Part 01:
Arithmetical Models of Logics Series.
‘Fractional-Boolean Arithmetic’:
A “Half-Way House” on the Road to
Dialectical Logic.
Dear Reader,
It
is my pleasure,
and my honor, as an elected member
of the Foundation Encyclopedia Dialectica [F.E.D.]
General Council, and
as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release by the F.E.D. General Council, Seldon’s commentaries on key Encyclopedia
Dialectica concepts of Seldonian Theory.
This 1st text in
this new such
series is posted herewith, together with
supporting text-images and diagrams
[Some E.D.
standard edits have been applied, in the version presented below, by the editors
of the F.E.D. Special Council for the Encyclopedia,
to the direct transcript of our co-founder’s
discourse].
Seldon
–
“Many are the
arithmetical models of formal logic, and of extensions of logic to beyond
formal logic, that call upon the unit interval of ordinary arithmetics, [0, 1], or that
call upon some portion thereof, as the “space” in which to restrict arithmetic
so as to conform it to the operations of the given logic to be modeled
arithmetically and algebraically.”
“Such
is the case with Leibniz, with Boole, and with others, some of whose works on
logics we plan to review in the course of this series.”
“When
we take Boole’s ‘arithmetic of “Election”
operations’, which we
denote by the ideogram W_E ,
with ‘pre-subscript’ ‘W ’ denoting the first-order-logic axioms-system(s) of
the numbers W º {0, 1, 2, 3, …}, as our
“Whole-Numbers”-based
first category or ‘«arché»-category’ for a logics-arithmetics’ dialectic,
with its WE number-space
restricted to {0, 1}, or, at best, to {0/1, 1/1, 0/0, 1/0}, we do not arrive at an
arithmetical model of formal logic which appropriates the entire ‘“ratio-nal
numbers”’-based, or Q-based,
unit-interval. We do get a
Boolean-arithmetic model of formal logic, whose calculations are fit for a
logic of deductive certainty – a [formal] ‘Certainty-Logic’.”
“We
solve for the next category of logics in this dialectic progression of
logics-categories, i.e., for the first contra-category to arché»-category W_E , by the
‘ideo-ontological’ category which we sign by W_Q , as the “Whole-Numbers”-based arithmetical model of categorial dialectical
logic.”
“However,
inescapably, ‘ideo-meta-genealogically’, ‘ideo-hereditarily’, the categorial-dialectical-logic-arithmetic,
W_Q , inherits
some of the restrictiveness of the Boolean formal-logic-«arché»-arithmetic, W_E , with regard
to any full appropriation of the full, Q-based
unit-interval for a logic.”
“With
our WQ solution to the
algebraic-unknown second-category-symbol, W_q EE, we get a space of multiple whole
unit-intervals; of multi-dimensional,
mutually perpendicular unit-interval, unit-length line-segments. But these unit-intervals are ‘a-tom-ic’
– i.e., ‘uncuttable’, ‘im-part-ible’
-- yielding an arithmetical-logical language in which no
unit-interval fractional components can be expressed.”
“This
solution does give us an algebra modeling a categorial
dialectical logic. In this model of
dialectical logic, the ontological “classes” or categories progressions
expressible include all of the categorial-self-&-other-combinatorial
potential or possible categorial-ontological existences for a given
stage of categorial-ontological self-expansion -- yielding a
dialectical-ontological ‘Possibilities-Logic’.”
“Again,
when we arrive at the third ‘ideo-ontological’ category, the
‘first uni-category’, or ‘dialectical synthesis category’, in this dialectic of
categories of arithmetical models of logics, we again inherit
something of our «arché»-category’s
restrictions against full unit-interval utilization.”
“This
third arithmetical model of logics, which we solve for as W_ q QE |-º A , represents a ‘dialectical synthesis category’, or ‘complex unity category’, of the W_E ‘thesis-category’ and/with the/its W_Q ‘antithesis-category’, by combining a
Boolean ‘‘‘coefficient’’’, denoted herein by e(t), inherited from W_E , which can
express the logical quantities “ALL” or “NONE” [but not the
logical quantity “SOME”], with a – this time [logically-]quantifiable
class or category quality-symbol, inherited from W_Q , and denoted
generically by uW,
so that the full generic unit of the
W_q QE ontological
categorial progressions is the product
e(t)·uW.”
“Thus,
e(t)·uW can express the
Actualization
[or the non-Actualization]
of a given ontological category or class, i.e., of the wth class, uW, in a given stage, t Î W, of the ontology-self-development of a given
ontological Domain – an ‘Actualizations-Logic’.”
“If
the ‘‘‘Boolean coefficient’’’ of class-unit uW at stage t, e(t), is 0, then the individuals represented by class-unit uW are absent – e.g., extinct – or do not Actualize [yet]
in ontological self-developmental stage t –
e(t)·uW = 0 ´ uW = u0
–
u0 being the symbol for the, Boolean, “origin” of the
WqQE |-º A space, as overall non-Boolean space, and
such that –
u0 ´ u0 = u0+0 = u0
–
so u02 = u01 = u0, conforming to the Boolean “Fundamental Law”, or “Law
of Duality”, a la the Boolean versions of –
0 ´ 0 = 02 = 0 and 1 ´ 1 = 12 = 1].”
“If
the ‘‘‘Boolean coefficient’’’ of ontic class uW at stage t, e(t), is 1, then the individuals represented by class uW are present
– e.g., fully-formed – that is, they do ‘‘‘Actualize’’’ in ontic self-developmental stage t –
e(t)·uW = 1 ´ uW = uW
“But
all of these arithmetical models of logics still exclude the utilization of the
potentially descriptively-valuable mathematical-semantic resources of an
arithmetical language that explicitly includes unit-interval fractions, and,
thereby, also includes the logical-quantity “SOME”, not just the
logical-quantities “NONE” and “ALL”, as so far.”
“That
is, so far, “NONE” of the logic-arithmetics in the ‘meta-system-atic’
dialectical ‘ideo-ontological’ categorial progression of axioms-systems of
logic-arithmetics so far recounted – W_E_3 ¶|-º
W_E__ ~+~ W_q EE ~+~ W_q EEE
¶|-º
W_E__ ~+~ W_q Q ~+~ W_q QE |-º
W_E__ ~+~ W_Q ~+~ W_A_
–
or –
‘Formal-Certainties-Logics’ ~+~
‘Ontological-Possibilities-Logics’ ~+~
‘Existential-Actualizations-Logics’
--
allow for an arithmetical modeling of the “SOME” logical quantity via the fractions of the Q_ axioms-systems-of-arithmetics’ unit-interval. [in the formulas above, ‘¶|-º’ denotes a partial-qualitative
[‘¶’], not-yet-univocal/ ‘‘‘named’’’
solution to the algebraic unknowns
of the W_ Q arithmetical
language in which those formulas are written, and ‘|-º’ denotes a full univocal solution – an
assertion [‘|-’] of a definition
[‘º’] for the unsolved
category-symbol on the LHS of this solution-relation-symbol, by the
category-symbol/‘‘‘name’’’ on its RHS. The symbol ‘~+~’ stands for ‘additions of categorial opposites’.].”
“In
original Boolean algebra, the ‘Boolean coefficient’ (0/1) functions as the logical-quantity “NONE”, signifying
that none of the class/category symbol that it
modifies/multiplies exists. The ‘Boolean
coefficient (1/1) functions as the logical-quantity “ALL”, signifying
that all of the class/category symbol that it modifies/multiplies
exists. The ‘Boolean coefficient (0/0)
functions as the logical-quantity “AMBIGUOUS”, signifying that none, some, or all
of the class/category symbol that it modifies/multiplies exists. The ‘Boolean coefficient (1/0) functions as the Boolean logical-quantity “INFINITY”
or “IMPOSSIBILITY”, signifying that class/category symbol that it
modifies/multiplies cannot exist given the premises. There is no symbol in the original Boolean
logic-language that distinctly, unambiguously, separately signifies just the
logical-quantity “SOME”.”
“The
following1 ‘Boolean-logical’
calculation, that Boole used to illustrate his method for the disambiguation of
his logical-arithmetic operation of division, illustrates the point noted just
above [note: in Boole’s language, (1 - x) stands for “not
x”. Thus, if x = 1, then (1 - x) =
(1 – 1) =
0; if x = 0, then (1 – x) =
(1 – 0) =
1.]:
Given the Boolean translation
of the definition “humans are rational
animals”, namely, the Boolean
logical-equation –
h = ra
– Boole derives the
definition for the class
“rational [beings]”
that is implied by the
h = ra
definition,
by a “logical-arithmetical division” operation, as follows.
By Boolean-algebraic manipulation,
Boole notes that –
r = h/a.
Boole then applies his “development” function to evaluate the
‘‘‘fraction’’’/division h/a, obtaining, on his
logical-equations RHS, logical-quantifications for all combinations of a
and not-a with h and not-h
–
r = (1/1)ah + (0/1)a(1-h) +
(0/0)(1-a)(1-h) +(1/0)(1-a)h
– meaning that the members
of the class of ‘‘‘rational beings consist of:
all animals [(1/1)
of them] that
are humans,
+
no animals [(0/1)
of them] that are not humans,
+
an indefinite
remainder [(0/1) of them; none, some, or all]
of beings neither animal nor human [android robots?],
+
the assertion: humans who
are not animals cannot exist/are IMpossible given the premiss-equation h = ra’’’.
1[based upon George Boole:
Selected Manuscripts in Logic and its Philosophy, Ivor
Grattan-Guiness and Gérard Bornet, editors, pp. 98-104.].
“A
way around this Boolean-logic-rooted, sub-unit-interval restrictiveness and
‘some-less-ness’ arises by way of considering the Boolean, W_E_ expressions of
the highly-problematic paradoxes immanent to formal logic. What we are seeking, thereby, is a dialectical, immanent critique of Boolean formal-logic-arithmetic, that lead us in the direction of at least a rudimentary version of a dialectical-logic-arithmetic.”
“The
Boolean solution of the Boolean logical equations which express the paradoxes
of formal logic are not strictly part of the original-Boolean
logic-arithmetic space – not part of that ‘Boolean arithmetic’. Those solutions are outside, and point beyond
the closure of, original-Boolean numbers-space and their analytical geometry.”
“However,
those simple solutions suggest an extension of Boolean arithmetic/algebra that
would encompass an explicit, distinct logical-quantity “SOME”, via the fractions
of the Q_ axioms-systems’ unit-interval.”
“Original
Boolean algebra is dualistic, radically “either/or”, strictly “black and white”,
1 or 0, “ALL” or “NOTHING”.
But the whole interior of the Q_ axioms-systems’ unit-interval is “gray”. It can be used to describe existences which
are “SOME”-times present,
and also “SOME”[-other]-times absent,
and to describe propositions which are “SOME”[-other]-times TRUE, and also “SOME”[-other]-times FALSE.”
“Boole
divides the meanings of his Boolean-arithmetical logical equations into two
classes, “primary” and “secondary”. His “primary”
class units are ‘‘‘existential’’’, addressing the existence or non-existence
of definite classes. The Boolean-value 1 stands for existence. The Boolean-value 0 stands for non-existence.”
“Boole’s
“secondary” proposition units are ‘meta-propositional’ units, each one made up
out of a heterogeneous multiplicity of two or more “primary” proposition units.”
“For
example “no animals that are not humans exist as rational beings.”, and “no
humans that are not animals can exist.”, are both “primary propositions”
for Boole. But the [meta-]proposition “If
no animals that are not humans exist as rational beings, Then no
humans that are not animals can exist.”, is a “secondary proposition” for
Boole, and, for such [meta-] propositions, the Boolean-value 1 stands for “TRUE”, and the Boolean-value 0 stands for “FALSE”.”
“Now
consider the paradoxes of formal logic, such as the ancient paradox, known as
the “pseudomenon” of Epimenides the Cretan – “I am a Cretan and all Cretans are liars”, or, in a more modern form of it – “This sentence is false”, or the Russell paradox “The
set of all sets that are not members of themselves is a member of itself”.”
“If
you take any one of them to be a command, programming your mind to think them
through, then you experience a ‘truth-value oscillation’, a rapid alternation
between the truth-values “TRUE” and “FALSE”, or Boolean 1 and 0.”
“Now,
Boole has a particular way of defining the truth-values “TRUE” and “FALSE” for
his “secondary propositions”. Boole ‘temporalizes’
his notion of propositional truth. Thus,
if t denotes the time/duration during which a proposition is true, then (1 – t)
denotes the time/duration during which a proposition is false.”
“Applying
this model to any of the formal paradox propositions, such as “This proposition
is false.”, takes the form of the Boolean equation for a formal contradiction,
‘t equals not-t’, or –
t = (1 –
t)
–
e.g., for |x| signifying the ‘“absolute
[truth-]value”’ of proposition
x –
|This proposition is false.| = |This proposition is true.|.”
“Now
if we, still confining our solution within the
unit-interval, allow ourselves to solve this algebraic equation
for the value of t, we
obtain the simple solution –
t = (1 –
t),
t + t = (1 –
t) + t,
2t = 1 + (–t
+ t),
2t = 1 + (0),
2t = 1,
2t/2 = 1/2,
t = ½.”
“Now,
of course, Boole wanted his algebraic models of “secondary propositions” to
solve as either t = 1 or t =
0, i.e., as always true – as “eternal
truths” -- or as never true; as “eternal falsehoods”.”
“However,
for the paradoxical propositions that are problematically immanent to formal
logic, their Boolean truth-value2 might be extended to include t = ½.
2[for a more complex derivation of a similar
point, see Nicholas Rescher, Many-Valued Logic, McGraw-Hill
Book Company, NY, 1969, pp. 89-90.].
“Should
this mean that such, paradoxical, or contradictory propositions are true for
the first half of eternity, and false for the second half? Or false for the first half of eternity, and
true for the second half?”
“It
might most often be more apt for applications if we interpreted t = ½ as signifying an ongoing alternation
between “true” and “false”, so that, in the aggregate, the t = ½ proposition is true half of the
time, and also false [the other] half of the time, that
is, is true “SOME” of the time, and also false “SOME” of the time.”
Propositions
that could be well-modeled in the way of this ‘fractional-Boolean’ extension of
the ‘original-Boolean’ language include “It is day time.” vs. “It is
night time.”, and “It is raining.” vs. “It is not raining.”, as
well as “The economy is expanding.” vs. “The economy is contracting.”,
and, of course, many others such.”
“Of
course, this strictly ‘‘‘half-and-half’’’ way of incorporating the
logical-quantity “SOME” into an extended form of Boolean arithmetic-algebra – ‘fractional-Boolean
arithmetic-algebra – does not
do justice to the ‘intermediate classes’, such as the class whose sub-classes
include twilight, dusk, and solar eclipses, each of which partakes of both day time and night time, but is neither day time nor night time, in the case of the mutually-negating
propositions:
“It
is day time.” vs. “It is night time.”
“This
‘fractional-Boolean dialectics’; this generic Boolean “unity of opposites”, or ‘equality
of opposites’:
t = (1 –
t)
– seems to be limited, in its modeling efficacy, to the simplest, most abstract
stage of dialectics-in-general, e.g., to a likeness to the beginning dialectic in
Hegel’s dialectical «Logik», his –
‘BEING ---) NOTHING ---) BECOMING
– dialectic3; i.e., to a kind of “half-way house” on the way to a
fuller formulation and representation of dialectical logics.”
3[for Hegel’s exposition of this, his opening
dialectic, see G.W.F. Hegel, Science of Logic, translated
by A. V. Miller, Humanity Books, Amherst, NY, 1969, pp. 79-108.].
“Note
that such ‘Boolean-dialectical’ equations can be formed, quite simply, which
model asymmetric
sharings of ‘times-true’, if we allow non-Boolean,
beyond-Q-unit-interval values –
uninterpretable for Boole’s logic-algebra – in the unsolved Boolean-algebraic equations4, that drop out of expression as the solution-seeking sequence
of these algebraic equations moves toward their solution-conclusion, e.g. –
[t = 1 – 2t] Þ [3t = 1] Þ [t = 1/3; 1 – t = 2/3];
[t = 2 – 2t] Þ [3t = 2] Þ [t = 2/3; 1 – t = 1/3],
–
and, more generally, given m
< (n+p) –
[nt = m –p t] Þ [(n+p)t = m] Þ [t = m/(n+p)].”
4[Boole himself engages this expedient,
conducting calculations for his probabilistic applications of his algebra of
logic with transitory coefficients that are > 1, values “uninterpretable” for Boole’s logical arithmetic. See, for examples, George Boole, An Investigation of the
Laws of Thought on which are Founded the Mathematical Theories of Logic and
Probabilities, Dover Publications, Inc., NY, 1958, pp. 279, 280, 281,
292, 293, 297, etc.].
“Moreover,
‘Boolean thinking’ is violence-prone and war-prone. It leads each side in a conflict to think
that its propositions have Boolean-value 1 – denoting ABSOLUTE, ETERNAL TRUTH – and that all
other, opposing sides’ propositions in this conflict have Boolean-value 0 – denoting ABSOLUTE, ETERNAL FALLACY.”
“In
actuality, probably/usually, each side’s propositions contain “SOME” of the
truth, but each, opposing, other side’s views also contain “SOME” of the
truth. Each side’s propositions are
likely part[ly]-true, but also part[ly]-false.
Such realizations can ‘acceleratedly’ conduce negotiations toward nonviolent
success in engineering a ‘‘‘dialectical synthesis’’’ or ‘‘‘complex unity’’’ of
the opposing views.”
“Of
course, the Boolean “Universe”, as modeled by the “closed” unit-interval, with
end-points 0 and 1 inclusive, may involve not just two, but
a multitude of classes and subn-classes. Even
the extended, ‘fractional
Boolean arithmetic’, Q_E_, gives no way to distinctly represent these classes
and sub-classes as explicit fractional-length line-segments within the
extended-Boolean unit-length line-segment, and no way to order them, to place
them in a sequence of such sub-segments.”*
*[One might hope to concretize the exact ‘SOME’-ness;
to ‘explicitize’ the ‘fractional density’, or ‘ratio-nal density’, or ‘sub-unit[-interval]
segmental length’ of each given, ‘Fractional-Boolean’, class, or even of each ‘subn-class’, contained within a given Boolean ‘‘‘universe
of discourse’’’.
One
might hope to do so by counting the number of members of each class [and even
of each ‘subn-class’] of that “Universe”,
by assigning to each such member, as its ‘‘‘metric’’’, the cardinal
value unity, ‘1’, [not
the Boolean unity value], then summing these cardinal units for each of that class’s
members to get the count of all of the members in that class, and then dividing
that class-members-count by the count of all members in that “Universe”
as a whole, so as to calculate “the percentage of the Universe” represented by
that class, and then assigning a sub-segment within the Q unit-interval, of the corresponding fractional length,
thus explicitly Q-quantifying
that class’s ‘SOME’-ness.
However,
even apart from the potential arbitrariness and bias of measuring each member
as cardinal 1 – e.g., counting the Sun the same as a sub-atomic “particle”
– one would need, in order to practically implement such a method, of rectilinearly
mapping, to a distinct percentage sub-segment of the Q unit-interval, as a fraction of 1, each class of that ‘‘‘universe of discourse’’’, one would have to have a recent ‘‘‘census’’’ of each class’s membership ready-to-hand].
“Even
‘extended/fractional Boolean arithmetic/-algebra’, Q_E_, can get us only so far into the Domain of an explicit, descriptively-apt formulation of
a veritable dialectical logic – can get us only to a “half-way house” on
that journey.”**
**[Given that both Boolean arithmetical/algebraic {0, 1}-logic, and another {0, 1} arithmetic – the base-2, binary arithmetic – reside at the conceptual and
theoretical basis of modern digital computers, one might wonder as to whether W_Q_ and Q_E_ have any implication(s) for future
computer-kinds. For example, might there
be a connection between the, stage one triad of W_Q_ logic, and the “three-valued-ness” of “quantum
computer” logic, which still conserves something of the Boolean {0, 1}, but also adds a ‘superpositional’ ‘‘‘complex unity’’’ of 0 and 1 as third value, in a kind of “quantum-mechanical” dialectical
synthesis by superposition of Boolean 0 and 1?].
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¡ENJOY!
Regards,
Miguel
Detonacciones,
Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];
Elected Member, F.E.D. General Council;
Participant, F.E.D. Special Council for Public Liaison;
Officer, F.E.D. Office of Public Liaison.
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