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Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
For a consistent set of axioms, the statements bearing the “t" value are called theorems, lemmas, corollaries, and propositions. Though many inconsistencies in using these words are encountered, the following rules may be suggested:a theorem is an important true statement;a lemma is a true statement, serving, however, as an auxiliary tool to prove a certain theorem or theorems;a proposition is (in fact) a theorem which is not important enough to be called a theorem. This suggests that the name theorem be used rather rarely to emphasize especially important key results;finally, a corollary is a true statement, derived as an immediate consequence of a theorem or proposition with little extra effort.Lowercase letters will be used to denote statements. Typically, letters p, q, r, and s are preferred. Recall once again that a statement p is a sentence for which only one of the two values “true" or “false" can be assigned.
Propositional Calculus
Published in Janet Woodcock, Software Engineering Mathematics, 1988
We can also use previously performed derivations in this way. If we have performed a derivation of the form S h If, and we are attempting another derivation in which we we have already obtained all the sentences of S, then we can write down W without reproducing the entire derivation. Results introduced into proofs and derivations are called lemmas. These may be proved elsewhere, or taken for granted without a formal proof. Remember that your proofs and derivations are only as reliable as the weakest link, and leaving lemmas unproved is a notorious way of introducing errors.
Determinants
Published in Jeff Suzuki, Linear Algebra, 2021
When building a proof, mathematicians often rely on lemmas: results that minor in and of themselves, but are important because they lead to more general results. Activity 6.25 gives:
… Damned if you don’t
Published in Journal of Responsible Innovation, 2018
The overall purpose of the discussion paper is to identify the DSC as a model for another dilemma, that of societal alignment, and to frame the DSA as a way of advancing informed conversation about RRI. I am exceptionally sympathetic to this agenda, having used the Collingridge dilemma as a starting point for some of my own work (see, for example, Guston [2014] and below) and having contributed to the RRI agenda for more than a decade (beginning with Guston [2004],2 which takes on the issue of RRI in public institutions like universities). My fundamental issue with the discussion paper, therefore, is that while societal alignment is a difficult and important challenge, a challenge is not the same thing as a dilemma. Far be it for me to dispute the Oxford English Dictionary, but I think its definitions as quoted by Ribeiro and coauthors obscure more than they enlighten. A dilemma is not just a choice between undesirable alternatives. It is rather a choice between two lemmas (or lemmata) – which are mid-level, proven propositions in a logical argument. Thus, they are not merely undesirable, but they are importantly in some logical conflict with each other. It is not simply, ‘would you prefer bad 1 or bad 2?’, but more like, ‘you must choose either impossibility B or impossibility not B.’ That’s a dilemma worth having. And while I believe that we’d be damned for not pursuing societal alignment, I don’t believe we’d be damned if we do. Indeed, I believe that such an admittedly difficult but not illogical pursuit is the only way we’ll get the kind and quality of innovation we need to salvage our current situation.