Logical equivalence in the context of "Double negation elimination"

In logic, a rule of replacement is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a whole logical expression, a rule of replacement may be applied to only a particular segment. Within the context of a logical proof, logically equivalent expressions may replace each other. Rules of replacement are used in propositional logic to manipulate propositions.

Common rules of replacement include de Morgan's laws, commutation, association, distribution, double negation, transposition, material implication, logical equivalence, exportation, and tautology.

In logic, a strict conditional (symbol: ${\displaystyle \Box }$, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic. For any two propositions p and q, the formula p → q says that p materially implies q while ${\displaystyle \Box (p\rightarrow q)}$ says that p strictly implies q. Strict conditionals are the result of Clarence Irving Lewis's attempt to find a conditional for logic that can adequately express indicative conditionals in natural language. They have also been used in studying Molinist theology.

Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value. In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number.

Depending on the construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom), or may be a theorem proven from the construction. There are many equivalent forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness (completeness as a metric space).

where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with", P and Q are any given logical statements, and ${\displaystyle \neg P\lor Q}$ can be read as "(not P) or Q". To illustrate this, consider the following statements:

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that every infinite sequence of points has limiting values. For example, the real line is not compact since the sequence of natural numbers has no real limiting value. The open interval (0,1) is not compact because it excludes the limiting values 0 and 1, whereas the closed interval [0,1] is compact. Similarly, the space of rational numbers ${\displaystyle \mathbb {Q} }$ is not compact, because every irrational number is the limit of the rational numbers that are lower than it. On the other hand, the extended real number line is compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.

One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence ⁠1/2⁠, ⁠4/5⁠, ⁠1/3⁠, ⁠5/6⁠, ⁠1/4⁠, ⁠6/7⁠, ... accumulate to 0 (while others accumulate to 1). Since neither 0 nor 1 are members of the open unit interval (0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering ${\displaystyle \mathbb {R} ^{1}}$ (the real number line), the sequence of points 0,  1,  2,  3, ... has no subsequence that converges to any real number.

A trilemma is a difficult choice from three options, each of which is (or appears) unacceptable or unfavourable. There are two logically equivalent ways in which to express a trilemma: it can be expressed as a choice among three unfavourable options, one of which must be chosen, or as a choice among three favourable options, only two of which are possible at the same time.

The term derives from the much older term dilemma, a choice between two or more difficult or unfavourable alternatives. The earliest recorded use of the term was by the British preacher Philip Henry in 1672, and later, apparently independently, by the preacher Isaac Watts in 1725.