Category Archives: linear logic

The Four Binary Operators of Linear Logic, Part 2

Ordinarily, inference rules in natural deduction are written using a horizontal line, with the known, true, assumed or proven things written above the line and the inferred things written below the line. Here I’ve taken the artistic liberty to use diagonal lines instead of horizontal ones, and so tried to represent the introduction rules for the four binary operators of Linear Logic. In order to fit additive disjunction “plus” into this schema, I’ve broken the inference rule diagonal and written the duplicate inferred introduction below only once. I’m sure no self-respecting logician would do such a thing.

Further Reading:

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

https://plato.stanford.edu/entries/logic-linear/

https://en.wikipedia.org/wiki/Natural_deduction

https://equivalentexchange.blog/2012/04/17/the-four-binary-operators-of-linear-logic/

https://equivalentexchange.blog/2011/03/09/j-y-girards-linear-logic/

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

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.

Notes:

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.

References:

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

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

http://iml.univ-mrs.fr/~girard/Articles.html

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

https://www.academia.edu/10495057/On_Trascendental_syntax_a_Kantian_program_for_logic

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

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Four Dimensional Space-time

sq_4d_spacetime

Here’s a simple fourfold I’ve been ignoring just because it’s so trivial, but that triviality can be deceiving. Space-time as formulated in special relativity has four dimensions: three of space and one of time. Our everyday experience shows us the three dimensions of space: length, width (or breadth), and depth (or height), but time is a different kind of thing because we cannot see or move forward and backward through time with our eyes or body, like we can along the axes of space.

Personally, only our memory and imagination can let us range through time. Of course, after the invention of language and more recent technologies, the spoken word, writings, photographs, audio recordings, and videos can also be used. But it’s not the same as shifting one’s gaze along the length of something or moving one’s body across a width.

So, we can move semi-freely through the three spatial dimensions but our movement in time seems to be fixed into a relentless forward motion that we have no control over. And because gravity pulls us down onto the surface of the world, one of the spatial dimensions (depth or height) is more limiting than the other two.

sq_ll2Thus another interesting comparison to this fourfold is to that of linear logic. One observation is that length and width can be considered reversible but depth and time can be considered somewhat irreversible. That’s not true of course, but because of gravity it is easier to descend than to ascend, and it’s far easier to move into the future than into the past. But we can see into the distant past, just not our own, as we turn our telescopes to the heavens.

Space without time could have four or even higher dimensions, but we have no empirical evidence that it is so. Mathematically, however, we can easily construct multidimensional spaces. One representation of four dimensional space is by using quaternions, which have four dimensions to the complex numbers’ two. Tuples of real numbers or even vector spaces can also be used. However, the geometry of space-time is not Euclidean; it is described by the Minkowski metric.

Novels about characters living in different numbers of spatial dimensions are an interesting way to learn and think about them. The very first was Flatland by Edwin Abbott Abbott, about a being limited to two dimensions that learns about a third outside his experience when a three dimensional being comes to visit. Just recently I’ve finished reading Spaceland by Rudy Rucker, about an ordinary human person limited to the three dimensions of space that learns about the fourth dimension by similar reasons.

Links:

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

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

http://en.wikipedia.org/wiki/Four-dimensional_space

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

http://en.wikipedia.org/wiki/Flatland_%282007_film%29

http://en.wikipedia.org/wiki/Spaceland_%28novel%29

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Relations All the Way Down

There is nothing to be known about anything except an initially large, and forever expandable, web of relations to other things. Everything that can serve as a term of relation can be dissolved into another set of relations, and so on for ever. There are, so to speak, relations all the way down, all the way up, and all the way out in every direction: you never reach something which is not just one more nexus of relations.

— Richard Rorty from Philosophy and Social Hope

sq_ll2

The ancient Greek philosopher Empedocles somehow reasoned that the world was made entirely from four basic elements: fire, earth, water, and air. Science as we know it has disproved this from being the case, but this idea still has a rich symbolic meaning even today that informs our popular culture.

A recent philosophical stance called “ontic structural realism” argues that science suggests that only relations between things are of lasting importance, that is the structural relationships within and between things, not the things themselves that bracket the relations. What we call a quark for instance is just the relations it has with other quarks and the other entities that have relationships with quarks. Perhaps then the world consists of “relations all the way down”, instead of stopping at some point on the lowest level with the things that constitute the world.

If this is so, what if the world was made completely from four basic relations, instead of four basic things? sq_structure_functionCould they be something like the four binary operators of linear logic? I have likened these four basic operators of Linear Logic to my fourfold Structure-Function, where in addition to Structures, we also have Functions, Actions, and Parts. But these three other relations are also structural, in that only the relation something has to another something makes it structural, functional, actional, or a part of a something.

Book Description for Every Thing Must Go:

Every Thing Must Go argues that the only kind of metaphysics that can contribute to objective knowledge is one based specifically on contemporary science as it really is, and not on philosophers’ a priori intuitions, common sense, or simplifications of science. In addition to showing how recent metaphysics has drifted away from connection with all other serious scholarly inquiry as a result of not heeding this restriction, they demonstrate how to build a metaphysics compatible with current fundamental physics (“ontic structural realism”), which, when combined with their metaphysics of the special sciences (“rainforest realism”), can be used to unify physics with the other sciences without reducing these sciences to physics itself. Taking science metaphysically seriously, Ladyman and Ross argue, means that metaphysicians must abandon the picture of the world as composed of self-subsistent individual objects, and the paradigm of causation as the collision of such objects. Every Thing Must Go also assesses the role of information theory and complex systems theory in attempts to explain the relationship between the special sciences and physics, treading a middle road between the grand synthesis of thermodynamics and information, and eliminativism about information. The consequences of the author’s metaphysical theory for central issues in the philosophy of science are explored, including the implications for the realism vs. empiricism debate, the role of causation in scientific explanations, the nature of causation and laws, the status of abstract and virtual objects, and the objective reality of natural kinds.

Stuff I need to read:

James Ladyman, Don Ross / Every Thing Must Go: Metaphysics Naturalized

http://plato.stanford.edu/entries/structural-realism/

Jason D. Taylor / Relations all the way down? Exploring the relata of Ontic Structural Realism

http://hdl.handle.net/10402/era.27885

http://www.alcor.org/magazine/2011/01/27/every-thing-must-go-metaphysics-naturalized/

http://larvalsubjects.wordpress.com/2010/08/10/relations-all-the-way-down/

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My Dear Aunt Sally

sq_my_dear_aunt_sallyWhen I took first year algebra in school, I learned the rule “My Dear Aunt Sally” as a mnemonic for the order of applying binary operations in algebraic expressions. “My Dear” meant to perform multiplication and division first. “Aunt Sally” meant to perform addition and subtraction next and last. Most of us have learned some variation of this rule. I see that it has now been enlarged to “Please Excuse My Dear Aunt Sally” to include parentheses and exponentiation, and to perform these two first before the original and now last four.

Why remark about this simplistic and even obsolete rule? Note the similarity between this fourfold of binary arithmetic operators and the four binary linear logic operators. In each there are two operators for combining: addition and multiplication in arithmetic, and the conjunctive operators with and tensor in linear logic. In each there are two operators for separating: subtraction and division in arithmetic, versus the disjunctive operators plus and par in linear logic. In each there are two rules for attraction and two rules for repulsion.

In addition, the double duality of the four arithmetic operators is revealed, as in arithmetic addition and subtraction are duals, and multiplication and division are duals. In linear logic, with and plus are duals, and tensor and par are duals. Can arithmetic be simulated by linear logic, or vice versa? Is linear logic equivalently exchangable with arithmetic? I don’t think so but perhaps some expert can tell us.

References:

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

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Fourfolds and Double Duals

For every aspect of the world that someone has thought to analyze into its components, it probably has been suggested to divide it into four parts. I suggest that many of the things that have a fourfold form, are a fourfold in the same way. Not in the trivial cardinal sense, but in a deeper structural sense. They are a combination of two dualities, a double dual if you will, such that one dual operates as interior and exterior, or true and false, a duality of opposites, and another (dual) dual operates as one and many, or unity and multiplicity, or discreteness and continuity, a duality of numeracy.

I have been gathering fourfolds for a time, and have written about some more than others. Some have been around a long time, and others I’ve been inspired to fashion. I have tried to orient them all in the same way to accentuate their common deep structure. For example, everything in the left position in the diagrams have a commonality across fourfolds, as does everything in the lower, upper, and right positions. The four ancient symbols shown in the diagram above represent the four elements of alchemy: fire, earth, air, and water.

Because of these relations between the fourths and the halves of these fourfolds, I have choosen the name “Equivalent Exchange”. In addition, the fourfolds themselves might be “equivalently exchangable” with each other because of their common deep structure. For many reasons, I believe that the best exemplar for these fourfolds is the recent logical system known as Linear Logic, which has two combining binary operators and two dividing binary operators.

Others before me have reached similar hypotheses about fourfolds in general, and I am grateful for their scholarship. I hope others will follow, and I’m sure they will present their findings more eloquently and convincingly than I have.

References:

http://en.wikipedia.org/wiki/4_%28number%29

http://www.samuel-beckett.net/Penelope/four_symbolism.html

http://www.unterzuber.com/4our.html

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The Curry-Howard Correspondence

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.

http://en.wikipedia.org/wiki/Curry-Howard_correspondence

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