might be based on elements of the pattern
governed by "A".
8. Quaternary logical concept on early Buddhism
This is not a new
challenge, although it may now appear more dramatic to some.
An intriguing point of departure is a classic
Buddhist text entitled the "Brahmajala Sutta" (The Discourse on the All-Embracing Net of
Views). This appears to be unique in endeavouring to
map out as a system the complete set of fundamental viewpoints. It is the first
sutta in the entire collection of the Buddha's
discourses in the Pali Tipitaka.
Its importance stems from its primary objective, namely the exposition of a
scheme of 62 cases designed to include all possible views (past and future) on
the central concern of speculative thought, the nature of the self in relation
to the world.
This teaching
reflects insights formulated 2,500 years ago. It is not the intention to be
strictly faithful to it. Rather the intention is to be guided by a method which
is based on a fundamental patterning principle in the text itself. The logic of
that pattern is to a high degree tetra-Iemmic in the
sense that it provides for four logical alternatives concerning any thesis: A,
not-A, A and not-A, neither A nor not-A. It will be
argued that these reflect a progressive complexification
of understanding to whatever domain they are applied. Applied to this exercise,
it is proposed to: start with the constraints of the text (A); to ignore those
constraints, treating the text metaphorically, in search of a patterning
principle (not-A); to endeavour to fit the pattern to
the text with any necessary adjustments (A and not-A); and then to explore the insights
implied by that result, reaffirming the pattern, but unconstrained by the
limitations of the particular
representation (neither A nor
not-A).
Multi-valued logics and fuzzy logic have been considered
better alternatives to bivalent systems for handling vagueness. Truth (and
falsity) in fuzzy logic, for example, comes in varying degrees. Consider the
following statement.
The apple on the desk is red.
Upon observation, the apple is a pale shade of red. We
might say it is "50% red". This could be rephrased: it is 50% true
that the apple is red. Therefore, P is 50% true, and 50% false. Now consider:
The apple on the desk is red and it is not red.
In other words, P and not-P. This violates the law of
noncontradiction and, by extension, bivalence. However, this is only a partial
rejection of these laws because P is only partially true. If P were 100% true,
not-P would be 100% false, and there is no contradiction because P and not-P no
longer holds.
However, the law of the excluded middle is retained,
because P and not-P implies P or not-P, since "or" is inclusive. The
only two cases where P and not-P is false (when P is 100% true or false) are
the same cases considered by two-valued logic, and the same rules apply.
Of course, it may be stated that bivalence must always be
true, and that multi-valued logic is simply by definition a vague state of
perception. That is, multi-valued logic is a convenient way of saying, "This
instance has not been observed in enough detail to determine the truth value of
P." In other words, if a pale apple is 50% red (where red is noted as
P), then P can be said to be 100% true, noting that bivalence makes little
delineation as to the nature of not-P aside from the given, meaning that the apple
might very well be 50% white as well (when white is noted as not-P), meaning
that P and not-P can both be true, but in separate instances, as they both
exist as separate colours, which combine in a larger instance set in perhaps an
unobservable, exceedingly subtle way to create the shade of pale red. In this
case, the apple might be set S, which consisted of P and not-P to greater or
lesser or equal respective degrees, as long as it is acknowledged that P and
not-P are separate instances within a set instance. In this way, bivalence
simply states that white cannot be red, and makes no claim about the colour of
the set instance, to which is applied multi-value logic, in which case
multi-value logic is simply derivative of bivalence as well.