One of the first and most successful compromises was to develop languages based predominately on modus ponens, i . e . IF-THEN rules.
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Can I describe modus ponens as an algebraic operation in this structure ?-talk [ + ] 06 : 10, 12 June 2006 ( UTC)
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It can be shown that with only these three axiom schemata and " modus ponens ", one can prove all tautologies of the propositional calculus.
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Also, from the first element of, last element, as well as modus ponens, is a consequence, and so R \ in \ Gamma.
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On the other hand, DT is so useful for simplifying the syntactical proof process that it can be considered and used as another inference rule, accompanying modus ponens.
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Here, because of the simple syntax of pure Prolog programs, " implied by " can be understood very simply as derivability using modus ponens and universal instantiation alone.
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We describe here a Hilbert system with nine axioms and just the rule modus ponens, which we call the one-rule axiomatisation and which describes classical equational logic.
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Most axiomatic systems have only the rule of modus ponens ( and sometimes substitution ), so it requires only verifying the validity of the axioms and one rule of inference.
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However, if a formal system is introduced where modus ponens is simply a rule of inference defined by the system, then it can be abided by simply because it is so.
40.
Enderton, for example, observes that " modus ponens can produce shorter formulas from longer ones ", and Russell observes that " the process of the inference cannot be reduced to symbols.
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