Download 1 The calculus of “predicates”

1
The calculus of “predicates”
This is part of a longer thesis advancing a refutation of strong AI. To download the thesis visit: Poincare’s thesis
For an introduction to the work as a whole visit: Introduction to Poincare’s thesis by Peter Fekete
Up to now the subject of our enquiry has been the classical propositional calculus and its

n
correspondent model, the class of all finite Boolean lattices: B : B  2 , n 
.
The predicate calculus
adds to the language of the propositional calculus: names of individuals belonging to some domain or
universe of discourse; variables standing for these names (ranging over the domain), predicate symbols,
and quantifiers. In first-order logic there are also function symbols, but we concentrate for present just
on the predicate part of the calculus. A typical formula of the predicate calculus is just Pa , where P
stands for a predicate and a for an individual [Defined, chap. 2 / 2.1]; the standard example is,
“Socrates is mortal” where a is an alternate name for Socrates (“Socrates” is a name of Socrates) and P
is the predicate “... is mortal”, which might from a philosophical point of view be said to denote the
property of mortality1. Such a reading is an example of an application of predicate logic – and another
instance of the attempt to “force” natural language into the confines of formal logic. In mathematical
logic any pretension to be dealing directly with natural language is immediately dropped and we
decide from the outset that our individuals shall be mathematical entities of some kind – numbers or
sets. The predicates are number-theoretic or set-theoretic predicates – for example, “... is even” or “...
is less than ...”.
We can introduce such predicates into a finite language. For example, if the base set is given
by A  1,2,3,4 where 1,2,3,4 now really are numbers and no longer mere partitions of space, then
each subset of A shall define a predicate: 2,4 is the predicate “... even number in A”. Two predicates
shall be regarded as being identical if they share the same extension. [Chap.2, Sec. 1.3.1] The symbol
1
denotes the predicate “... is a member of A and is identical to 1.” Thus, we see automatically that
in this form the following result: -
1.1 (+) Result
In the formal analytic predicate calculus all predicates can be eliminated in terms of
propositions.
Proof
1
In this essay I am not concerned with the ontology of properties or properties, and neither assume them, nor
discount them.
If P  "is a member of 1 " and a  1 , then Pa is the proposition 1  1 and
represents an atom,  1 , in the Boolean algebra defined over A, which is
isomorphic to 24 ; that is P  A   24 .
Predicate calculus also embraces rules for the use of quantifiers; universal instantiation is illustrated
by the rule: -
 Pa
  x  Px
But the quantifier is eliminable in favour of a list,  x  Px  P 1  P 2  P 3  P 4  ... . This rule reduces in
the finite case of P  A   24 to  P 1  P 2  P 3  P 4   P 1  P 2  P 3  P 4  , which is a “mere tautology”.
The use of predicates in the finite calculus of predicates (a form of formal analytic logic) is nothing
more than a façon de parler, or at best a tool of convenience, and that there are no true predicates in
this calculus. Surprisingly, the same principle extends to the infinite case as well, provided that we
allow for infinite lists and infinite meets and joins, which we do when we claim that an infinite lattice
is complete.
[Defined 4.9.4]
Even if the lattice is not complete in the sense that every infinite
collection of lattice points has both a join and meet, it may still allow for some infinite joins and
meets. Thus, in the logic that is built over the lattice there may be quantifiers, but in the lattice there
are no distinct points that correspond to quantifiers that do not represent points already existing in
the lattice. Quantifiers serve to distinguish, that is mark out and identify, certain points in the lattice;
they do not create them.
In scaling up from a finite to an infinite lattice we have stepped from a model of a lattice in
which there finite meets and joins to one in which there exist at least some infinite meets and joins, if
not all of them, as the lattice may be incomplete.
1.2 On the nature of a true predicate logic
In Aristotle’s logic subject and predicate are said to be “terms”, but the distinction between them is
founded on ontology and not grammar: a subject is a term denoting an individual, and a predicate is a
term denoting a universal.
Thus, the Aristotlean subject/predicate distinction cannot be divorced
from a theory of judgement, and all the mentalistic “baggage” that could attach itself to such a theory;
it is a form of the logic of intensions not extensions. This constitutes a true predicate logic. 2 Modern
formal predicate calculus is not a true predicate logic.
2
For confirmation of this historical perspective consider the following observations made by G.H.R. Parkinson
in his introduction to the logic of Leibniz: “He [Leibniz] states the subject/predicate distinction. He next
proposes, as a task for inventive logic, the problem of determining all the possible predicates of any given
subject, and all the possible subjects of any given predicate.” A term is either a subject or predicate. “He is
clearly using ‘term’ in the traditional sense of the subject or predicate of a proposition, and the fact that he
speaks in this context of an alphabet of human thoughts indicates that he regards such terms as concepts. The
analysis, then, is one of concepts; stated roughly, Leibniz’s view is that every concept is either ultimate and
indefinable, or is composed of such concepts. The indefinable concepts are called by Leibniz ‘first terms’, and
a list of these constitutes what he was later to call the ‘alphabet of human thoughts’, for derivative concepts
are formed from first terms in much the same way as words are formed from the letters of the alphabet.
Leibniz proposes to regard the first terms as constituting the first of a series of classes; the second class of the
series consists of the first terms arranged in groups of two; the third class, of the first terms arranged in groups
Consider atoms of a finite Boolean lattice – taking 24 as our paradigm. These atoms arise
from the conception of a division of space 0,1 into disjoint mutually exclusive parts.
0
1
1
2
{
1
}
{
2
}
3
4
{
3
}
{
4
}
The atoms are propositions that name these partitions. In the predicate calculus our focus of interest
has already shifted to natural numbers, sets and the continuum (points). The atoms are now points in
space. Thus, dimension makes its appearance, and we require variables to distinguish coordinates in
some space
k
or
k
.
The expression x  y when it appears in algebra affirms that the first
coordinate x is numerically equal to the second coordinate y. It is a statement about vectors. Thus,
x  y , or perhaps more perspicuously, x 1  x 2 , makes its appearance when it is implicit that the
domain is some 2- or n-dimensional space, and the individuals are pairs or n-tuples. That is why
x  y defines a diagonal set as a subspace of the universe of discourse, which is implicitly the
Cartesian product X  X of some base set X. Likewise,  x  Px is related to a projection function from
a space X n onto its (first) coordinate axis. The atoms of
2
correspond to points of ordered pairs,
and we can enumerate them: -
 0,0  ,  0,1 ,  0,2 , ... , 1,0  , 1,1 , 1,2 , ...
By diagonalisation these may be combined into a single list.
partitions of this
2
Predicates may be represented as
space (which is the domain of discourse), thus: -
0,0, 0,1, 0,2, ... , 1,0, 1,1, 1,2, ...
The atomic propositions take the form: 0,0  0,0 .3 This can be abbreviated to just
0,0 , which
also serves as the name of the atom. The predicate x  y or x  y  x , y  : x  y is the diagonal set,
0,0 , 1,1 , 2,2 , ...

[See
2.5

below].
An
example

R  x , y    x , y  : y  x 2   0,0  , 1,1 , 2,4 , ... . The predicate, Py   x  y  x 2

of

a
relation
is,
enumerates the image

2
set of this relation: Py   x  y  x  0,1,4,9,16, ... , so it is the projection of the relation R onto the
y-axis. [See 2.2 below]
1.3 Quantifiers
of three; and so on.” (Parkinson [1966] xvii.) Parkinson goes on to state that in the De Arte Combinatoria
Leibniz “follows Hobbes by regarding predication in terms of addition or subtraction, or he follows Aristotle
and scholastic tradition by speaking of the mind ‘compounding and dividing’.” The identity theory is a more
‘mature’ concept where “Leibniz’s view that to assert a proposition is to say that once concept is included in
another – that is, his ‘intensional’ view of the proposition.” (Parkinson [1966] xiv.)
3
Atoms are asserted contingently, so the expression  0,0    0,0  means 0,0 is contingently given: or
“Consider a model in which 0,0 is given.”
Now we have to interpret quantifiers. For a finite join,  x  Px  P1  P2  ...  Pn , corresponds to a finite
set.
In an infinite lattice a predicate may correspond to an infinite set.
No predicate (join) is
necessary, except 1. The impossible predicate is 0.
Existential quantifiers pick out points in a lattice, given contingently, and hence also their
filters.
The “natural” scope of the existential quantifier in the arbitrary statement
 x  Px
is that
which is may be contingently asserted (possibility). By contrast the scope of the universal quantifier in
 x  Px
is that which is necessary. If  x  Px is true, then P is a necessary predicate whose scope is
the whole set of atoms (the base set). In other words   x  Px is a synonym of   x  x  x and is a
name of 1.
Strictly,   x  Px cannot be asserted contingently; if it is asserted at all, it must be
asserted as a necessary proposition, that is, as a name of 1.
 x  x  x
is necessarily false, and is a
name of 0.
1.4 Rule for generalization
The rule for the introduction of the universal quantifier4 is: -

  x   x
where  is any well-formed formula of the logic.
This rule is partly responsible for the illusion that analytic logic is not vacuous; it appears to allow for
the deduction of a universal generality  x   x from finite information. One only has to examine the
rule to see that this must be a false impression. The statement  could only be true if  was a
name of 1, since it is affirmed categorically, that is, without premise. So  x   x  1 , and the inference
reduces to 1  1 .
In practice, “substantive” uses of generalization appear in results such as,
 x  Fx  Gx  ”   x  Fx   x Gx  ,
and are “useful” in the manner in which tautologies in general
are useful. Strictly speaking, there is no contingent meaning to   x  Px and all instances of this
formula are disguised names of 1.
1.5 Example
By the principle of dilution, the inference   x  Px    x  Fx  Gx  should be
interpreted as:   x  Fx  Gx 
4

  x  P1x  P1x  P2x  ...

  x  P1x  P1x  P2x  ...

  x  P1x  P1x 

  x  1

1
For example, see Mendelson [1979] p. 60.
Here also I will not allow  x   x unless it quantifies over an infinite domain. Then: -
 x  Px  Pa1  Pa2  Pa3  ...
where the list on the right-hand side is infinite.
1.6 Lattice inferences involving quantifiers
It would be useful to interpret in terms of the lattice the meaning of the valid inferences: -
 x  Fx  Gx  ”   x  Fx    x Gx 
 x  Fx    x Gx    x  Fx  Gx 
 x  Fx  Gx  ”   x  Fx    x Gx 
 x  Fx  Gx    x  Fx    x Gx 
But not conversely
But not conversely
Firstly I observe that all the inferences involving only the universal quantifier are strictly pseudoinferences lying in the pseudo-lattice of 1. This follows from the observation that  x  Fx is strictly a
name of 1, so any joins and meets of propositions of this type belong not to the lattice of joins (and
meets) of atoms, but to the pseudo-lattice of 1. The inferences
x  Fx  Gx  ”  x  Fx   x Gx 
x  Fx   x Gx   x  Fx  Gx 
But not conversely
belong to the pseudo-lattice, whereas the inferences
 x  Fx  Gx  ”   x  Fx    x Gx 
 x  Fx  Gx    x  Fx    x Gx 
belong to the proper lattice.
But not conversely
xFx  Gx)  xFx)  xGx)
xFx)
xGx)
xFx)  xGx)
xFx  Gx)
Consequence in the
pseudo-lattice of 1
xFx  Gx)
xFx)  xGx)
xFx)
xGx)
1
1.7 Logic of “identity” – Equations
The introduction of = into the predicate calculus concerns relations on the ideals of the lattice. Each
ideal corresponds to an equation. First, an illustration: -
1.8 Example
In the lattice 24 : 1 = {1,2,3,4}
{1,2,3}
p q
 p q
{1,3,4}
p  q
{1,2,4}
 p q
{2,3,4}
p  q
{1,2}
p
{1,3}
q
p q
{2,3}
p q
p  q
{1}
{2,4}
q
 p q
{2}
{3}
p
{3,4}
 p  q
{4}
0
let
p  x  y  t  3,4  1,2 , q  y  z  t  2,4  1,3 and x  z  t  2,3  1,4 .
Then
p  q   x  y    y  z   t  2,3,4  1 and
 x  y    y  z  xx  z
corresponds to the
inference p  q p  q . This is equivalent to the relation that the filter generated by the
node p  q is contained as a subset in the filter generated by the node p  q .
1.9 (+) Proposition
Equation logic is the logic of filters.
Proof
In disjunctive normal form a node in a lattice is a disjunct of propositions.
Assume that to each node p there is a conjunction
p  i1  ...  aik
where k is an ordinal and each  i is an atom. We need to show that there is a
relation on the set of all filters that is an equivalence relation and hence justifies
the introduction of the equality symbol into the logic. Reflexivity and symmetry
will be the easy case.
It is the transitive relation that is needed:
x  y  y  z  x  z . Putting
p  x  y  i1  ...  ai   t  ai1   ...  t  ai  

q  y  z   j1  ...  a j   t  a j1   ...  t  a j 



Then p  q  t  ai1   ...  t  ai    t  a j1   ...  t  a j  .
The statement x  z
will be equivalent to a sub-conjunction of this list. To an equation x  y there
corresponds a subset of the power set of the universe of discourse, which also
corresponds to a node in the lattice. Denote the filter generated by this node by
Fx  y . We have, for example: -
Equation
Partition
x y
   x  y 
 1,2
p
Filter
F   x  y      P V  :    
1,2 ,1,2,3 ,1,2,4 ,1,2,3,4
The general relation on which equation logic is based is: -
F  x  z   F  x  y     y  z 
The inclusion is strict if
x  y . In the preceding above ( 24 ) we have
x  z  p  q - this is based on the universe V  1,2,3,4 . In a larger universe
the statement x  z is equivalent to an unspecified disjunctive normal form.
Introduction of = into the language creates a distinguished point in the lattice
corresponding to a filter.
Thus, axioms governing the = symbol are based upon an existing relation in the lattice. The lattice
induces
a
relation
on
filters,
and
the = symbol creates a convenient language to describe
this relation. It maps back that relation to a relation between lattice points. The relation between
lattice points appears to be not a law of the lattice, but this is an appearance only. To each lattice
relation there corresponds an algebraic lattice law, though the correspondence is a function of the size
of the universe and the interpretation of the equation in that universe: -
Lattice
p x y
Filters
 Fx  y

x  y  y  x  x  z 

Fx  z  Fx  y  Fy z
pull back
induces
The axioms governing = do not correspond to points in the lattice – they are distinguished elements
from the set of logical laws.
There is only one way to add the axioms.
They are not contingent
structures but alternative descriptions of the structure of the lattice. Adoption of the axioms of = are
forced by the axioms of the lattice. It is the only consistent way to extend the lattice to include in the
language the = sign. The relation already exists and all that is supplied is a name.
This is part of a longer thesis advancing a refutation of strong AI. To download the thesis visit: Poincare’s thesis
For an introduction to the work as a whole visit: Introduction to Poincare’s thesis by Peter Fekete