Notes from the underground
barang dot sg
Updated 2 December 2023
What’s a logical paradox?

1. Beyond belief
2. Swallowing the conclusion
3. Rejecting the reasoning
4. Hempel’s ravens
5. Descartes’s dream
6. The bicycle
7. The heap
8. Logical fatalism
9. Newcomb’s paradox
10. The unicorn
11. References

11. References

If you’d like a more thorough introduction to the subject, try Mark Sainsbury’s Paradoxes (2009) or Doris Olin’s Paradox (2003).

Sainsbury is more succinct while Olin is more relaxed, but both explore numerous logical paradoxes in some depth. Olin also has a handy guide for newcomers. (Sainsbury) (Olin)

The passage from Tycho Brahe about a projectile on a moving ship is from his Astronomical Letters (1596). It is mentioned in a paper by Christopher M. Graney, found online here.

Carl G. Hempel introduced the paradox of the ravens in ‘Studies in the Logic of Confirmation,’ Mind (1945). W. V. Quine’s thoughts are found in his ‘Natural Kinds,’ in Nicholas Rescher (ed.), Essays in Honor of Carl G. Hempel (Springer 1969).

René Descartes discussed the dreaming problem in his Meditations on First Philosophy (1641), a classic work of philosophy, studied by students all over the world.

Zhuangzi’s butterfly dream is depicted in The Complete Works of Chuang Tzu (Columbia Univ. Press 1968), trans. Burton Watson. Both Zhuangzi and Plato lived in the 4th century B.C., with Plato likely being a little older.

For more on Nick Bostrom on whether we might be brains in vats, see this website. As for David Chalmers’s latest ruminations, see his recent book, Reality+ (W. W. Norton 2022).

Daniel Dennett’s claim that reality simulation is “computationally intractable” is found in the Prelude to his 1991 classic, Consciousness Explained.

G. E. Moore explored various responses to external-world scepticism throughout his career. The sort of response discussed above may be found in ‘Four Forms of Scepticism,’ in his Philosophical Papers (George Allen & Unwin 1959).

For Rudolf Carnap’s use of the verifiability criterion of meaning, see his 1928 pamphlet, ‘Pseudoproblems in Philosophy,’ which was written for a general audience.

For Hilary Putnam’s views, see ‘Brains in a Vat,’ Chapter 1 of his Reason, Truth and History (Cambridge 1981).

I first saw the bicycle “paradox” in one of Martin Gardner’s recreational math books, but it appears as far back as Cyril Pearson’s The Twentieth Century Standard Puzzle Book (1907), found online here.

The paradox of the heap may be traced as far back as Eubulides of Miletus in the 4th century B.C., who belonged to a school of philosophers in Megara, a city that still exists in modern Greece.

W. V. Quine’s reflections on vagueness are found in his paper, ‘What Price Bivalence?,’ Journal of Philosophy (1981). For Kenton Machina, see ‘Truth, Belief and Vagueness,’ Journal of Philosophical Logic (1976).

Michael Dummett’s crisp analysis of logical fatalism is found in ‘Bringing About the Past,’ Philosophical Review (1964).

For Chrysippus and the Idle Argument, see the aforementioned Stanford encyclopedia entry, which draws in turn on Susanne Bobzien’s Determinism and Freedom in Stoic Philosophy (Oxford 1998).

William Newcomb (pictured) was a physicist at the Lawrence Livermore Laboratory at the University of California. The definitive introduction to his paradox, ‘Newcomb’s Problem and Two Principles of Choice,’ is found in Nicholas Rescher (ed.), Essays in Honor of Carl G. Hempel (Reidel 1969). It was penned by the philosopher Robert Nozick, who was a friend of a friend of Newcomb.

Raymond Smullyan’s cheeky proof of the existence of a unicorn may be found in his book, What is the Name of This Book? (1978).