Kant’s Reflective Perspectives on Experience

April 27, 2012 Martin K. Jones 2 Comments

The web site of Stephen R. Palmquist has a great wealth of material on fourfolds in relation to Kant’s as well as his own philosophy. From my own initial reading of his extensive material I have tried to choose a canonical Kantian fourfold which has the most relevance to my project.

The fourfold shown above Dr. Palmquist calls Kant’s “reflective perspectives on experience”. Consisting of the logical, the empirical, the transcendental, and the hypothetical, these facets bear a close analogical likeness to many of the fourfolds presented here.

Logical: Analytic a priori
Transcendental: Synthetic a priori
Hypothetical: Analytic a posteriori
Empirical: Synthetic a posteriori

Dr. Palmquist also has many of his own books available on his web site for the interested reader. I will certainly be returning to his web site in the future for much enjoyable study.

References:

http://www.hkbu.edu.hk/~ppp/

http://www.hkbu.edu.hk/~ppp/ksp2/KCR3.htm

[*7.68, *7.84]

fourfolds, linear logic, logic

The Four Binary Operators of Linear Logic

April 17, 2012 Martin K. Jones 14 Comments

The four binary operators for Linear Logic can be described by their logical sequents, or inference rules (shown above in the table). Note that in the rules for the operators, the operator appears below and not above the horizontal line. Thus the sequents can be considered in two ways: moving from top to bottom, the operator is introduced, and moving from bottom to top, the operator is eliminated. As operators are introduced the rules deduce a formula containing instances of them, and as operators are eliminated the rules construct a proof of that formula without them. Also note that in the sequents the symbols A and B denote formula in our logic, and the symbols Γ and Δ are finite (possibly empty) sequences of formulae (but the order of the formulae do not matter), called contexts.

The rule for the operator & (with), additive conjunction, says that if A obtains along with the context Γ, and if B obtains also along with Γ, then A & B obtains along with Γ.

The rule for the operator ⊕ (plus), additive disjunction, says that if A obtains along with the context Γ, or if B obtains along with Γ, then A ⊕ B obtains along with Γ.

The rule for the operator ⅋ (par), multiplicative disjunction, says that if the combination A,B obtains along with the context Γ, then A ⅋ B obtains along with Γ.

The rule for the operator ⊗ (tensor), multiplicative conjunction, says that if A obtains along with context Γ, and if B obtains along with context Δ, then A ⊗ B obtains along with the combined context Γ,Δ.

There are several immediate observations we can make about these rules.

First, note that for operators & and ⅋, they are reversible. That is, if one obtains A & B or A ⅋ B, then one knows exactly what the previous step had to be to introduce the operator. In contrast, ⊕ and ⊗ are not reversible. If one has A ⊕ B, one doesn’t know if we started with A or with B. If one has A ⊗ B, one doesn’t know what was the context of A or what was the context of B.

For this reason, I will consider & and ⅋ to be rational, and ⊕ and ⊗ to be empirical.

Second, note that all of the parts of the multiplicative sequents for ⅋ and ⊗ are the same above and below the horizontal line. In contrast, the parts of the additive sequents for & and ⊕ are different above and below the line. For &, a duplicated context is eliminated even as the operator is introduced. For ⊕, a new formula is introduced along with the introduction of the operator.

For this reason, I will consider & and ⊕ to be discrete, and ⅋ and ⊗ to be continuous.

References:

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

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

http://www.uni-obuda.hu/journal/Mihalyi_Novitzka_42.pdf

[*6.38, *6.40]

Computation, fourfolds, Space and Time

A Digital Universe

April 12, 2012 Martin K. Jones 1 Comment

A digital universe – whether 5 kilobytes or the entire Internet – consists of two species of bits: differences in space, and differences in time. Digital computers translate between these two forms of information – structure and sequence – according to definite rules. Bits that are embodied as structure (varying in space, invariant across time) we perceive as memory, and bits that are embodied as sequence (varying in time, invariant across space) we perceive as code. Gates are the intersections where bits span both worlds at the moments of transition from one instant to the next.

— George Dyson, from Turing’s Cathedral

The Four Freedoms

April 9, 2012 Martin K. Jones 1 Comment

From Wikipedia:

The Four Freedoms were goals articulated by US President Franklin D. Roosevelt on January 6, 1941. In an address known as the Four Freedoms speech (technically the 1941 State of the Union address), he proposed four fundamental freedoms that people “everywhere in the world” ought to enjoy:

Freedom of speech
Freedom of worship
Freedom from want
Freedom from fear

His inclusion of the latter two freedoms went beyond the traditional US Constitutional values protected by its First Amendment, and endorsed a right to economic security and an internationalist view of foreign policy. They also anticipated what would become known decades later as the “human security” paradigm in social science and economic development.

References:

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

http://www.americanrhetoric.com/speeches/fdrthefourfreedoms.htm

Update:

Homepage

https://en.wikipedia.org/wiki/Franklin_D._Roosevelt_Four_Freedoms_Park

[*7.76]

fourfolds, Mathematics, Science

Systems Dynamics

April 2, 2012 Martin K. Jones 1 Comment

Another interesting fourfold that I discovered while reading mathematical physicist John C. Baez’s blogs Azimuth and This Week’s Finds in Mathematical Physics concerns the notions of system dynamics and bond graphs. These concepts generalize the fourfold of the basic electronic components into other types of physical systems, such as mechanics, hydraulics, and to some extent even thermodynamics and chemistry.

The types of systems that can be modeled by system dynamics are described by two variables that vary functionally over time and their corresponding integrals. These four functions can be thought of as flow and effort and their respective integrals displacement and momentum.

	Displace- ment	Flow	Momentum	Effort
Mechanics of translation	Position	Velocity	Momentum	Force
Mechanics of rotation	Angle	Angular velocity	Angular momentum	Torque
Electronics	Charge	Current	Flux	Voltage
Hydraulics	Volume	Flow	Pressure momentum	Pressure
Thermo-dynamics	Entropy	Entropy flow	Temperature momentum	Temperature
Chemistry	Moles	Molar flow	Chemical momentum	Chemical potential

Relational Models Theory

March 9, 2012 Martin K. Jones 2 Comments

The Relational Models Theory (RMT) of Alan Fiske is a framework for social relations that details four major types of interpersonal interactions: Communal Sharing (CS), Authority Ranking (AR), Equality Matching (EM), and Market Pricing (MP). These four types can also be combined and nested to form more complex social relations. Thus they are presented to be the elementary building blocks out of which all social relations are made.

John Bolender expounds on Fiske’s theory, by arguing that the four relationships are each described by the inherent mathematical symmetry of the social relation, and that all four can be ordered: in one way by inclusion (in the sense of mathematical group theory) and the opposite way by symmetry breaking (usually considered in physics). The symmetries are viewed as transformations which when applied, like a reorganization of the relationship, do not alter the substance or content of the relationship. CS is more symmetric than AR, which in turn is more symmetric than EM, which lastly is more symmetric than MP. For inclusion, the symmetries of MP are included in the symmetries of EM, those of EM in AR, and AR in CS. For symmetry breaking, a symmetry of CS can be broken to form AR, which in turn has a symmetry that can be broken to form EM, and so on to MP. Thus CS > AR > EM > MP in terms of transformations that preserve symmetry.

As we move from CS to MP, we add increasing structure to a social relation, a greater number of constraints. Additionally, consider symmetry operations in general in the context of equivalent exchange: symmetry by definition is an exchange of participants in a relation, a permutation such that the relation itself is unchanged, that is, equivalent.

References:

Alan Page Fiske / Structures of social life: the four elementary forms of human relations

John Bolender / The Self-Organizing Social Mind

http://www.rmt.ucla.edu/

http://fuquaccl.wordpress.com/tag/relational-models-theory/

http://www.sscnet.ucla.edu/anthro/faculty/fiske/RM_PDFs/Fiske_Haslam_Four_Basic_Bonds_2005.pdf

http://www.sscnet.ucla.edu/anthro/faculty/fiske/relmodov.htm

http://ndpr.nd.edu/news/24644-the-self-organizing-social-mind/

http://www.iep.utm.edu/r-models/

Notes:

Some changes: replacing the diagram and removing reference to J.-Y. Girard’s infernos of semantics. Also the very important insight from IEP: “The symmetries of solid matter form a subset of the symmetries of liquid matter which form a subset of the symmetries of gaseous matter which form a subset of the symmetries of plasma.”

[*6.110]

fourfolds, linear logic, logic, Philosophy

J.-Y. Girard’s Transcendental Syntax

March 7, 2012 Martin K. Jones 1 Comment

With the recent release of the paper found at the link below, logician Jean-Yves Girard has updated his program for a transcendental syntax to version 2.0. The first version was available last year only in French, but this new manuscript is available in English. Girard is known for his refinement of classical and intuitionistic logic, Linear Logic, and his exploration into the mechanisms of logic, Ludics.

In this new paper, Girard describes four levels of semantics, his infernos: alethic, functional, interactive, and deontic. They descend into the depths of meaning, and thus are numbered from -1 to -4. The negatively first, alethic, is the layer of truth or models. The negatively second, functional, is the layer of functions or categories. The negatively third, interaction, is the layer of games or game semantics. The negatively fourth, deontic, is the layer of normativity or formatting.