Category Archives: logic

fourfolds, linear logic, logic

Linear Logic and the Laws of Excluded Middle and Noncontradiction

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If LEM is the Law of Excluded Middle and LNC is the Law of Non-contradiction then

  • Classical Logic preserves both LEM and LNC
  • Intuitionistic Logic preserves LNC, but rejects LEM
  • Co-Intuitionistic Logic preserves LEM, but rejects LNC
  • Linear Logic broadly preserves neither, but narrowly preserves and rejects them with its pairs of conjunctive and disjunctive logical operators

Above is shown the four operators of Linear Logic and the statements for their preservation and rejection of LEM and LNC.

Further Reading:

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

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

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

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

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

Pete Wolfendale / Essay on Transcendental Realism
(at PhilPapers)

https://t.co/hoHWUOhQE0

https://plato.stanford.edu/entries/contradiction/

https://ncatlab.org/nlab/show/paraconsistent+logic

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Computation, fourfolds, linear logic, logic, Mathematics

Linear Process Algebra

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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

Further Reading:

Vaughan Pratt / Linear Process Algebra

Click to access bhub.pdf

Click to access lpa.pdf

Click to access bud.pdf

https://www.researchgate.net/publication/2663060_Chu_Spaces_A_Model_Of_Concurrency

https://www.researchgate.net/publication/222310260_Types_as_Processes_via_Chu_spaces

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

https://dblp.org/pid/p/VRPratt.html

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eightfolds, fourfolds, logic, Philosophy

John Buridan’s Octagon of Opposition

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Medieval logician John (or Jean) Buridan was a scholar of Aristotle, and wrote many works of commentary and elaboration on Aristotelian philosophy. Several items in logic and philosophy are tied to Buridan (such as Buridan’s Bridge and Buridan’s Ass) but he may now be more widely known for his Octagon which combines Aristotle’s Square of Opposition with a Square of Modality.

Below and to the right is a fourfold diagram of Aristotle’s Square of Opposition. The modern universal and existential (or particular) qualifiers are ∀ (meaning All) and ∃ (meaning Some), respectively. Also in these diagrams, ¬ means logical Not.

  • ∀ S are P
  • ∃ S are P
  • ∀ S are ¬ P
  • ∃ S are ¬ P

Next I show a fourfold of modal operators and their equivalents. The modern modal symbols are □ (meaning Necessarily) and ◇ or ◊ (meaning Possibly).

  • □ P ≡ ¬ ◊ ¬ P
  • ◊ P ≡ ¬ □ ¬ P
  • ¬ □ P ≡ ◊ ¬ P
  • ¬ ◊ P ≡ □ ¬ P

 

Further Reading:

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

https://plato.stanford.edu/entries/buridan/

https://plato.stanford.edu/entries/medieval-syllogism/#JohnBuri

The Art of the Syllogism

https://1000wordphilosophy.com/2018/12/08/possibility-and-necessity-an-introduction-to-modality/

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

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

https://plato.stanford.edu/entries/modality-varieties/

https://lirias.kuleuven.be/retrieve/517366

Click to access Demey_DWMC2015_Buridan_Avicenna_slides.pdf

Click to access Buridan_Octagon.pdf

Click to access hughes-buridan.pdf

https://en.wikipedia.org/wiki/Buridan%27s_ass

https://en.wikipedia.org/wiki/Buridan%27s_bridge

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

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fourfolds, logic, Politics, Science

Our Demarcation Problem

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I have a foreboding of an America in my children’s or grandchildren’s time — when the United States is a service and information economy; when nearly all the manufacturing industries have slipped away to other countries; when awesome technological powers are in the hands of a very few, and no one representing the public interest can even grasp the issues; when the people have lost the ability to set their own agendas or knowledgeably question those in authority; when, clutching our crystals and nervously consulting our horoscopes, our critical faculties in decline, unable to distinguish between what feels good and what’s true, we slide, almost without noticing, back into superstition and darkness.

— Carl Sagan from The Demon-haunted World

As science is confused with pseudo-science, as real news is conflated with fake, we need much better ways to judge the truth of the information we require to be good citizens. Unfortunately, in this age of nontraditional television networks, kitchen-sink cable, and internet news sources, our information sources can be subverted by entities that wish to bend our mindset to their agenda, rather than giving us measured and reasonable knowledge. When these entities wish to fracture and divide our polity, our social fabric strains and unravels.

Here are four (or five minus one) distinctions for information or knowledge claims, based upon their type of warrant, or context of truthfulness. Three of them are modalities from Kant’s doctrine of judgments, and I suggest that Dialectic could reasonably be added to them, but I do not know if they form a complete set or not. I would suppose they can be ordered by their level of assurance, from low to high. Another more scientific option might be Probablistic instead of Dialectic, based upon measurements or even theoretical arguments. Certainly there must be something between a bald assertion or the questionable and the certain.

  • Assertoric: assert to be true or false without (inherent) proof
  • Problematic: assert as possibly true (or false)
  • Dialectic: philosophically reasoned as true or false (qualified?)
  • Probabilistic: quantified or theoretically argued as mostly true or false
  • Apodictic: clearly provable as true (or false) or logically certain

From Wikipedia:

Apodictic propositions contrast with assertoric propositions, which merely assert that something is (or is not) true, and with problematic propositions, which assert only the possibility of something being true. Apodictic judgments are clearly provable or logically certain. For instance, “Two plus two equals four” is apodictic. “Chicago is larger than Omaha” is assertoric. “A corporation could be wealthier than a country” is problematic. In Aristotelian logic, “apodictic” is opposed to “dialectic,” as scientific proof is opposed to philosophical reasoning.

For example, the president’s language (“many say”, “everyone knows”, “we’ll see”) is full of assertoric and problematic claims (to be extremely generous), and perhaps that’s the limit of his ability. I don’t think he could manage part of a measured dialectical argument if pressed, and if he manages an apodictic statement it would be like a clock that tells the time correctly twice a day. To have the head of the executive branch of our government to be so untrustworthy in providing information and knowledge hurts us all, and misleads those that takes his words at face value.

And then there are the news sources that cater to the president and his followers. Perhaps they present some warranted information, but mix plenty of misleading punditry in to tickle the fancy of unquestioning minds. As a result we have citizens who only digest information from sources that appeal to their sensibilities. Some of these news sources disseminate their fabrications via a flood in social media and the internet, because our ability to stifle them is almost nonexistent. And when these news sources originate from foreign countries wanting to influence us for their own purposes, how is it that they are allowed to continue?

In truth, people can be misled on scientific topics like the coronavirus and COVID-19, vaccinations, face masks, climate change or global warming, environmentalism and pollution, pseudoscience, and political topics like mail-in voting, Russian meddling with the 2016 and 2020 elections, conspiracy theories such as QAnon, etc. The lists seem almost endless.

Further Reading:

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

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

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

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

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

https://plato.stanford.edu/entries/kant-judgment/

Immanuel Kant: Logic

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

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

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

https://en.wikipedia.org/wiki/Russian_interference_in_the_2016_United_States_elections

https://en.wikipedia.org/wiki/Russian_interference_in_the_2020_United_States_elections

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

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logic, sixteenfolds

The Art of the Syllogism

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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.

Further Reading:

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

http://changingminds.org/disciplines/argument/syllogisms/categorical_syllogism.htm

http://www.philosophypages.com/lg/e08a.htm

https://plato.stanford.edu/entries/medieval-syllogism/

Also:

Vaughan Pratt / Aristotle, Boole, and Categories (PDF, October 12, 2015)

Click to access PrattParikh.pdf

Vaughan Pratt / Aristotle, Boole, and Chu: Duality since 350 BC (Slides, August 12, 2015)

Click to access PrattABCD.pdf

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fourfolds, logic

Four Valued Logic

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Logic is not as absolute as we would like it to be. For example, linear logic breaks down normal logic into a realm of substructurality. There seems to be several ways to consider expanding classical two-valued logic to four values.

Let this post be a placeholder for considering expansion of classical two-valued logic to four values. For example, one might have:

  • True
  • False
  • Both
  • None

Further Reading:

https://en.wikipedia.org/wiki/Four-valued_logic

J. Michael Dunn / Two, Three, Four, Infinity: The Path to the Four-Valued Logic and Beyond
https://link.springer.com/chapter/10.1007/978-3-030-31136-0_6

Katalin Bimbo & J. Michael Dunn / Four-valued Logic
https://projecteuclid.org/euclid.ndjfl/1063372199

J. Ulisses Ferreira / A Four-Valued Logic
https://doi.org/10.5121%2Fcsit.2017.71206

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fourfolds, logic, Metaphysics, Philosophy

Schopenhauer’s Four Laws of Thought

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The first three of Arthur Schopenhauer’s Four Laws of Thought are pretty much the same as the classical three laws of thought. Schopenhauer added a fourth law that was basically for his Principle of Sufficient Reason.

  • Identity
  • Non-contradiction
  • Excluded middle
  • Sufficient reason

These Four Laws are often given in two flavors: the first, in fairly concrete terms of subjects and predicates, and the second, more glib in terms of existence and being and such (isness).

  • A subject is equal to the sum of its predicates. Everything that is, exists. (Identity)
  • No predicate can be simultaneously attributed and denied to a subject. Nothing can simultaneously be and not be. (Non-contradiction)
  • Of every two contradictorily opposite predicates one must belong to every subject. Each and every thing either is or is not. (Excluded middle)
  • Truth is the reference of a judgment to something outside it as its sufficient reason or ground. Of everything that is, it can be found why it is. (Sufficient reason)

The phrase ‘it can be found’ sounds like a constructive method rather than a mere existence proof, but the common theological technique that combines both by saying “everything happens for a reason” avers the reason to an ineffable deity. (I bet Schopenhauer would have disliked this view because from what I understand he was an atheist.)

Moving on, I would like to represent these four laws in even more concrete terms of logical expressions. In the following attempt, let a, b be subjects (or objects), and P, Q be predicates (or qualities):

  • ∀a (a ≡ ∀P P(a))
  • ∀a ¬∃P (P(a) ∧ ¬P(a))
  • ∀a ∀P (P(a) ∨ ¬P(a))
  • ∀a ∃b (b → a)

When detailed in this way, these four laws don’t seem very complete, or don’t quite form a unity, as implication and equivalence are each in only one of them. Even though it doesn’t help that criticism, perhaps one can succinctly say:

  • Things can be reduced to (all) their qualities.
  • Qualities are disjoint from their opposites.
  • Qualities and their opposites are sufficient.
  • Things are entailed by some thing (possibly same).

In addition, I quite liked this Goodread review which aligns Aristotle’s Four Causes with Schopenhauer’s Fourfold Root. So then:

  • From Parts : Material Cause : Becoming : Identity
  • For Functions : Final Cause : Knowing : Non-contradiction
  • Into Structures : Formal Cause : Being : Excluded-middle
  • By Actions : Efficient Cause : Acting : Sufficient reason

Further Reading:

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

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

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

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

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

https://plato.stanford.edu/entries/sufficient-reason/

https://plato.stanford.edu/entries/schopenhauer/

https://en.wikipedia.org/wiki/Quantifier_(logic)

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

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

https://www.goodreads.com/review/show/1328220242

https://equivalentexchange.blog/2010/08/12/aristotles-four-causes/

https://equivalentexchange.blog/2013/09/09/schopenhauers-fourfold-root-of-the-principle-of-sufficient-reason/

https://equivalentexchange.blog/2013/10/09/things-happen/

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Notes:

At some point, I need to understand the difference between the law of the excluded middle and the principle of bivalence.

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

https://philosophy.stackexchange.com/questions/3268/what-is-the-difference-between-law-of-excluded-middle-and-principle-of-bivalence

https://thephilosophyforum.com/discussion/1840/the-principle-of-bivalence-and-the-law-of-the-excluded-middle-please-help-me-understand

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Computation, fourfolds, logic, Philosophy, religion, sixteenfolds

The Arcane Arts of Ramon Llull : the Dignities

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Let us calculate!

— Gottfried Wilhelm Leibniz

Oh, Ramon Llull, where have you been all my life? I’m sure he’s been there all along, death now over seven hundred years in the past, just like always. His legacy seems at first glance to be quite the essence of medieval religion and scholastic philosophy, but still significantly and obscurely different to be enticing to this one. And on further examination, much more.

My schema above has little to do with his grand elaborate figures, except for listing the sixteen attributes he called “dignities”. Llull’s diagrams are full of clock-like wheels within wheels, complicated tableau, and combinatorial patterns. He wished to create a universal model to understand reality, and who wouldn’t want to discover the same? It is said that his methods are akin to an early computer science, and I’m just now starting to understand why.

The magister based the substance of his methods on his Christian faith, although he converted in midlife from Islam. Living in Barcelona, it was probably a good place to make such a change, but felt his calling was to convert others as well, so traveling he went. The methods he developed to convince others of their errors in belief were quite remarkable, as were the volume of his writing.

Like Gottfried Wilhelm Leibniz, who lived four hundred years later and was influenced by him, Llull wished to automate reasoning. But instead of building mechanical devices, Llull built computers from paper and ink, rulers and drawing compasses, scissors and glue. And instead of numbers as the smallest tokens of his computer, he used abstractions (i.e. words) that he felt would be understood by everyone in exactly the same way.

For example, he enumerated these sixteen dignities or aspects of his Christian diety, although sometimes he used the first nine. His constructions allowed one to pose questions and then obtain answers mechanistically that would be convincing to all observers of the correctness of the result. Too bad he was ultimately stoned to death while on his missionary work, although he lived to be eighty two.

Llull’s devices remind me of some of my pitiful charts and diagrams, and make me wonder if I may either adapt some of his techniques to my own use, or be inspired to develop others. I suspect I have locked myself into limitations by my approach, or are these constraints to my advantage? It might be hard to have spinning elements, but I can envision sliding elements like Napier’s Bones, origami-style folding and pleating, and even physical constructions like linkages and abacuses.

Now a martyr within the Franciscan Order, Llull’s feast day is June 30, which I’ve now missed. I hope to remember him to repost or improve on this by next year.

Further Reading:

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

https://plato.stanford.edu/entries/llull/

http://trepanatus.blogspot.com/2007/06/ramon-llull.html

https://history-computer.com/Dreamers/Llull.html

https://publicdomainreview.org/2016/11/10/let-us-calculate-leibniz-llull-and-computational-imagination/

http://www.ncregister.com/blog/astagnaro/bl.-raymond-lull-and-the-worlds-first-computer

https://dangerousminds.net/comments/the_13th_century_thinking_machine_of_ramon_llull

The memory wheel

https://www.google.com/search?q=ars+magna+llull&tbm=isch

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fourfolds, logic, sixteenfolds

The Mouse’s Tale

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No apologies to Lewis Carroll.

Further Reading:

https://en.wikipedia.org/wiki/The_Mouse%27s_Tale

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eightfolds, logic, Mathematics

The Lambda Cube

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More or less from Wikipedia:

In mathematical logic and type theory, the λ-cube is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new way of making objects depend on other objects, namely

    1. terms allowed to depend on types, corresponding to polymorphism.
    2. types depending on terms, corresponding to dependent types.
    3. types depending on types, corresponding to type operators.

The different ways to combine these three dimensions yield the 8 vertices of the cube, each corresponding to a different kind of typed system.

So in the diagram above, we have emblazoned the names of these type systems ordered from lower left to upper right:

  • λ→: the simply typed lambda calculus, our base system
  • λ2: add 1. above to λ→, giving what is also known as System F or the Girard–Reynolds polymorphic lambda calculus
  • λP: add 2. above to λ→
  • λ_ω_: add 3. above to λ→
  • λP2: combine 1. and 2., λ2 and λP
  • λω: combine 1. and 3., λ2 and λ_ω_
  • λP_ω_: combine 2. and 3., λP and λ_ω_
  • λC: combine 1., 2., and 3., giving the calculus of constructions

Further Reading:

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

http://www.rbjones.com/rbjpub/logic/cl/tlc001.htm

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

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

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