2 Analytic logic
and Boolean lattices
This
is part of a longer thesis advancing a refutation of strong AI. To download the thesis visit: -
For an introduction to the work as a
whole visit: -
Introduction
to Poincare’s thesis by Peter Fekete
Modern formal logic is a rigorous
expression of the syllogistic logic of Aristotle. Regarding the foundation of his logic,
Aristotle wrote: -
Whenever three
terms are so related to one another that the last is contained in the middle
as in a whole, and the middle is either contained in or excluded from
the first as in or from a whole, the extremes must be connected by a perfect
syllogism. By a middle term I mean one
that is itself contained in another and contains another in itself;
this term also becomes middle by position.
By extremes I mean both that term which is itself contained in
another and that in which another is contained. (Aristotle [1964 / c.400 BC] p.
8)
My underling. The extract illustrates how the idea of
syllogistic logic arises from an analogy with containment in space, and
is developed from the relation of part to whole.
If A is contained in B
and B in C then A must be contained in C.
If A is contained in B
and B is wholly separate from C, then the A cannot
be contained in C. To say that A is contained in C is a contradiction. Every tautology is founded on a relation of
containment in space. To illustrate
this, consider the tautology, 
. This concerns two
propositions, p and q, each of which can be negated to give 
and 
. Let there be a
partition of space to which these four propositions, 
apply. We cannot have 
, for that is as much as to say that a partition is not
contained in itself, which violates geometric intuition. Likewise, 
is impossible. All other combinations of are contingently
possible. Hence, the space may be
divided into four partitions: -
As indicated, let us identify these
partitions with sets 
. Here the numerals 1,
2, 3, 4 signify nothing more than arbitrary distinct names for the
undifferentiated content of each partition – whatever state of affairs it is in
the world that makes the propositions true.
The do not necessary stand for numbers. Let us call these four partitions atoms
of the space generated by contingent propositions p and q.
The proposition p corresponds
to the union of partitions 
, which is 
and 
corresponds to the
partition 
. Hence, since
partition 
is contained in partition , if 
is true then must be true; this
gives 
and the tautology follows. There are 16 combinations of p, q
corresponding to 16 partitions of the space represented by 
.
2.1 Geometric
definition of a lattice
A poset [Chap. 2, Sec. 2.5.2] L is called a lattice
if for every 
. Let 
and 
. The element 
is called the “join”
of x and y and the element 
is called the “meet”
of x and y.
2.2 Largest and
smallest elements
The symbol 1
(bold typeface) denotes the largest element in the lattice and is the join of
all the atoms. Likewise, the symbol 0
denotes the smallest element of the lattice, which is the meet of all the
atoms.[1]
From this definition we can construct
the model of the propositional logic of p and q, which is
a Boolean lattice – that is, a complemented, distributive lattice.
2.3 Boolean
lattice of two propositions
This lattice shall
be denoted, 
, where 
and
(Cartesian
product) Here 
denotes the complement
of p.
The properties of are significant
because they are inherited by all finite Boolean lattices.
2.4
Definition, Boolean lattice / algebra
A Boolean
lattice, also called a Boolean algebra, is any structure
, subject to the axioms:
is
a distributive lattice[2]
I shall be using the terms lattice
and algebra interchangeably, but it is useful to appreciate their
different nuances. The term lattice
emphasises the structural aspect – an abstract and rigid object that is
visualised in the above diagram by the points and the lines joining them. The term algebra emphasises the
relation to the language used to describe this structure. So we need both terms. This lattice / algebra is also related to the logic built over the lattice. Formal analytic logic is a structure built
for the purpose of conducting inferences, whereas the lattice is a structure
conceived as a collection of relations given a priori. Analytic logic is the application of the
analytic properties of a lattice to the purpose of inference. It is customary to use the same symbols for
the join 
and meet 
in the lattice as
those used for the logical
operations of disjunction
and conjunction
. This
is an abuse
of notation nonetheless, because
of the intimate relation between the logic and the lattice, it is natural and
convenient. The lattice complement 
corresponds to
negation 
in logic. The top and bottom elements, 1 and 0
are said to be “distinguished elements”; this just simply means that they are
marked out in our language by distinct symbols; we do not use 
for 1; it would
be inconvenient.[3] In the discussion that follows any general
property of is inherited by all
finite Boolean algebras.
Concerning
Boolean algebras, there is an important principle of duality: -
2.5 Principle of
duality
Every Boolean
algebra has an isomorphic dual formed by interchanging 0 for 1
and 
for 
at every lattice
point.
This
is part of a longer thesis advancing a refutation of strong AI. To download the thesis visit: -
For an introduction to the work as a
whole visit: -
Introduction
to Poincare’s thesis by Peter Fekete