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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.
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. Conditional proofs exist linking several otherwise unproven conjectures, 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.
As an example of a conditional proof in symbolic logic, suppose we want to prove A → C (if A, then C) from the first two premises below:
| 1. | A → B | ("If A, then B") |
| 2. | B → C | ("If B, then C") |
| 3. | A | (conditional proof assumption, "Suppose A is true") |
| 4. | B | (follows from lines 1 and 3, modus ponens; "If A then B; A, therefore B") |
| 5. | C | (follows from lines 2 and 4, modus ponens; "If B then C; B, therefore C") |
| 6. | A → C | (follows from lines 3–5, conditional proof; "If A, then C") |
In propositional logic, affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement under certain assumptions, and invalidly inferring its converse, even though that statement may not be true under the same assumptions. This arises when the consequent has other possible antecedents.
Conditional may refer to:
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 and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
In propositional logic, modus ponens, also known as modus ponendo ponens, implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q.P is true. Therefore Q must also be true."
In propositional logic, modus tollens (MT), also known as modus tollendo tollens and denying the consequent, is a deductive argument form and a rule of inference. Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.
In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. it is impossible for the premises to be true and the conclusion to be false.
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P. For the categorical proposition All S are P, the converse is All P are S. Either way, the truth of the converse is generally independent from that of the original statement.
In classical logic, a hypothetical syllogism is a valid argument form, a deductive syllogism with a conditional statement for one or both of its premises. The term originated with Theophrastus.
In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement " if and only if ", where is known as the antecedent, and the consequent.
In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than to David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology. More subtle distinctions may exist; for example, propositions may implicitly depend upon non-logical axioms. In that case, sequents signify conditional theorems in a first-order language rather than conditional tautologies.
In mathematical logic, a sequent is a very general kind of conditional assertion.
The material conditional is an operation commonly used in logic. When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.
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 the consequent. In some contexts, the consequent is called the apodosis.
The barbershop paradox was proposed by Lewis Carroll in a three-page essay titled "A Logical Paradox", which appeared in the July 1894 issue of Mind. The name comes from the "ornamental" short story that Carroll uses in the article to illustrate the paradox. It existed previously in several alternative forms in his writing and correspondence, not always involving a barbershop. Carroll described it as illustrating "a very real difficulty in the Theory of Hypotheticals". From the viewpoint of modern logic it is seen not so much as a paradox than as a simple logical error. It is of interest now mainly as an episode in the development of algebraic logical methods when these were not so widely understood, although the problem continues to be discussed in relation to theories of implication and modal logic.
In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" to the truth of "Not-B implies not-A", and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that
A premise or premiss is a proposition—a true or false declarative statement— an argument to prove the truth of another proposition called the conclusion. Arguments consist of a set of premises and a conclusion.
Connexive logic names one class of alternative, or non-classical, logics designed to exclude the paradoxes of material implication. The characteristic that separates connexive logic from other non-classical logics is its acceptance of Aristotle's thesis, i.e. the formula,
In proof theory and constructive mathematics, the principle of independence of premise states that if φ and ∃x θ are sentences in a formal theory and φ → ∃x θ is provable, then ∃x is provable. Here x cannot be a free variable of φ, while θ can be a predicate depending on it.