Adam I. Gerard

Restrictionism and the Four Corners of Logic

I recently read Graham Priest's excellent work The Fifth Corner of Four. Context: The connection between Eastern (Indian, Buddhist, Chinese Folk, and Confucian) philosophies and issues within the Western philosophical tradition have been frequently discussed:

Of greatest relevance is the ancient religious and philosophical notion called the catuskoti. Any statement may be:

  1. True only
  2. False only
  3. Both True and False
  4. Neither True nor False

An astute reader of this blog and those familiar with my work will immediately recognize the interesting connection to the Restrictionist approach advocated by Solomon Feferman, Saul Kripke, and myself (Solomon Feferman and Saul Kripke were both "Nobel Prize Winners" in Philosophy - like Mathematics there's no Nobel Prize proper for Philosophy there's the Wolf Prize and Fields Medal. In Philosophy, there's the Rolf Schock Prize.) - specifically, the "strange phenomenon" and oddity described and reprised from Page 12 here:

"There is another cost associated with the proposal that is worth mentioning. The following sets are consistent interpretations of FOLT* - they are consistent with constraints 14-29 imposed in section 2.2: {位,卢T饾憮(位), 卢F饾憮(位), ...}, {位,卢T饾憮(位), F饾憮(位), ...}, {卢位,T饾憮(位), F饾憮(位), ...}, and {卢位,T饾憮(位), 卢F饾憮(位), ...}. The first set assigns the liar sentence a truth-value of 1 but requires us to say of the liar that it is neither true nor false. The second set assigns the liar sentence a truth-value of 1 but requires us to say of the liar that it is not true and false. The third set says that the liar is both true and false while assigning the liar sentence a truth-value of 0. The fourth set says of the liar that it true and not false while assigning the liar sentence a truth-value of 0. This feature is a well-known consequence of any classical solution utilizing Kripke-Feferman."

Note: my solution (Truth Grounding and the Liar) is the only specific implementation of Kripke-Feferman - it specifies the constraint, what the condition is exactly, creates a formal definition for all known Alethic Paradox (not just the Liar Sentence), etc.

I previously argued that this apparent drawback (the main criticism against Fefereman's Axiomatic Theory of Truth for which he won the Rolf Schock Prize in great part) is really a plus:

"Is the only formal solution that respects (here, is "consistent with") the empirical observation that people have offered four prima facie compelling kinds of solutions to the Liar Paradox in the first place (that the Liar Sentence is True, that it is False, that it is Neither True Nor False, and that it is Both True and False) that have framed the entire historical debate since Antiquity."

But we see at once that not only is the above correct, the semantic oddity I mentioned back in 2012-2014 is in fact also the catuskoti.

Aristotelian Syllogism or Catuskoti

Recollect that the Aristotelian Syllogism forms the basis of modern Logic and Maths (along with the Socratic Method).

Classical (Aristotelian) Logic requires of all sentences:

  1. That they respect Bivalence: every sentence is either true or false.
  2. That they respect Non-contradiction: no sentence and its negation can both be true.

This is prima facie incompatible with the catuskoti. Historically, Western thinkers have dismissed or outright rejected Eastern logic systems as incoherent, backwards, or misguided.

What It Means

In taking a step and reflecting deeply on the above, I've arrived at the following initial conclusions:

  1. Graham Priest is really on to something (and I emphatically reiterate that he is one of the best Logicians around today). His compelling questions and criticism have inspired quite a bit of internal consideration and the fruits are being borne out above.
  2. The supposed superiority of Western Logic and Philosophy (to all others) is almost certainly a white-washed, dogmatic, lie.
  3. Moreover, the supposed incompatibility of say-Indian Logic and Western Aristotelian Logic can be demonstrably shown false - they appear to be a singular system that's continuous with each other.
  4. There are "regions" or pockets of logical phenomena. This should come as no surprise to those familiar with modern topological and spatial treatments of logical concepts. (See Restrictionism and Univalent Foundations and Homotopy Type Theory for more.)
  5. "Eastern" (India, The Middle East, Levant, Tranxsonia, Euphrates, Egyptian, and major ancient cultures) philosophy is ascendant in the 21st century. Western Philosophy has curled back around on itself (Skepticism, the Death of God, Logical Nihilism, Structuralism) and found itself walking the path of "Eastern Philosophies" for the first time.

Again, I emphasize the approach above is at once Hegelian (since it sublates supposedly Eastern and Western philosophical antipathies), Nietzschean since we have gone at the marble statues with our mallets once more, deeply structuralist, and holistic since it unites seemingly distinct systems into one.