Category Archives: logic

The Art of the Syllogism

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/

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Four Valued Logic

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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Schopenhauer’s Four Laws of Thought

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:

  • Material Cause : Becoming : Identity
  • Final Cause : Knowing : Non-contradiction
  • Formal Cause : Being : Excluded-middle
  • 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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The Arcane Arts of Ramon Llull : the Dignities

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

http://www.ramonllull.net/sw_studies/studies_original/compsale.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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The Mouse’s Tale

No apologies to Lewis Carroll.

Further Reading:

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

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The Lambda Cube

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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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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Four Forms Make a Universe, Part 2

This is a continuation of my last entry. Above is a different representation of the LICO alphabet, with the letters turned 45 degrees counter-clockwise, and rearranged into a symmetric pattern. The letters seem to arise more naturally in this orientation, but then Schmeikal rotates them into his normal schema.

And to the right is a diagram of the logical expressions that correspond to the letters above.

After making these new diagrams, I became inspired and made a few other figures to share with you.

These two versions, with triangles instead of line segments, and also with borders between adjacent triangles removed:

 

 

 

 

 

And these two versions, with quarter circles, and also with edges between adjacent quarter circles removed:

 

 

 

 

 

 

Further Reading:

https://equivalentexchange.blog/2019/03/12/four-forms-make-a-universe/

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Four Forms Make a Universe

How could I not love a paper with this title? I’ve struggled with it for a bit, and I’ve only managed a couple of diagrams relating the author’s LICO (Linear Iconic) alphabet made up of 16 letters. However, I see that there are a few other papers by Schmeikal available on ResearchGate that look easier to understand. But also however, the first one says to read the “Four Forms” paper first!

At any rate, I present a sixteen-fold of the LICO alphabet, and another of the binary Boolean operators that are in a one-to-one mapping with LICO. There is much to understand from these papers, including much syncretism between various mathematical sixteen-folds, so please forgive me if I don’t explain it all with immediate ease. However, I believe it is well worth the effort to understand.

(Please note that the characters of the LICO alphabet are oriented so that the bottoms of the letters are downward, but the Boolean operators are oriented so that the bottoms of the equations are towards the right angles of the triangles.)

The title comes from the result that four elements of LICO can reproduce the other twelve via linear combinations. These four forms are 1) Boolean True (A or ~A), 2) A, 3) B, and 4) A=B. These are within the interior right-hand triangles in the LICO diagram. Of course, it is well known from Computer Science that the NAND operator (~A or ~B) can also generate all other fifteen operators, but this is by multiple nested operations instead of simple Boolean arithmetic. There are several other “universal” binary gates that can do this as well.

Two other representations that have four elements that can generate the other twelve via linear combinations come from CL(3,1), the Minkowski algebra. These representations are called “Idempotents” and  “Colorspace vectors”. Because of this algebra’s association with space and time in relativity, Schmeikal claims that LICO has ramifications in many far-ranging conceptualizations.

Further Reading:

Bernd Schmeikal / Four Forms Make a Universe, in Advances in Applied Clifford Algebras (2015), Springer Basel (DOI 10.1007/s00006-015-0551-z)

https://link.springer.com/article/10.1007/s00006-015-0551-z

At http://www.researchgate.net:

Bernd Schmeikal / Free Linear Iconic Calculus – AlgLog Part 1: Adjunction, Disconfirmation and Multiplication Tables

Bernd Schmeikal / LICO a Reflexive Domain in Space-time (AlgLog Part 3)

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

https://www.allaboutcircuits.com/technical-articles/universal-logic-gates/

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The Marriage of Opposites, Part 3

Everything is dual; everything has poles; everything has its pair of opposites; like and unlike are the same; opposites are identical in nature; but different in degree.

— From The Kybalion by The Three Initiates

There are trivial truths and the great truths. The opposite of a trivial truth is plainly false. The opposite of a great truth is also true.

— Niels Bohr

I have mentioned the alchemical notion of the “Marriage of Opposites” several times (here and here). When opposites marry, what happens as a result? Do they cancel one another out, leaving just a boring average as result? Do they explode in a fiery conflagration, like matter and anti-matter releasing energy? Or do they create a new thing, something that is greater than the sum of their parts?

If opposites annihilate each other, what is the result, emptiness or a void? It is often said that nature abhors a vacuum (“horror vacui”), but I think it is far more true that the mind does. In dualistic thinking, everything that is not one thing must be its opposite. Not good is bad, not happy is sad, not black is white.

In classical logic, the Law of the Excluded Middle says that for any proposition “p”, either it is true or its negation “not p” is true. Thus, “p or not p” is necessarily true, a tautology. Similarly, their combination “p and not p”, cannot ever be true, and so is necessarily false. If one can assume “not p” and derive a contradiction, then “p” must be true (reductio ad absurdum).

In intuitionistic logic, one cannot deduce “p” simply from the falsity of “not p” (that is, “not not p”), one must actually prove that “p” is true. So “p or not p” may still be uncertain, if we don’t know how to prove “p”. However, “p and not p” is still false, based on the falsity of “not p”.

In the viewpoint of Dialetheism, it is offered that there are truths whose opposites are also true, called “true contradictions”. Dialetheisms cannot exist in formal logics because if “p and not p” is true, then you can deduce anything you want and your logic breaks down. Nonetheless, much thought through the years has been dedicated to dialetheisms and their ilk. Please see the recent work by philosopher Graham Priest.

When one considers something and its opposite at the same time, how can you reach an agreement between them? In magnetism, opposite charges attract and like charges repel. All too often, opposite viewpoints vigorously repel each other instead of reaching a happy medium. Each viewpoint considers the other “false” and so they push away at each other, instead of meeting halfway in compromise.

If there is empirical evidence supporting one viewpoint and not the other, and both parties can agree to it, then problem solved. But if viewpoints are more like ideologies, and one side shows evidence that the other side dismisses, what then? Are we only left to agree to disagree? That doesn’t seem like a long term solution.

In this blog I have insinuated but not stated explicitly that a marriage of opposites can often be achieved by combining it with another pair of opposites. Rather than meeting in the middle to a void or an annihilation, one can reach the other side by “going around” the danger, by way of intermediates. Much like Winter reaches Summer by passing through Spring and Summer reaches Winter via Fall, this type of structure is found everywhere in human thinking.

In fact, many systems of pluralistic philosophies are built on fourfolds instead of dualities. For example, see the work of Richard McKeon, specifically this paper.

Further Reading:

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

https://en.wikipedia.org/wiki/Horror_vacui_(physics)

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

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

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

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

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

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

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

http://www.richardmckeon.org/

Click to access McK-PhilosophicSemantics&Inquiry.pdf

https://www.quora.com/Is-Graham-Priest-sincere-and-serious-about-his-stance-on-dialetheism-I-have-difficulty-empathizing-with-such-a-position-What-should-I-do-to-better-understand-this-position-Do-other-philosophers-respect-this-position/answer/Toni-Kannisto?share=368d7909&srid=5ofmf

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