Paradoxes and Their Resolutions

Foreword

Paradoxes and their Resolutions is a ‘thematic compilation’. It collects in one volume the essays that I have written in the past (over a period of some 27 years) on this subject.

It comprises expositions and resolutions of many, though far from all, ancient and modern paradoxes, including: the Protagoras-Euathlus paradox (Athens, 5^th Cent. BCE), the Liar paradox and the Sorites paradox (both attributed to Eubulides of Miletus, 4^th Cent. BCE), Russell’s paradox (UK, 1901) and its derivatives the Barber paradox and the Master Catalogue paradox (also by Russell), Grelling’s paradox (Germany, 1908), Hempel's paradox of confirmation (USA, 1940s), and Goodman’s paradox of prediction (USA, 1955).

I also here present and comment on some of the antinomic discourse found in some Buddhist texts (namely, in Nagarjuna, India, 2^nd Cent. CE; and in the Diamond Sutra, date unknown, but probably in an early century CE).

Despite its title, note well, the present book is not intended as an exhaustive study; there are many paradoxes it does not mention or treat.[1]

By ‘paradox’ is here, of course, meant apparent double paradox. A paradoxical proposition has the self-contradictory hypothetical form ‘if P, then not-P’, or ‘if not-P, then P’, where P is any sort of proposition. Such a proposition, taken alone, is not antinomic, i.e. in breach of the laws of thought, simply because there is a formal way out of it: ‘if P, then not-P’ implies the categorical proposition ‘not-P’; and ‘if not-P, then P’ implies the categorical proposition ‘P’. A single paradox, then, constitutes logically legitimate discourse. Very different is a double paradox, i.e. a claim that both ‘if P, then not-P’ and ‘if not-P, then P’ are simultaneously true for a given instance of P. Such a claim is diametrically opposed to the law of non-contradiction, since it concludes that both the categorical propositions ‘not-P’ and ‘P’ are simultaneously true for a given instance of P.

Clearly, resolving (double) paradoxes is essential to logic, ontology and epistemology, since to accept any such antinomy would put human knowledge, and indeed the cognitive faculties that make it possible, in grave doubt. Skeptics relish paradoxes, because they maliciously wish to invalidate human knowledge and the human mind. Defenders of human reason are therefore obligated to confront every such challenge, and neutralize it convincingly. More positively, paradoxes are great opportunities to learn something new about the way we think or what we believe, and to discover and correct our errors. If we did not encounter the paradox, we might remain unaware of our errors; the paradox opens the door to our correcting them.

‘Resolution’ of paradox consists in showing that the apparent double paradox is in fact, for some specific reason, only apparent; i.e. it is illusory, not real. To find the resolution, one generally needs to examine the underlying or surrounding discourse very carefully, and uncover where in it one made a mistake. One may have relied on some overly vague notion or fallen into equivocation or made some unjustified assumption or whatever. Resolution of paradox is generally not a mechanical process, but requires considerable perspicacity and reflection. The main paradoxes are far from easy to dissolve; each one requires due consideration and special treatment. Often, after they are discovered by some logician or philosopher, they remain unresolved for a long time.

As regards the ordering of the present collection, I would like to make the following clarification. In my past thematic compilations (namely, The Laws of Thought, The Self, Ethics, and Theology), I have generally opted for a chronological presentation. However, in the present case, the ordering has been determined by didactic as well as chronological considerations.

• I have placed as the opening chapter an essay, “The vanity of the tetralemma,” drawn from my A Fortiori Logic (2013). The next two chapters, “Clarifying contradiction” and “Clarifying negation,” are drawn from my book Ruminations (2005). These first three essays serve to, at the outset, focus attention on and reaffirm the laws of thought.
• Next, I insert, as introduction to the subject of paradoxes, the essay “Paradoxes,” drawn from my earliest work, Future Logic (1990).

• This is followed by some early thoughts about paradoxes in general and the Liar paradox in particular, in the essay “The Liar paradox (early),” drawn from Future Logic and Ruminations. At this point, I insert my more advanced analysis of this paradox in “The Liar paradox (redux),” drawn from A Fortiori Logic.

• Next, in the essay “The Russell paradox (early),” also drawn from Future Logic and Ruminations, I present some early thoughts on the Russell paradox, as well as on certain derivatives of it, namely the barber paradox, the book catalogue paradox and Grelling’s paradox. After that, I insert my more advanced analysis of this paradox in “The Russell paradox (redux),” drawn from A Fortiori Logic.
• Thereafter, I have put the essays “Hempel’s paradox of confirmation” and “Goodman’s paradox of prediction,” both drawn from my Logical and Spiritual Reflections (2008-9). Then the essays “The Sorites paradox is contrived” and “Protagoras vs. Euathlus resolved,” both drawn from a book I have yet to complete and publish (working title: “Topics in Logic, Philosophy, and Spirituality,” partly published online in 2017).
• Finally, the present volume comprises two essays, “Buddhist antinomic discourse” and “More Buddhist antinomic discourse,” drawn respectively from my Buddhist Illogic (2002) and from Ruminations and Logical and Spiritual Reflections.

As for the relative importance of the paradoxes here presented and dealt with, the following may be said.

To my mind, the liar paradox and the Russell paradox are especially important, in view of their potential impact on logic theory; and I particularly recommend my most recent reflections concerning them (in the essays labeled “redux”). Although I had treated these two paradoxes in earlier writings (here labeled “early”), I managed in 2013, when I was in the last stages of writing my book A Fortiori Logic[2], to go much more deeply into them. I was at the time at the top of my form, intellectually at my most experienced and mature, and could therefore see things a lot more clearly than ever before.

In the redux essay on the liar paradox, I show that this paradox is a many-headed hydra, which cannot be explicated only with reference to self-reference, as commonly done, but involves a variety of problems that all need to be addressed. In the redux essay on the Russell paradox, I show that this class-logic conundrum is not due, as customarily assumed, to some difficulty with the idea of non-self-membership, but is due, on the contrary, to the impossibility of self-membership; this is a new and definitive resolution of the paradox, and I detail its many implications.

The modern paradoxes of Hempel and Goodman are also of significance to logic theory, in that they raise doubts in relation to induction; but their resolutions are relatively easy. The ancient sorites paradox is also logically interesting, in that it shows the importance of intellectual and verbal precision in discourse; but as a paradox it is easily resolved. The earlier Protagoras-Euathlus paradox does not have great logical importance, except perhaps to teach us that when endeavoring to resolve paradoxes we should not get caught up in logically irrelevant issues (in this case, legal ones).

As regards Buddhist antinomic discourse, dealing with it is important at this time in history, because Buddhism is fashionable in some circles, and some of its doctrines are there received in too dogmatic a spirit.

If in the future I write additional essays on paradoxes, I will hopefully include them in expanded editions of the present volume.