Example 1. Simple Axiom System
Let, where, are defined as follows:
- The alpha set, is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:
- Of the three connectives for conjunction, disjunction, and implication (, and ), one can be taken as primitive and the other two can be defined in terms of it and negation . Indeed, all of the logical connectives can be defined in terms of a sole sufficient operator. The biconditional can of course be defined in terms of conjunction and implication, with defined as .
Adopting negation and implication as the two primitive operations of a propositional calculus is tantamount to having the omega set partition as follows:
- An axiom system discovered by Jan Łukasiewicz formulates a propositional calculus in this language as follows. The axioms are all substitution instances of:
- The rule of inference is modus ponens (i.e., from and, infer ). Then is defined as, and is defined as .
Read more about this topic: Propositional Calculus
Famous quotes containing the words simple, axiom and/or system:
“But the whim we have of happiness is somewhat thus. By certain valuations, and averages, of our own striking, we come upon some sort of average terrestrial lot; this we fancy belongs to us by nature, and of indefeasible rights. It is simple payment of our wages, of our deserts; requires neither thanks nor complaint.... Foolish soul! What act of legislature was there that thou shouldst be happy? A little while ago thou hadst no right to be at all.”
—Thomas Carlyle (1795–1881)
““You are bothered, I suppose, by the idea that you can’t possibly believe in miracles and mysteries, and therefore can’t make a good wife for Hazard. You might just as well make yourself unhappy by doubting whether you would make a good wife to me because you can’t believe the first axiom in Euclid. There is no science which does not begin by requiring you to believe the incredible.””
—Henry Brooks Adams (1838–1918)
“The intellect is vagabond, and our system of education fosters restlessness. Our minds travel when our bodies are forced to stay at home. We imitate; and what is imitation but the travelling of the mind?”
—Ralph Waldo Emerson (1803–1882)