A conditional proof is a
proof that takes the form of asserting a
conditional, and proving that the
antecedent of the conditional necessarily leads to the
consequent.
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 a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. 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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
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 class of complexity theory. There is a large number of interesting tasks (see ''
List of NP-complete problems''), 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 has many consequences already proven.
Symbolic logic
As an example of a conditional proof in
symbolic logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, suppose we want to prove A → C (if A, then C) from the first two premises below:
See also
*
Deduction theorem
*
Logical consequence
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more stat ...
*
Propositional calculus
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
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