Basis Properties
Since all axioms and rules of the calculus are schemata, derivation is closed under substitution:
- If then
where σ is any substitution (of formulas using only implication).
The implicational propositional calculus also satisfies the deduction theorem:
- If, then
As explained in the deduction theorem article, this holds for any axiomatic extension of the system consisting of axioms 1 and 2 above and modus ponens.
Read more about this topic: Implicational Propositional Calculus
Famous quotes containing the words basis and/or properties:
“The basis of our governments being the opinion of the people, the very first object should be to keep that right; and were it left to me to decide whether we should have a government without newspapers, or newspapers without a government, I should not hesitate a moment to prefer the latter.”
—Thomas Jefferson (1743–1826)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)