Arthur Schopenhauer | Equivalent eXchange

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

Schopenhauer’s Four Classes that comprise his Fourfold Root of the Principle of Sufficient Reason are diagrammed above. Becoming, Knowing, Being, and Willing are these classes.

From Wikipedia:

In his Translator’s Introduction to Schopenhauer’s The World as Will and Representation, E. F. J. Payne concisely summarized the Fourfold Root.

Our knowing consciousness…is divisible solely into subject and object. To be object for the subject and to be our representation or mental picture are one and the same. All our representations are objects for the subject, and all objects of the subject are our representations. These stand to one another in a regulated connection which in form is determinable a priori, and by virtue of this connection nothing existing by itself and independent, nothing single and detached, can become an object for us. …The first aspect of this principle is that of becoming, where it appears as the law of causality and is applicable only to changes. Thus if the cause is given, the effect must of necessity follow. The second aspect [knowing] deals with concepts or abstract representations, which are themselves drawn from representations of intuitive perception, and here the principle of sufficient reason states that, if certain premises are given, the conclusion must follow. The third aspect of the principle is concerned with being in space and time, and shows that the existence of one relation inevitably implies the other, thus that the equality of the angles of a triangle necessarily implies the equality of its sides and vice versa. Finally, the fourth aspect [willing] deals with actions, and the principle appears as the law of motivation, which states that a definite course of action inevitably ensues on a given character and motive.