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:
“Most young black females learn to be suspicious and critical of feminist thinking long before they have any clear understanding of its theory and politics.... Without rigorously engaging feminist thought, they insist that racial separatism works best. This attitude is dangerous. It not only erases the reality of common female experience as a basis for academic study; it also constructs a framework in which differences cannot be examined comparatively.”
—bell hooks (b. c. 1955)
“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)