"calculus" and "formal system"
a calculus is a formal system that consists of
- a set of syntactic expressions (well-formed formulæ or wffs),
- a distinguished subset of these expressions (axioms), plus
- a set of formal rules that define a specific binary relation on the space of expressions.
Formal systems in mathematics consist of the following elements:
- A finite set of symbols (i.e. the alphabet), that can be used for constructing formulas (i.e. finite strings of symbols).
- A grammar, which tells how well-formed formulas (abbreviated wff) are constructed out of the symbols in the alphabet. It is usually required that there be a decision procedure for deciding whether a formula is well formed or not.
- A set of axioms or axiom schemata: each axiom must be a wff.
- A set of inference rules.
what is the difference between a formal system, and a calculus, then? I think they are the same?
"calculus" and "logic system"
"Calculus" appears in "propositional calculus" and "first-order predicate calculus", which are also called "propositional logic" and "first-order logic" respectively. So I thought "calculus" and "logic" mean the same, and "a logic" is, according to http://en.wikipedia.org/wiki/Formal_system#Logical_system,
A logical system or, for short, logic, is a formal system together with a form of semantics, usually in the form of model-theoretic interpretation, which assigns truth values to sentences of the formal language, that is, formulae that contain no free variables.
But then I saw "calculus" also appears in "lambda calculus", which is also a formal system. I think a lambda calculus isn't a logic system, right? What does "calculus" mean in "lambda calculus"?
- Furthermore "calculus" can also mean computational real analysis for first-year college students.