A conditional proof is a
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
that takes the form of asserting a
conditional
Conditional (if then) may refer to:
*Causal conditional, if X then Y, where X is a cause of Y
*Conditional probability, the probability of an event A given that another event B has occurred
*Conditional proof, in logic: a proof that asserts a co ...
, and proving that the
antecedent of the conditional necessarily leads to the
consequent
A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...
.
Overview
The assumed antecedent of a conditional proof is called the conditional proof assumption (CPA). Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion
necessarily follows. The validity of a conditional proof does not require that the CPA be true, only that ''if it were true'' it would lead to the consequent.
Conditional proofs are of great importance in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. Conditional proofs exist linking several otherwise unproven
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...
s, so that a proof of one conjecture may immediately imply the validity of several others. It can be much easier to show a proposition's truth to follow from another proposition than to prove it independently.
A famous network of conditional proofs is the
NP-complete
In computational complexity theory, a problem is NP-complete when:
# it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...
class of complexity theory. There is a large number of interesting tasks (see ''
List of NP-complete problems
This is a list of some of the more commonly known problems that are NP-complete when expressed as decision problems. As there are hundreds of such problems known, this list is in no way comprehensive. Many problems of this type can be found in ...
''), and while it is not known if a polynomial-time solution exists for any of them, it is known that if such a solution exists for some of them, one exists for all of them. Similarly, the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
has many consequences already proven.
Symbolic logic
As an example of a conditional proof in
symbolic logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, suppose we want to prove A → C (if A, then C) from the first two premises below:
See also
*
Deduction theorem In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs—to prove an implication ''A'' → ''B'', assume ''A'' as an hypothesis and then proceed to derive ''B''—in systems that do not have an ...
*
Logical consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is on ...
*
Propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
References
* Robert L. Causey, ''Logic, sets, and recursion'', Jones and Barlett, 2006.
* Dov M. Gabbay, Franz Guenthner (eds.), ''Handbook of philosophical logic'', Volume 8, Springer, 2002.
{{DEFAULTSORT:Conditional Proof
Logic
Conditionals
Mathematical proofs
Methods of proof