In general, one can create arbitrary complicated cycles and check that they can lead either to contradictions or to infinite regress; but in either case, the expression fails to converge to a definite proposition.
An indefinite proposition is said to be true if it yields a true proposition (Łukasiewicz says 'judgement' for a definite proposition) for all values of its variables, it is false if it yields a false judgement for all values, and is neither true nor false if it yields true judgements for some values and false judgements for others.
If the Hessian matrix is negative definite, the proposition will be proved.
Since ℒe is positive definite by [1, Proposition III.2.2] and c i ≠ 0, it follows 〈e, c i 〉 > 0 and ||c i || > 0. Thus, from (2.1) we have that λmax(a k ) → ∞ when, which implies that the result (i) holds; Similarly, for any i ∈ {1,..., r}, and hence, λmin(a k ) → -∞ when, which implies that the result (ii) holds.
The correct analysis of propositions containing definite descriptions has been the subject of considerable philosophical controversy.
From this standpoint, statements of geometry and arithmetic were necessarily true propositions with definite empirical content.
This method of analysis, first introduced by Russell in his article "On Denoting" (1905), translates propositions containing definite descriptions (e.g., "the present king of France") into expressions that do not the purpose being to remove the logical awkwardness of appearing to refer to things (such as the present king of France) that do not exist.
Not definitive, just definite.
The definite truth-values of the propositions about possible events do not make them necessary, many of these being acts of the will which as a free cause chooses between alternatives without being determined.
Propositions and propositional functions are unlike definite descriptions and classes in that there are no explicit definitions of them in PM.
