Sentence examples for axiomatic proof from inspiring English sources
The form of the work as a whole is therefore a mixture of axiomatic proof and philosophical narrative.
Although an arithmetic semantics is not possible, in (Fitting 2008) a possible world semantics, and an axiomatic proof theory, was given for a version of \(\mathsf{LP}\) with quantifiers ranging over justifications.
Another benefit is that by specifying models with respect to which the axioms and rules of the system are demonstrably complete, we provide a mechanism for producing counter-examples (counter-models), as well as another method, besides e.g. an axiomatic proof procedure, for seeing what conclusions we might be able to draw from a given premise set.
Around 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented.
That really calls for an axiomatic proof, which would be quite do-able.
I would urge the authors to try and do the axiomatic proof for their environment or to just stay away from axiomatics altogether.
One of the fundamental using of the Boolean operations "OR, AND" and "NOT" are axiomatic and Algebraic proofs.
The mathematical theory (developed by the formalists) to cope with proofs about an axiomatic theory T is called proof theory, or metamathematics.
As will appear later (see below Axiomatization of PC), the question whether a sequence of formulas in an axiomatic system is a proof or not depends solely on which formulas are taken as axioms and on what the rules are for deriving theorems from axioms, and not at all on what the theorems or axioms mean.
Employing an axiomatic system in which proof was by direct reduction and indirect reduction or reductio ad impossibile, Aristotle was able to reduce all syllogisms to those of the first figure.
This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error estimators.
