The paradox
In the story, Uncle Joe and Uncle Jim are walking to the barber shop. They explain that there are three barbers who live and work in the shop—Allen, Brown, and Carr—and some or all of them may be in. We are given two pieces of information from which to draw conclusions. Firstly, the shop is definitely open, so at least one of the barbers must be in. Secondly, Allen is said to be very nervous, so that he never leaves the shop unless Brown goes with him.
Now, according to Uncle Jim, Carr is a very good barber, and he wants to know whether Carr will be there to shave him. Uncle Joe insists that Carr is certain to be in, and claims that he can prove it logically. Uncle Jim demands this proof.
Uncle Joe gives his argument as follows:
Suppose that Carr is out. We will show that this assumption produces a contradiction. If Carr is out, then we know this: "If Allen is out, then Brown is in", because there has to be someone in "to mind the shop." But, we also know that whenever Allen goes out he takes Brown with him, so as a general rule, "If Allen is out, then Brown is out". The two statements we have arrived at are incompatible, because if Allen is out then Brown cannot be both In (according to one) and Out (according to the other). There is a contradiction. So we must abandon our hypothesis that Carr is Out, and conclude that Carr must be in.
Uncle Jim's response is that this conclusion is not warranted. The correct conclusion to draw from the incompatibility of the two "hypotheticals" is that what is hypothesized in them (that Allen is out) must be false under our assumption that Carr is out. Then our logic simply allows us to arrive at the conclusion "If Carr is out, then Allen must necessarily be in".
Discussion
In modern logic theory this scenario is not a paradox. The law of implication reconciles what Uncle Joe claims are incompatible hypotheticals. This law states that "if X then Y" is logically identical to "X is false or Y is true" (¬X ∨ Y). For example, given the statement "if you press the button then the light comes on", it must be true at any given moment that either you have not pressed the button, or the light is on.
In short, what obtains is not that ¬C yields a contradiction, only that it necessitates A, because ¬A is what actually yields the contradiction.
In this scenario, that means Carr doesn't have to be in, but that if he isn't in, Allen has to be in.
Simplifying to Axiom 1
Applying the law of implication to the offending conditionals shows that rather than contradicting each other one simply reiterates the fact that since the shop is open one or more of Allen, Brown or Carr is in and the other puts very little restriction on who can or cannot be in the shop.
To see this let's attack Jim's large "contradictory" result, mainly by applying the law of implication repeatedly. First let's break down one of the two offending conditionals:
- "If Allen is out, then Brown is out"
- "Allen is in or Brown is out"
| - (¬A ⇒ ¬B)
- (A ∨ ¬B)
| |
Substituting this into
- "IF Carr is out, THEN If Allen is also out Then Brown is in AND If Allen is out Then Brown is out."
| - ¬C ⇒ ( (¬A ⇒ B) ∧ (¬A ⇒ ¬B) )
| |
Which yields, with continued application of the law of implication,
- "IF Carr is out, THEN if Allen is also out, Brown is in AND either Allen is in OR Brown is out."
- "IF Carr is out, THEN both of these are true: Allen is in OR Brown is in AND Allen is in OR Brown is out."
- "Carr is in OR both of these are true: Allen is in OR Brown is in AND Allen is in OR Brown is out."
| - ¬C ⇒ ( (¬A ⇒ B) ∧ (A ∨ ¬B) )
- ¬C ⇒ ( (A ∨ B) ∧ (A ∨ ¬B) )
- C ∨ ( (A ∨ B) ∧ (A ∨ ¬B) )
| |
- note that : C ∨ ( (A ∨ B) ∧ (A ∨ ¬B) ) can be simplified to C ∨ A
- since ( (A ∨ B) ∧ (A ∨ ¬B) ) is simply A
And finally, (on the right we are distributing over the parentheses)
- "Carr is in OR Either Allen is in OR Brown is in, AND Carr is in OR Either Allen is in OR Brown is out."
- "Inclusively, Carr is in OR Allen is in OR Brown is in, AND Inclusively, Carr is in OR Allen is in OR Brown is out."
| - C ∨ (A ∨ B) ∧ C ∨ (A ∨ ¬B)
- (C ∨ A ∨ B) ∧ (C ∨ A ∨ ¬B)
| |
So the two statements which become true at once are: "One or more of Allen, Brown or Carr is in", which is simply Axiom 1, and "Carr is in or Allen is in or Brown is out". Clearly one way that both of these statements can become true at once is in the case where Allen is in (because Allen's house is the barber shop, and at some point Brown left the shop).
Another way to describe how (X ⇒ Y) ⇔ (¬X ∨ Y) resolves this into a valid set of statements is to rephrase Jim's statement that "If Allen is also out ..." into "If Carr is out and Allen is out then Brown is in" ( (¬C ∧ ¬A) ⇒ B).
Showing conditionals compatible
The two conditionals are not logical opposites: to prove by contradiction Jim needed to show ¬C ⇒ (Z ∧ ¬Z), where Z happens to be a conditional.
The opposite of (A ⇒ B) is ¬(A ⇒ B), which, using De Morgan's Law, resolves to (A ∧ ¬B), which is not at all the same thing as (¬A ∨ ¬B), which is what A ⇒ ¬B reduces to.
This confusion about the "compatibility" of these two conditionals was foreseen by Carroll, who includes a mention of it at the end of the story. He attempts to clarify the issue by arguing that the protasis and apodosis of the implication "If Carr is in ..." are "incorrectly divided". However, application of the Law of Implication removes the "If ..." entirely (reducing to disjunctions), so no protasis and apodosis exist and no counter-argument is needed.
In logic, a logical connective is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective can be used to join the two atomic formulas and , rendering the complex formula .
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.
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites".
In mathematical logic, Russell's paradox is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, as he told Hilbert and Richard Dedekind by letter.
The barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell as an illustration of the paradox, though he attributes it to an unnamed person who suggested it to him. The puzzle shows that an apparently plausible scenario is logically impossible. Specifically, it describes a barber who is defined such that he both shaves himself and does not shave himself, which implies that no such barber exists.
Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F". The paradox requires only a few apparently innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic.
"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal Mind, is a brief allegorical dialogue on the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race. In Carroll's dialogue, the tortoise challenges Achilles to use the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails, because the clever tortoise leads him into an infinite regression.
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.
A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic, which reject the principle of explosion.
Conditional sentences are natural language sentences that express that one thing is contingent on something else, e.g. "If it rains, the picnic will be cancelled." They are so called because the impact of the main clause of the sentence is conditional on the dependent clause. A full conditional thus contains two clauses: a dependent clause called the antecedent, which expresses the condition, and a main clause called the consequent expressing the result.
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.
An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause. In some contexts the antecedent is called the protasis.
The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally they are taken as laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc. However, such classical ideas are often questioned or rejected in more recent developments, such as intuitionistic logic, dialetheism and fuzzy logic.
The drinker paradox is a theorem of classical predicate logic that can be stated as "There is someone in the pub such that, if he or she is drinking, then everyone in the pub is drinking." It was popularised by the mathematical logician Raymond Smullyan, who called it the "drinking principle" in his 1978 book What Is the Name of this Book?
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
Connexive logic is a class of 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,
