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(pr), Axiom. Axiom Simp.
Axiom A1 of [Margaris], p.49. One of the 3 axioms of propositional calculus. The 3 axioms are also given as Definition 2.1 of [Hamilton] p. 28. This axiom is called Simp or "the principle of simplification" in Principia Mathematica (Theorem *2.02, p. 100) because "it enables us to pass from the joint assertion of phi and psi to the assertion of phi simply."
Propositional calculus (axioms ax1 through ax3 and rule ax-mp) can be thought of as assertnig formulas that are universally "true" when their variables are replaced by any combination of "true" and "false". Propositional calculus was first formalized by Frege in 1879, using as his axioms (in addition to rule ax-mp) the wffs ax1, ax2, [pm2.04], con3, nega, and negb. Around 1930 Lukasiewicz simplified the system by eliminating the third (which follows from the first two, as you can see by looking at the proof of [pm2.04]) and replacing the last three with our ax3. (Thanks to Ted Ulrich for this information.)
The theorems of propositional calculus are also called tautologies. Tautologies can be proved very simply using truth tables, based on the true/false interpretation of propositional calculus. To do this, we assign all possible combinations of true and false to the wff variables and verify that the result always evaluates to true. This is called the semantic approach. Our approach is called the syntactic approach, in which everything is derived from axioms. A metatheorem called the Completeness Theorem for Propositional Calculus shows that the two approaches are equivalent and even provides an algorithm for automatically generating syntactic proofs from a truth table. Those proofs, however, tend to be long, and the much shorter proofs that we show here were found manually.
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