Aristotle (Αριστοτἑλης)

• [Aristotle, De Interpretatione]
• [Aristotle, Pr. Anal.]
• [Lemmon and Scott, 1977]

Notes

Aristotle's presentation in "De Interpretatione" ([Aristotle, DE]) and ([Aristotle, Pr. Anal.]) is prose, not actually symbolic logic. Lemmon and Scott [Lemmon and Scott, 1977, Historical Introduction] give a presentation in modern symbology with this page (mostly follows)

Since the discussion was not in the form of a minimal axiomatic basis, the list below has redundancy.

Based on

(Some changes in notation to match the rest of these pages. -JH)

• Operators
  • Ip : p is impossible.
  • Mp : p is possible. (if p is assumed, it has no impossible consequences)
  • Qp : p is not necessary, and has not impossible consequences
  • ~p : p is false.
  • p>q : p implies q
  • p == q : p is (logically) equivalent to q
  • p&q : p and q are both true.
• Rules
  • Uniform substitution (US) [Assumed]
  • Modus Ponens for > (MP>) [explicit]
• Axioms
  • 1) Lp>Mp
  • *2) Mp>M~p (not in final system, Causes Modal collapse (Lp==p) with axiom 1)
  • 3) Lp>p
  • 4) p>Mp
  • 5) Ip == L~p
  • 6) Ip == ~Mp
  • 7) Qp == (~Lp & ~Ip)
  • 8) Mp == ~Ip
  • 9) Qp == Q~p (This replaces what he originally used axiom 2 for)
  • 10) Qp > Mp
  • 13) L(p>q) >L(Mp > Mq)
  • 14) L(p>q) >L(Lp > Lq)
  • 21) Lp == ~M~p (LS states as implicit)
  • 22) Mp == ~L~p (LS states as implicit)

As reported in [Lemmon and Scott, 1977, Historical Introduction]

Basis for

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This page was last modified on August 6th, 2006.