Logical Paradoxes: Definitions and Explanations
A paradox is an inconsistent conclusion drawn from sound reasoning. Logic tries to determine good rules for reasoning. Here are 3 possible logical paradoxes
A paradox happens when one is lead to contradictory, inconsistent, or otherwise impossible conclusions by sound reasoning. In philosophy, paradoxes can be categorized by the sort of impossibility they result in. Logical paradoxes result in logically inconsistent conclusions. Other sorts of inconsistency might be metaphysical, epistemic, or rational in nature. Logical paradoxes can occur when thinking about the nature of mathematics.
This is a list of some possible logical paradoxes with definitions and explanations. However, no explanations of attempted solutions are given. The list isn't meant to be complete – other logical paradoxes might be found. Also, the definitions are meant to be introductory. Further thinking about the paradox may lead to defining it in a more refined way. Embedded links lead to further reading.
What is a Logical Paradox?
In a logical paradox one is lead to a contradictory or inconsistent conclusion by apparently sound reasoning about logic. Logic is concerned with identifying good rules of inference. Good rules of inference are ones which preserve truth: they shouldn't allow us to reason from true premises to a false conclusion. Since that's the goal (or at least a central goal) of logic, we should be very surprised to find logical paradoxes.
In fact, most logical paradoxes arise from reasoning not just about logic, but also set theory. Set theory aims at rules for collecting things into groups. For example, the axiom of extensionality says that two sets with the exact same members are really the same set. There's a number of different set theories depending on what rules they employ. Some mathematicians and philosophers believe that numbers can be defined as abstract sets. This leads to Russell's paradox.
The first paradox listed, known as the liar's paradox, isn't a logical paradox at all, even though it might be mistaken for one. The last paradox listed, Skolem's paradox, comes closest to being a truly logical paradox. It arises when we think about two particular proofs in logic and mathematics: Cantor's theorem about infinite sets and the Löwenheim-Skolem theorem.
Liar's Paradox
Here's an example of a liar's sentence: "This sentence is false." If the sentence is true, then it must be false. If it is false, then it must be true. This is a paradox of meaning rather than logic. Logic is concerned with laws of inference and the liar's sentence by itself infers nothing. Rather, the paradox arises when we try to determine whether the sentence is true. The truth value of a sentence is, perhaps, the most basic aspect of its literal meaning.
In Werner Herzog's film "The Enigma of Kasper Hauser", the idiot-savant Kasper is tested with a version of the liar's paradox by a philosopher. When challenged as to how he could determine whether the man giving him directions is from the liar's village (all of whom lie all the time), Kasper says he would ask the man if he is a tree frog. Presumably the liar would have to say yes. This is rejected as the wrong answer by the frustrated philosopher.
Russell's Paradox
Understanding Russell's paradox requires a basic understanding of sets. A set is just a collection of objects. Any property might denote a set of things – this is often included as a rule of set theory. For example, the property 'red' can denote the set of all the red things. Anything which is red is a member of this set. Since sets themselves are things (abstract things to be sure), we can also have sets of sets.
Now consider the property "not a member of itself". This is a property which sets can have. The set of red things isn't itself red. Therefore the set of red things belongs to the set of sets which are not members of themselves. Consider the set of even numbers between 1 and 9. This set has 4 members: {2, 4, 6, 8}. Since it is a set of 4, and 4 is an even number between 1 and 9, this set is a member of itself (note that this example presumes that sets are used to define numbers).
Now consider the set of all sets which are not members of themselves. Is this set a member of itself or not? If it is, then it has the property "not a member of itself" so it can't be a member of itself. If it is not, then it doesn't have this property, which means it must be a member of itself.
Bertrand Russell discovered this paradox while working on the philosophical foundations of mathematics. The paradox posed significant problems for Gottlob Frege, as well as for his own project to reduce mathematics to logic.
Skolem's Paradox
Skolem's paradox might come closest to being a truly logical paradox. It arises from two theorems of logic: Cantor's theorem which distinguishes between sets that are countably infinite and sets that are uncountably infinite and the Löwenheim-Skolem theorem which proved that if a sentence of first order predicate logic has a model for interpretation, then it has a model whose domain is countable. Combining these two results in a seeming paradox: predicate logic can be used to prove the existence of uncountable sets, but the formulas used for such a proof can be modeled with a countable set.
Cantor developed a new technique for comparing the size (cardinality) of sets. Two sets have the same cardinality if their members can be put in one-one correspondence with each other. This lead him to the surprising conclusion that there are two kinds of infinity: countable and uncountable. Sets that are countably infinite can be put in one-one correspondence with the natural numbers. Examples include the set of even numbers and the set of rational numbers. Uncountably infinite sets can not be put in a one-one correspondence with the set of natural numbers. Examples include the set of real numbers or the set of all subsets of natural numbers. The proof for Cantor's theorem relies on his axioms of set theory and these axioms can be represented in predicate logic.
The Löwenheim-Skolem theorem is about models used for interpreting formulas in predicate logic. A model is a domain of discourse and an interpretation function which assigns predicates to the members of that domain. This theorem says that any theory which has an infinite model also has a model which is countably infinite.
Here is a basic example of a single formula in predicate logic: ∀x(Px⊃Cx). As a quick example, let's construct a model which allows us to interpret this sentence as "all philosophers are crazy".
The basic elements of a formula are quantifiers, connectives, variables, singular terms and predicates. This formula uses all these elements except singular terms, which name individuals. Rather, it uses the variable x to range over individuals. ∀ is the universal quantifier so ∀x means all the individuals. ⊃ is a connective denoting the material conditional and might be read as 'if ... then ..." in natural English. Finally, the uppercase letters are predicates. In this case Px is read as "x is a philosophers" and Cx is read as "x is crazy". Putting this altogether, our formula can be read in English as "for each person, if that person is a philosopher then that person is crazy" which is just to say that all philosophers are crazy.
We start with a set of individuals in our universe of discourse. In other words, we define all the things that we are talking about. In this case, lets say that we are talking about all the people in the world. Each of these people would be denoted by a lower case letter, except x which we are using as a variable, in a formula.
Next, we define an interpretation function which assigns predicates to the members of the domain of discourse. This function says, of every person, whether that person is a philosopher and whether that person is crazy.
This sentence is true if every member of the domain who is assigned the predicate "is a philosopher" is also assigned the predicate "is crazy". Our formula is true just in case the interpretation function assigns both predicates whenever it assigns one of them.
We're now ready to see what appears to be a paradox. Cantor's theorem used axioms of set theory to prove that there are uncountable infinities. The axioms of set theory can be represented as formulas in first order predicate logic and we can use predicate logic to prove his theorem. If there's a model for these formulas, the Löwenheim-Skolem theorem tells us that there is a countable model. How can a countable model be used to prove the existence of an uncountable infinity?
Skolem's paradox doesn't present a significant challenge to working mathematicians. Some philosophers however, believe that the paradox poses problems for determining the foundations of mathematics. Hilary Putnum has even employed a version of the paradox in an argument against realism. He believes that the paradox shows that there can be no fact of the matter about what terms (such as proper names) and predicates (such as properties like 'red') refer to.
A Paradox is Different from Circular Reasoning
Paradoxes are not the same as circular reasoning. Justifying logical rules of inference is, perhaps, necessarily circular. For example, one rule of logical inference is known as "AND introduction". It says that if you can assert the proposition "P" and you can assert the proposition "Q", then you can assert the proposition "P AND Q". Any proof that this is a good rule of inference is likely going to use the rule itself. That the justification is circular in this way does not mean that it is a bad rule of inference; we need to start with something. You might say that rules of inference are self-justifying once we grasp them.
Both Russell's paradox and Skolem's paradox share a common trait: they seem to pose problems for the idea that mathematics can be reduced to logic and set theory. Neither of them arise from logical reasoning alone. Rather, both lead to apparently contradictory conclusions through reasoning about logic and set theory.
Sources
Bays, Timothy. "Skolem's Paradox". Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/paradox-skolem/
Bolander, Thomas. "Self Reference". Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/self-reference/#Para-sema
Irvine, A.D.. "Russell's Paradox". Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/russell-paradox/
Stoll, Robert. Set Theory and Logic New York, NY: Dover