Tag Archives: Vaughan Pratt

Computation, fourfolds, linear logic, logic, Mathematics

Linear Process Algebra

August 27, 2021 Martin K. Jones Leave a comment

One of computer scientist and Professor Emeritus Vaughan Pratt’s most recent conference papers is on “linear process algebra,” which relates several of his previous interests on linear logic, Chu spaces, concurrent processes, events and states, etc.

The paper opens with a nice overview of computer science research primarily concerned with concurrent processes. Computation itself divides into the aspects of logical and algorithmic, formal methods into the logical and algebraic, concurrent computation into operational and denotational, and then the author gives a brief list of models of processes by a variety of mathematical structures until he comes to his theme of using Chu spaces.

As an example, he presents processes as Chu spaces over the set K, where K = { 0, T, 1, X}, with names and meanings :

0: Ready
T: Transition
1: Done
X: Cancelled

and then details four binary operations as working in Chu spaces over K:

P ; Q: Sequence
P + Q: Choice
P || Q: Concurrence
P ⊗ Q: Orthocurrence

The Art of the Syllogism

August 27, 2020 Martin K. Jones 1 Comment

The syllogism is a logical system that was invented by Aristotle which deduces valid inferences from given premises. It is categorical in nature because each of two premises and the conclusion has an internal relationship of belonging or inclusion. Specifically, there is a major premise of a general nature and a minor premise that is usually specific, or of reduced generality. Both are combined deductively to reach or prove the conclusion.

Both premises and the conclusion deal with three categories two at a time, a subject term (S), a middle term (M), and a predicate term (P), joined by one of four binary inclusion relations. The major premise deals with M and P, the minor premise deals with S and M, and the conclusion with S and P. The four types of relations are denoted by the letters A, E, I, O (also a, e, i, o) and are described below. The premises may have M first or second, but the conclusion always has the S first and the P second.

S = Subject
M = Middle
P = Predicate

A = a = XaY = All X are Y
E = e = XeY = All X are not Y
I = i = XiY = Some X are Y
O = o = XoY = Some X are not Y

Major premise: MxP or PxM, x = a, e, i, or o
Minor premise: SxM or MxS
Conclusion: SxP

The distinction between the four Figures concerns the placement of the middle term M in each of the premises. In order to highlight this order, I’ve written them with ( and ) on the side of the relation where the M is.

Figure 1: MxP, SyM, SzP: (xy)z
Figure 2: PxM, SyM, SzP: x(y)z
Figure 3: MxP, MyS, SzP: (x)yz
Figure 4: PxM, MyS, SzP: x()yz

There are only 24 valid inferences out of all possible combinations, six for each of the four Figures (and some of these may be erroneous sometimes due to the existential fallacy). In addition, they were given mnemonic names in the Middle Ages by adding consonants around the vowels of the relations. And so the valid inferences and their names (or something close to it) are as follows (by my notation and in no special order):

(aa)a, B(arba)ra
(ea)e, C(ela)rent
e(a)e, Ce(sa)re
a(e)e, Ca(me)stres
a()ee, Ca(l)emes
(ai)i, D(ari)i
(a)ii, D(at)isi
(i)ai, D(is)amis
i()ai, Di(m)atis
(ei)o, F(eri)o
e(i)o, Fe(sti)no
(e)io, F(er)ison
e()io, Fre(s)ison
a(o)o, Ba(ro)co
(o)ao, B(oc)ardo
(aa)i, B(arba)ri
a()ai, Ba(m)alip
(ea)o, C(ela)ront
e(a)o, Ce(sa)ro
a(e)o, Ca(me)stros
a()eo, Ca(l)emos
(e)ao, F(el)apton
e()ao, Fe(s)apo
(a)ai, D(ar)apti

For example, (aa)a, or Barbara, is a syllogism of the form: All Y are Z; All X are Y; thus All X are Z.

The Duality of Time and Information, V3

October 1, 2017 Martin K. Jones Leave a comment

The states of a computing system bear information and change time, while its events bear time and change information.

— from The Duality of Time and Information by Vaughan Pratt

The most promising transformational logic seems to us to be Girard’s linear logic.

— from Rational Mechanics and Natural Mathematics by Vaughan Pratt

Here we have three duals:

Information – Time
States – Events
Bear – Change

Four Transformations of Chu Spaces

November 29, 2015 Martin K. Jones Leave a comment

Can mathematics help us reformulate Cartesian Dualism? I have previously tried to diagram some of computer scientist Vaughan Pratt’s notions, such as a Duality of Time and Information and the Stone Gamut. Another recent attempt is the diagram above of four transformations that issue out of his analysis of Chu Spaces. Pratt’s conceptualization of these generalized topological spaces led him to propose a mathematization of mind and body dualism.

The duality of time and information was actually an interplay of several dualities, such as the aforementioned time and information, plus states and events, and changing and bearing (or dynamic and static). The philosophical mathematization in his paper “Rational Mechanics and Natural Mathematics” leads to additional but somewhat different dualities, shown in the following table:

Mind	Body
Mental	Physical
States	Events
Anti-functions	Functions
Anti-sets	Sets
Operational	Denotational
Infers	Impresses
Logical	Causal
Against time	With time
Menu	Object
Contingent	Necessary

Pratt reveals two transformations that are “mental”: delete and copy, and two that are “physical”: adjoin and identify.

These four transformations are functions and their converses which:

Identify when the function is not injective.
Adjoin when the function is not surjective.
Copy when the converse is not injective.
Delete when the converse is not surjective.

Ordinarily we think of mind and body as being radically different in kind, but perhaps they are the same but merely viewed from a different perspective or direction. Recall what Heraclitus says, “the road up and the road down are the same thing”.

References:

https://en.wikipedia.org/wiki/Dualism_%28philosophy_of_mind%29

Click to access ratmech.pdf

http://chu.stanford.edu/

http://en.wikipedia.org/wiki/Chu_space

[*6.74, *6.75, *9.76]

Cartesianism, fourfolds, Mathematics, Mind-Body Dualism

The Stone Gamut

May 17, 2013 Martin K. Jones 1 Comment

Our thesis is that the category Set is the ultimate abstraction of body, and that Set^op, equivalent to the category of complete atomic Boolean algebras (i.e. power sets), which we shall advocate thinking of as antisets, is dually the ultimate abstraction of mind.

— From Chu Spaces: automata with quantum aspects by Vaughan Pratt

Reflecting an era of reduced expectations for the superiority of humans, we have implemented causal interaction not with the pineal gland but with machinery freely available to all classical entities, whether newt, pet rock, electron, or theorem (but not quantum mechanical wavefunction, which is sibling to if not an actual instance of our machinery).

— From Rational Mechanics and Natural Mathematics by Vaughan Pratt

http://en.wikipedia.org/wiki/Boolean_algebra

http://en.wikipedia.org/wiki/Distributive_lattice

http://en.wikipedia.org/wiki/Vector_space

http://en.wikipedia.org/wiki/Partially_ordered_set

http://en.wikipedia.org/wiki/Set_%28mathematics%29

References:

Click to access ph94.pdf

Click to access ratmech.pdf

http://chu.stanford.edu/

http://en.wikipedia.org/wiki/Chu_space

[*7.140]