Category Archives: Philosophy

John Buridan’s Octagon of Opposition

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

[*12.144]

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The Twelve Virtues of Aristotle

The Twelve Virtues of Aristotle:

  • Brave
  • Temperate
  • Generous
  • Munificent
  • High-minded
  • Ambitious
  • Patient
  • Friendly
  • Truthful
  • Witty
  • Modest
  • Indignant

Further Reading:

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

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

https://aesthetichealingmindset.wordpress.com/2011/06/12/4706/

https://www.cwu.edu/~warren/Unit1/aristotles_virtues_and_vices.htm

Aristotle’s 12 virtues: from courage to magnificence, patience to wit

https://www.google.com/search?q=twelve+virtues+of+aristotle&tbm=isch

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The Square of Epictetus

Here is a nice two-by-two matrix by engineer David E. Goldberg derived from the stoic philosophy of Epictetus. For each and every concern, it’s the cross product of one’s Activity (control vs. no control) and State of Mind (concerned vs. not concerned). And so we have the four combinations:

  • Concerned with Control: Accomplishment
  • Unconcerned with Control: Forgone Opportunity
  • Concerned but No Control: Needless Worry
  • Unconcerned and No Control: Peace of Mind

Of course the key is to dwell within the positive Accomplishment and Peace of Mind regions, and not to inhabit the negative ones of Needless Worry and Forgone Opportunity.

Further Reading:

http://entrepreneurialengineer.blogspot.com/2006/11/square-of-epictetus.html

https://www.linkedin.com/pulse/20141007110433-722904-worry-less-accomplish-more/

More Accomplishment, Less Worry: Follow the Epictetus Square

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

Epictetus

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Knowledge as Justified True Belief

Quid est veritas?

— Pontius Pilate

Explanations come to an end somewhere.

— Ludwig Wittgenstein

I read that Edmund Gettier died recently, famous for the problem of knowledge named after him. He argued that justified true belief may not really be knowledge, and gave counterexamples for it not being so.

Here I show a diagram for those three conditions (Justified, True, Belief) and their opposites (Baseless, False, Doubt). The Wikipedia page on the Gettier problem states simply that the solution for it is that the justification must be true also, as the belief.

I don’t want to be posing as Pilate, but it does seems that truth is the most difficult and elusive of the three criteria, conditioning as it must the other two. In this age of disinformation, you could do worse than read the articles below.

Further Reading:

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

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

https://plato.stanford.edu/entries/knowledge-analysis/

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

https://www.researchgate.net/publication/318135910_Justified_True_Belief_Plato_Gettier_and_Turing

[*10.88]

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Four Dimensions of Knowledge

What is the best way to educate, to teach and learn? Ideally, students shouldn’t merely memorize facts and recall them on demand, although retaining well accepted knowledge is important. Certainly, students need mental structures to organize these facts, so that they form associated groups of categories and classification. Additionally, methods are needed for accepting and rejecting facts, and procedures for organizing facts, although facts may often be revisited for truth, or to reorganize them, and so on.

Benjamin Bloom et al. developed a taxonomy for educators of six mental aspects for the acquisition of knowledge, which was revised later into six cognitive actions or processes by L. Anderson, B. S. Krathwohl and others. These six actions form a sort of food pyramid for knowledge, so that lower actions form a broader base for higher ones. Both taxonomies sort a dimensional hierarchy of knowledge operated by these aspects or actions from concrete to abstract: Factual, Conceptual, Procedural, and Metacognitive (the last added when revised).

In the revised taxonomy, “Remember” is the lowest cognitive action, described by “remember facts and basic concepts,” which deals with the Factual and Conceptual. Above it is “Understand,” described by “explain ideas or concepts,” and I imagine an idea can be a fact. Next is a procedural action “Apply: use information in new situations,” and in fact all six actions are procedural by being actions. At the top of the pyramid is “Analyze,” “Evaluate,” and “Create”. The Metacognitive dimension (“thinking about thinking”) is for thinking about these six actions and these four dimensions, as to how they are related and differ.

These taxonomies are well considered and there are many resources to investigate.

Further Reading:

http://edtheory.blogspot.com/2016/03/knowledge-dimensions-factual-conceptual.html

Bloom’s Taxonomy

Revised Bloom’s Taxonomy

https://www.schoolnet.org.za/teach10/resources/dep/thinking_frameworks/bloom_taxonomy_4.htm

Bloom’s Taxonomy Revised

https://www.goconqr.com/en/mindmap/12973371/four-types-of-knowledge

Pedagogy of book and chapter organization

Click to access Anderson-and-Krathwohl_Revised-Blooms-Taxonomy.pdf

Click to access krathwohl.pdf

Anderson, L., Bloom, B. S., Krathwohl, D., & Airasian, P. (2000). Taxonomy for learning, teaching and assessing: A revision of Bloom’s Taxonomy of Educational Objectives (2nd ed.). New York: Allyn & Bacon, Inc.

Anderson, L. W., Krathwohl, D. R., Airasian, P. W., Cruikshank, K. A., Mayer, R. E., Pin- trich, P. R., … & Wittrock, M. C. (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom’s taxonomy of educational objectives, abridged edition. White Plains, NY: Longman.

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

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

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

Concepts, Theory-Theory of

 

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

https://plato.stanford.edu/entries/innateness-cognition/

[*12.84]

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Kant’s Tables of Judgments and Categories

Even though Immanuel Kant’s tables of Judgments and Categories are each made up of four triples, both are divided into the same four headings: Quantity, Quality, Relation, and Modality. And as to the tripartite structure of their divisions, I can’t say I’m convinced of their coherence and completeness, c.f. Lovejoy’s article below.

Judgments

  • Quantity: Universal, Particular, Singular
  • Quality: Affirmative, Negative, Infinite
  • Relation: Categorical, Hypothetical, Disjunctive
  • Modality: Problematic, Assertoric, Apodictic

Categories

  • Quantity: Unity, Plurality, Totality
  • Quality: Reality, Negation, Limitation
  • Relation: Substance, Cause, Community
  • Modality: Possibility, Existence, Necessity

Further Reading:

Immanuel Kant: Logic

Kant, Immanuel

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

https://en.wikipedia.org/wiki/Category_(Kant)

Arthur O. Lovejoy / Kant’s Classification of the Forms of Judgment, The Philosophical Review, Nov. 1907, Vol. 16, No. 6, pp. 588-603

Click to access 2177294.pdf

 

Some researchers discussing Kant’s judgments that look interesting:

https://siucarbondale.academia.edu/RichardLanigan

https://www.degruyter.com/view/journals/semi/2019/227/article-p273.xml?language=en

http://www.mrc.uidaho.edu/~rwells/

http://www.mrc.uidaho.edu/mrc/people/rwells/techdocs/Mental%20Physics/Principles%20of%20Mental%20Physics/Chapters/

[*4.136, *7.114]

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Agent, Actor, Critic, and Spectator

To understand all (causally) is to forgive all (normatively). To understand all (normatively) is to forgive nothing, for it is to talk about norms, not people; about mathematics, in which there is no error and no responsibility, and not about mathematicians, who do make mistakes.

— Lewis White Beck

Here is a set of distinctions by philosopher Lewis White Beck that may have some relevance to thinking about human determination and action. Beck was most renowned as a scholar of Kant.

  • Agent: a person X who thinks they are responsible and free in choices and actions (essentially a “free” agent), with motives and reasons
  • Actor: a person X who thinks they are an agent but is judged by a Spectator Y to not be free and instead determined by external causes
  • Spectator (diagrammed as Viewer): a person Y who observes any X and explains their actions exclusively by causal terms, and so judges them to be an Actor
  • Critic: a person Z who is in a position to agree with X that X is indeed an Agent or a Spectator Y that X is instead an Actor

A great deal of language revolves around explaining our behavior and that of others, both for the benefit or detriment to ourselves and others. All of us would usually rather be free and responsible agents than causally determined actors, but certainly we are not entirely free to act no matter how little we are not. Philosophical discussion of human action continues to be commonly cast into terms such as agency, action, intention, and even authenticity.

Beck later returned to these ideas for his Ernst Cassirer lectures, and a book of them was published in 1975. By the title and a short description, it appears that Beck reduced the fourfold to a mere dual, in order to simplify as well as elaborate his ideas (but I cannot say for sure as I have not read it). Still, I like the original fourfold scheme that in my mind bears likenesses to others presented here (for example, Modal Verbs.)

Further Reading:

Lewis White Beck / Agent, Actor, Spectator, and Critic, The Monist , April, 1965, Vol. 49, No. 2, pp. 167-182
http://www.jstor.com/stable/27901588

Lewis White Beck / The Actor and the Spectator: foundations of the theory of human action (1975)

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

https://en.wikipedia.org/wiki/Agency_(philosophy)

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

https://en.wikipedia.org/wiki/Action_(philosophy)

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

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

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

https://en.wikipedia.org/wiki/Authenticity_(philosophy)

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

[*4.40, *4.108, *7.38, *9.66]

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Flowing and Falling

Some philosophers say everything is process. From the cosmic scale of the universe to the submicroscopic scale of atoms, physical forces marshal matters and energies to and fro. In between these at the human scale, biology is ruled by flows of energy from the sun and nutrients from the environment as well as from other living beings. Dynamical forces such as temperature, weather and tides also affect biology and even the cultural processes of higher lifeforms.

At the scale of the solar system, gravity collects gases to ignite stars and form planets. Once stars reach their limits to burn, gravity can collapse them to dense cinders and black holes or even to rebound and spread their atomic matters in novas and supernovas. Even light spread by the stars can be gathered together into gravity wells such as black holes. Stars in turn are gathered into galaxies by the gravity of black holes and even unknown dark matters.

At the scale of atoms and molecules, temperature differentials and water can partition certain types of elementary constituents to form membranes and segregate insides from outsides. If an inside is protected sufficiently, then there is time and the conditions to harbor and perpetuate the delicate structures and processes that form cells. Cells can even gather together and continue as multicellular communities, or only temporarily to fruit and disperse again as simple creatures known as slime molds.

At the individual human or societal scale, there are flows for nutrients and excreta, materials for habitation and the manufacture of tools, distributed energies such as electricity, fossil fuels, and information for learning, work, and civic participation. Even speech and writing can be thought as flows of information. But just as flows of nutrients and materials and energies can prove toxic to biological health or ecologies, so can information.

For two-dimensional dynamical systems, certain common elements can be mapped out: sources and sinks, saddles and centers. Sources have flows out from a point or region, and sinks have flows in. Saddles have a roughly stationary center, due to balanced flows in and out at (not necessarily) right angles. Centers are circular vortexes about a stationary point or region. Sources and sinks can spiral, saddles can twist, centers can become eccentric or elliptical.

For example, think about everyday weather forecasts. The atmosphere is relatively thin compared to the earth and so the flows of air can be considered two-dimensional, at least at the ordinary strata of human habitation. There are air pressure highs and lows (sources and sinks), and air temperature cold and warm fronts (usually not saddles though), stationary fronts (centers?), and even circulations (hurricanes are spiraling sinks I guess).

Ordinary, human-sized change has conditioned many of our intuitions and insights about the way the universe works. Heraclitus famously said that all was change, and so he thought fire was the primal element. His predecessor Thales thought that water was instead the basic element, and it is pretty mutable also. Lucretius, inspired by Empedocles, thought none of the four classical elements were foundational, but all were composed of tiny bits that fell and bounced against each other through an endless void.

As earth is in opposition to air, and fire to water, the seasonal changes of temperature and moisture were considered by Hippocrates. Heat gains dominion over cold in Spring and Summer, but cold replaces it in Fall and Winter. Similarly wet and dry quarters cycle through the seasons. These oppositions gave rise to the theory of the four temperaments or humourism. Even to this day these considerations have inspired various theories of personality, like the Myers-Briggs Assessment.

Is everything a struggle of opposites? Empedocles, already mentioned, thought love and strife were the relations that respectively attracted and repelled all matter in their dance and change. Now we know that things fall towards the earth, not for the love of it, but because of the shape of space that the earth’s mass makes. Heat flows into the cold because both even up. Order dissolves into chaos since the latter is more likely, unless fed by other sources of order turning to disorder.

Is everything a flow between opposites? Light spreads out and diminishes into darkness, but gravity gathers matter together. Enough gravity can even gather light and bind it into the darkness of a black hole. A drop of ink spreads out in a glass of water, never to return to that inky state, unless the glass sits and the water evaporates until only a drop remains. Even epidemics and pandemics can be thought to be flows of transmission and contagion. Here, the small becomes the large, and the few the many.

Further Reading:

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

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

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

Also note that these four classifications are somewhat analogous to four valued logic: True is Source, False is Sink, None is Center, Both is Saddle.

[*12.52, *12.56]

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The Four Philosophy Ensemble

A woman came up to me and said
“I’d like to poison your mind
With wrong ideas that appeal to you
Though I am not unkind”
She looked at me, I looked at something
Written across her scalp
And these are the words that it faintly said
As I tried to call for help…

— From Whistling in the Dark, by They Might Be Giants

  • The Cynic
  • The Realist
  • The Optimist
  • The Apathetic

Further Reading:

https://tvtropes.org/pmwiki/pmwiki.php/Main/FourPhilosophyEnsemble

The Utopian vs. the Dystopian

https://en.wikipedia.org/wiki/Flood_(They_Might_Be_Giants_album)

https://genius.com/They-might-be-giants-whistling-in-the-dark-lyrics

Notes:

Since these pair up nicely (Realism and Apathy, Optimism and Cynicism), perhaps one could combine them as Optimistic Realism, Optimistic Apathy, Cynical Realism, Cynical Apathy.

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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:

  • 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/

[*11.196, *11.197]

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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