Archive for the ‘logic’ Category

Laws of Form

September 19, 2016

sq_laws_of_formGeorge Spencer-Brown, author of Laws of Form, recently passed away.

I’ve tried to appreciate this work in the past, but couldn’t really get started. I recently ran across the following four terms associated with the work,

  • Compensation
    -> (())
  • Cancellation
    (()) ->
  • Condensation
    ()() -> ()
  • Confirmation
    () -> ()()

Compensation and Cancellation are both considered Order, and Condensation and Confirmation are both considered Number. Number and Order are distinguished by Distinction, and the pairs of the two distinctions are distinguished by Direction.

I understand Laws of Form starts with “Draw a distinction.” Perhaps I would say “Draw a distinction, then draw a distinction of that distinction.”


For my further reading:


Compensation (+2) (Pairs of parentheses)
Cancellation (-2) (Involutory?)
Condensation (-1) (Idempotence)
Confirmation (+1)



The Semiotic Square

July 29, 2016

sq_greimasFrom Wikipedia:

The semiotic square, also known as the Greimas square, is a tool used in structural analysis of the relationships between semiotic signs through the opposition of concepts, such as feminine-masculine or beautiful-ugly, and of extending the relevant ontology.


In an earlier post I combined an unusual representation of the semiotic square with that of the Tetralemma. Instead of using that one, please use this one instead.




Propositions as Types

February 14, 2016

sq_propositions_as_types3For almost 100 years, there have been linkages forged between certain notions of logic and of computation. As more associations have been discovered, the bonds between the two have grown stronger and richer.

  • Propositions in logic can be considered equivalent to types in programming languages.
  • Proofs of propositions in logic can be considered equivalent to programs of given type in computation.
  • The simplification of proofs of propositions in logic can be considered equivalent to the evaluation of programs of types in computation.

The separate work of various logicians and computer scientists (and their precursors) can be paired:

  • Gerhard Gertzen’s work on proofs in intuitionistic natural deduction and Alonzo Church’s work on the simply typed lambda calculus.
  • J. Roger Hindley and Robin Milner’s work on type systems for combinatory logic and programming languages, respectively.
  • J. Y. Girard and John Reynold’s work on the second order lambda calculus and parametric polymorphic programs, respectively.
  • Haskell Curry’s and W. A. Howard’s work on the overall correspondence between these notions of proofs as programs or positions as types.

Logic and computation are the sequential chains of efficient causation and actions. Propositions and types are the abstract grids of formal causation and structures. Proofs and programs are the normative cycles of final causation and functions. Simplification and evaluation are the reductive solids of material causation and parts.


Philip Wadler / Propositions as Types, in Communications of the ACM, Vol. 58 No. 12 (Dec 2015) Pages 75-85.

Preprint at

Also see:




J.-Y. Girard’s Transcendental Syntax, V2

March 11, 2015

sq_transcendental_syntaxMeaning is use.

– Ludwig Wittgenstein

If people never did silly things nothing intelligent would ever get done.

– Ludwig Wittgenstein

The latest two preprints by logician J.-Y. Girard continue his program for transcendental syntax, divided into deterministic and nondeterministic. He defines transcendental syntax as the study of the conditions of the possibility of language: to begin by discovering the preliminary suppositions in the creating of a logical sentence such as a proposition or deduction.

What are the presuppositions for using propositions? Girard claims the main one is the balance between the creation and the use of words, which is at the heart of meaning. But the notion that a proposition has a meaning that is well defined is prejudice, albeit one that allows us identify the terms of a sentence and thus to perform deductions.

Girard wants instead to find inner explanations of logical rules: explanations based on syntax instead of a semantics that correlates to a mandated “reality”. To emphasize this, he gives the term Derealism as another expression for transcendental syntax. Logical rules should have a normative aspect because of their utility, so his project appears to be one of pragmatism. Others have said that Linear Logic is the logic of the radical anti-realist.

Girard divides all of logical activity into four blocks that weave together: the Constat, the Performance (please forgive my shortening on the diagram above), L’usine (factory), and L’usage (use). These four blocks are partitioned by Kant’s analytic-synthetic and a priori-a posteriori distinctions. The analytic is said to have “no meaning”, that is, “locative”. The synthetic is said to have “meaning”, that is, “spiritual”. The a priori is said to be “implicit”, and the a posteriori is said to be “explicit”.
transcendental_syntax_tableCan we find all the explanations we need to create logic internally? If so, perhaps it is only because of how the brain works, like how John Bolender posits that social relations described by the Relational Models Theory are created out of symmetry breaking structures of our nervous systems, which are in turn generated by our DNA. A realist would certainly say that our understanding of logical rules is enabled but also limited by our brains, whereas an idealist would say that our minds could “transcend” those limits. But it seems pragmatic to say that the mind is what the brain does.

I believe a closer analogy for the fourfold of Transcendental Syntax is to Hjelmslev’s Net than to Kant’s Analytic-Synthetic Distinction. If so, then Performance and L’usage are Content (Implicit), whereas Constat and L’usine are Expression (Explicit). Performance and Constat are Substance (Locative), and L’usine and L’usage are Form (Spiritual). Hjelmslev was a linguist that developed a theory of language as consisting of only internal rules.

Or even to analogy with Aristotle’s Four Causes, which is how I’ve arranged the first diagram: the Constat is the Material cause, the Performance is the Efficient cause, L’usine is the Formal cause, and L’usage is the Final cause. Material and Efficient causes are often considered mere matter in motion, which could be Locative, or meaningless (physical). Formal and Final could be Spiritual, or meaningful, as patterns of matter and motion, respectively.


How can we know that a given named term is the same as another one in a different part of our formula? Rather than using names, or linking them through semantics or a well-defined meaning, we can tie terms together by their locations in our sentences and deductions.


J.-Y. Girard / Transcendental syntax 1: deterministic case (January 2015 Preprint)

J.-Y. Girard / Transcendental syntax 2: non deterministic case (February 2015 Preprint)

V. Michele Abrusci, Paolo Pistone / On Transcendental Syntax: a Kantian Program for Logic?

[*8.122, *8.123]


The Rational Structure of Inquiring Systems

February 26, 2015

sq_engelhartWhat are the components of consciousness? In the dissertation of L. Kurt Engelhart we see a fourfold used to analyze the texts and bodies of work of both scientists and philosophers, a hermeneutical tool if you will. This tool is also styled by concepts of “systems theory”, and requires the exposition of the aspects of Content, Control, Process, and Purpose of the authors. These match closely the Four Causes of Aristotle, which are the causes of made things or the explanations of how and why they came about: material, efficient, formal, and final. In fact, this close association was one of the main reasons I dove into the world of fourfolds. Texts are made things, after all.sq_causes

Making is so fundamental to what we do, that humans have been called “Homo Faber”, man the maker. We make tools, stories, culture, and even our concept of self. What if I turned this tool onto my own work, the writings and images found here? Perhaps that will be the project of another analyst, if my efforts warrant. What if I applied this tool to Engelhart’s project? That would be interesting indeed.

Another fourfold Engelhart presents is that of the domains of conscious experience, or the self itself as system. sq_four_alsThis fourfold consists of the Real, the Actual, the Ideal, and the Literal, but my version is in disagreement with Engelhart’s as to the classification of integrative and differentiative for the Ideal and the Literal. My assignments match the conjunctive and disjunctive properties of the operators of Linear Logic. Also left out is the Universal and how it supersedes the Actual as we make a complete turn. I like my version because it is similar to Richard McKeon’s Things, Thoughts, Words, and Actions. Also T. S. Eliot’s Falls the Shadow.

Of course this is just a brief gloss of the rich ideas presented in Engelhart’s work. Another of his key concepts is that of wholeness, which I have completely omitted. I hope to return and write a better review at a later time. I’m glad to see that Engelhart’s dissertation is now available as a Kindle book for the low, low price of $1. It is much easier to read in this format! From the Amazon Book Description:

This study describes, as a single systemic model of inquiry, the context common to conscious experience of the phenomenon of inquiry. Data are the published texts of selected contemporary writers relevant to the question. The problem is to define a common systemic structure of inquiry in a context of consciousness. Research verifies that a specific structure is common to these writers and that their respective views are converging on this same structure.

Identifying a common structure involves reducing the textual descriptions of the writers to their systemically relevant essentials. Defining the essential elements and describing a reduction method depends heavily on theory of metaphor and metaphorical evolution. A history of the metaphorical structure relevant to inquiry is described and this structure is used as a basis for finding structure in the selected texts. Texts researched include evolutionary biology, sociology, psychology (phenomenology), and philosophy. This work replicates that done by Talcott Parsons in experimentally describing a voluntaristic theory of action. A wholistic theory of inquiry is described using the same systemic scheme.

The metaphysical approaches to inquiry of realism and idealism have converged on a common theoretical structure for describing inquiry. Commonalities emphasize systemic structure comprising the elements of function: purpose, process, content, and control. It has been necessary to distinguish between affectual and instrumental purposes, and between organic and mechanical function. The ontological essentiality of the structure reveals a necessary logical relationship between function, systemicity, wholeness, and rationality in human understanding. Continuing research in philosophy is crucial to expanding our understanding of the ontological and epistemological structural essentials of consciousness.

Human inquiry during the last century has specialized in the material realm of realism, objective description, and mechanical explanation. A wholistic theory of inquiry does not discount the contributions of realism-based science or idealism-based philosophy, but expands the horizons of each to include the other. Where mathematics provides essential tools for mechanical explanation, organic explanation still lacks abstract structural tools for describing conscious organic, including human, behavior. The intent of a wholistic theory of inquiry is to provide conceptual tools that support disciplined inquiry into conscious behavior.

References and Links:

L. Kurt Engelhart / Wholeness and the Rational Structure of Inquiring Systems: A Dissertation


I removed some text about the “Book of Nature”, because it needed more work. This mentioned the systems theory adage “the purpose of a system”, which can also tie into “meaning as use”. I also missed seeing an obvious thought that inquiry is making.

[*2.188, *3.104, *8.134]


Human Limitations

July 10, 2013

birth_death_nature_nurture Everything existing in the universe is the fruit of chance and necessity.


What are humans limited by? We are limited by time and by space.

We are limited in time because we are mortal beings, bound to the past by our birth and to the future by our death, delimiting the short interval of our lifetimes.

We are limited in space because we are finite beings. At each point of time we are constrained by what we are and what we might become. We are bound both by our actuality and our possibility. Our nature defines our actuality and our nurture sets our possibility.

Certainly, we can transcend our limitations, but this just means they weren’t really the true limits after all. We can’t say for certain what our real limits are, so that’s a good thing.

This fourfold also reminds me of the following modal square of opposition. At birth, much is possible. At death, all becomes impossible. Nurture is contingent and Nature is necessary.




The Curry-Howard Correspondence

August 21, 2012

The Curry-Howard Correspondence reveals a close correspondence between the constituents of Logic and of Programming. Also known as the Formulas as Types and the Proofs as Programs interpretations.

Existential Quantification (∃) of Logic corresponds to the Generalized Cartesian Product type (∑) of Programming. Universal Quantification (∀) of Logic corresponds to the Generalized Function Space type (∏) of Programming. Conjunction (⋀) of Logic corresponds to the Product type (×) of Programming. Disjunction (⋁) of Logic corresponds to the Sum type (+) of Programming.

There are associations between the Curry-Howard Correspondence and the fourfolds of the Square of Opposition, Attraction and Repulsion, and of course Linear Logic.


The Square of Opposition

May 21, 2012

Some readers may think I’ve never met a fourfold I didn’t like. However, there are several that I haven’t presented here because they don’t seem to play well with the others. The Square of Opposition, created by Aristotle, is one such fourfold. The four logical forms of the square are relations between a subject and predicate, S and P, and supposedly exhaust the possibilities of belonging: Some S are P, Some S are not P, All S are P, and No S are P (or All S are not P).

In the diagram I have removed the S and P, and the logical forms become spare and like a Zen Koan or nursery rhyme: Some Are, Some Are Not, All Are, and None Are (or All Are Not). By doing so, they resonate more brightly with the other fourfolds and how they are presented herein. Now, the logical forms can be about existence, or the subject and predicate withdraw and become implicit to the thought.

Also, consider the four edges between the four logical forms, and label the common terms. Then the important fourfold of Are, Are Not, Some, and All is shown.


Compare and contrast the Square of Opposition to the Tetralemma and the Semiotic Square.

The 3rd World Congress on the Square of Opposition is soon to convene. May the meeting be rewarding!


[*4.84, *5.82, *7.70, *7.90]


The Four Binary Operators of Linear Logic

April 17, 2012

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.


[*6.38, *6.40]


J.-Y. Girard’s Transcendental Syntax

March 7, 2012

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.

Interestingly, these four levels are in good agreement with Richard McKeon’s schema for philosophical semantics, represented by the fourfold of reality, method, perspective, and principle.