Is the Principle of Contradiction a Consequence of \(x^{2}=x\)?


According to Boole it is possible to deduce the principle of contradiction from what he calls the fundamental law of thought and expresses as \(x^{2}=x\). We examine in which framework this makes sense and up to which point it depends on notation. This leads us to make various comments on the history and philosophy of modern logic.

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Beziau, J. Is the Principle of Contradiction a Consequence of \(x^{2}=x\)?. Log. Univers. 12, 55–81 (2018).

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Mathematics Subject Classification

  • Primary 03A05
  • Secondary 00A30
  • 01A55
  • 03B53
  • 03B22
  • 03B05
  • 03B10
  • 03G05


  • Boole
  • Principle of contradiction
  • Laws of thought
  • Symbolic logic
  • Subtraction
  • Universal logic
  • Boolean algebra
  • Square of opposition