Necessity and sufficiency in the context of "Affirming the consequent"

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.

The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.

Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.

The lemma was proven (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935. It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure.

In law and insurance, a proximate cause is an event sufficiently related to an injury that the courts deem the event to be the cause of that injury. There are two types of causation in the law: cause-in-fact, and proximate (or legal) cause. Cause-in-fact is determined by the "but for" test: But for the action, the result would not have happened. (For example, but for running the red light, the collision would not have occurred.) The action is a necessary condition, but may not be a sufficient condition, for the resulting injury. A few circumstances exist where the but-for test is ineffective (see But-for test below). Since but-for causation is very easy to show (but for stopping to tie your shoe, you would not have missed the train and would not have been mugged), a second test is used to determine if an action is close enough to a harm in a "chain of events" to be legally valid. This test is called proximate cause. Proximate cause is a key principle of insurance and is concerned with how the loss or damage actually occurred. There are several competing theories of proximate cause (see Other factors). For an act to be deemed to cause a harm, both tests must be met; proximate cause is a legal limitation on cause-in-fact.

The formal Latin term for "but for" (cause-in-fact) causation, is sine qua non causation.

The principle of sufficient reason or PSR states that everything must have a sufficient reason. It is similar to the idea that everything must have a cause, a deterministic system of universal causation. A sufficient reason is sometimes described as the coincidence of every single thing that is needed for the occurrence of an effect. The principle is relevant to Munchausen's trilemma, as it seems to suppose an infinite regress, rather than a foundational brute fact. The principle was articulated and made prominent by Gottfried Wilhelm Leibniz. Arthur Schopenhauer wrote On the Fourfold Root of the Principle of Sufficient Reason.

A transcendental argument is a kind of deductive argument that appeals to the necessary conditions that make experience and knowledge possible. Transcendental arguments may have additional standards of justification which are more demanding than those of traditional deductive arguments. The philosopher Immanuel Kant gave transcendental arguments both their name and their notoriety.

A proof is sufficient evidence or a sufficient argument for the truth of a proposition.

The concept applies in a variety of disciplines,with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition. In any area of mathematics defined by its assumptions or axioms, a proof is an argument establishing a theorem of that area via accepted rules of inference starting from those axioms and from other previously established theorems. The subject of logic, in particular proof theory, formalizes and studies the notion of formal proof. In some areas of epistemology and theology, the notion of justification plays approximately the role of proof, while in jurisprudence the corresponding term is evidence,with "burden of proof" as a concept common to both philosophy and law.

Necessary condition analysis (NCA) is a research approach and tool employed to discern "necessary conditions" within datasets. These indispensable conditions stand as pivotal determinants of particular outcomes, wherein the absence of such conditions ensures the absence of the intended result. For example, the admission of a student into a Ph.D. program necessitates a prior degree; the progression of AIDS necessitates the presence of HIV; and organizational change necessitates communication.

The absence these conditions guarantees the outcome cannot occur, and no other condition can overcome the lack of this condition. Further, necessary conditions are not always sufficient. For example, AIDS necessitates HIV, but HIV does not always cause AIDS. In such instances, the condition demonstrates its necessity but lacks sufficiency. NCA seeks to use statistical methods to test for such conditions.