In my last post, "My Unwarranted Notes of Philosophy," I attempted to define a lot of things that are necessary for any philosophical discourse. I wrote about a priori and a posteriori propositions. You may easily remember that a priori propositions are either definitions or deductions, but a posteriori propositions come with the need for observational evidence to justify them.
For example, the proposition "420 is an even number" is a synthetic a priori proposition, and if you know the definitions of "420" and "even number," deducing the truth of the proposition is not very difficult. Now consider "All ravens are black." The claim is clearly a posteriori and relatively difficult to justify.
In terms of the properties of being raven and being black colored. Four possible observations are there:
1. Observing a black raven is a direct evidence to confirm "There exists one black raven." But it is only a supporting evidence or example (not an evidential proof) for "All ravens are black".
2. Observing a non-black raven is a direct contradictory evidence (counter-example) disproving "All ravens are black".
3. Observing a black non-raven entity is an irrelevant evidence since it neither confirm nor contradict "All ravens are black".
4. But observing non-black no-raven entities seems to be indirectly supportive of "All ravens are black." Since via contrapositive reasoning (rule of contraposition) "All ravens are black" is equivalent to "All non-black entities are non-ravens."
To understand how contraposition works, consider that entities are either black or non-black. If all ravens are black then all ravens belong to the set of all black things. So all non-black entities can not be ravens as there are no non-black ravens. This is how contraposition works.
But the observational justification of the contrapositive proposition is intuitively irrelevant. This is the raven paradox. It asks how can intuitively irrelevant supportive evidence (like a red robbin which is a non-black non-raven) be a logically valid supportive evidence?
If "All ravens are black" is true, then no contradictory evidence (counter-example) can be found.
If "All ravens are black" is false, then contradictory evidence (counter-example) can be found.
So instead of searching for evidence or example, one must search for contradictory evidence or counter-example. The process of searching only for evidence is of limited use since whenever counter-evidence is found, the evidence found no longer holds useful. One can call it as a falsification approach to this paradox.
The paradox is only there for "All ravens are black" and similar propositions like "there are no green stars". Such propositions describe some rule or universality about the property of something. The paradox is not there for propositions describing particular cases like "This raven is black" or "Some monkeys have red face." Hence there is no need for falsification for particular cases. Searching for direct evidence suffices for particular cases.
But actually, it turns out that falsification is not enough to solve this paradox since the paradox actually arises from the introduction of contrapositive. But before we resolve it, we must discuss first-order logic.
In the classical first-order theory of logic (also called predicate logic), a proposition is resolved into subject and predicate. The subject is the description of a thing and the predicate is the description of that thing's properties. A predicate works like a mathematical function that takes in things as variables and gives the truth value of the proposition as the output of the function on a given quantifier. A quantifier is the description of for how many things the given predicate is true or false.
If a predicate P is true for variable (thing) x then the proposition is written as P(x) which means that x has property defined in the predicate P.
Consider "All ravens are black" and "Some ravens are black". While it is easy to identify "ravens" as the subject and "are black" as the predicate in the above sentences, the difference lies in the quantification. Quantification is of two kinds: universal-when we talk about all counts of a thing and existential-when we talk about one or some count of a thing. Formally the above sentence are stated as "For all ravens, they are black" and "There exists ravens that are black". Well it is a wierd way of writing but it is formal atleast in first order logic.
Universal quantifier - “for all” (∀) - True when predicate is true for all variables. Here ∧ represents logical AND.
∀x(P(x)) = P(x1) ∧ P(x2) ∧ P(x1) ∧ ...
Existential quantifier - “there exists” (∃) - True when predicate is true for at least one variable. Here ∨ represents logical OR.
∃x(P(x)) = P(x1) ∨ P(x2) ∨ P(x1) ∨ ...
From De Morgan’s Laws:
Here ¬ is logical NOT.
¬∃x(P(x)) = ∀x(¬P(x))
¬∀x(P(x)) = ∃x(¬P(x))
A domain is the set of all variables for which the predicate is defined. A domain is necessary to determine the truth value of any predicate, and a predicate is not defined for variables outside its domain.
To highlight how the rules of inference in propositional logic does not make sense in the different domains consider the following three implications:
1. If Paris is in France then french people are from France (true implies true).
2. If Paris is in England then english people are from France (false implies false).
3. If Paris is in England then french people are from France (false implies true).
The problem here is that the implications are irrelevant despite being logically valid. There is no connection between the antecedent (the proposition after if and before then) and the consequent (the proposition after then) in any of the above sentences. Paris being in any country does not prove anything about french or english people. The properties of the thing "Paris" does not say anything about the properties of french people. The reason they are irrelevant is that the implication in propositional logic only deals with the truth value of a complete proposition not the things described in those propositions. Below is a table of some important first-order rules of inference and fallacies:
| Rule / Fallacy | Logical Formula | Example | Counter-Example (if applicable) |
|---|---|---|---|
| Existential Instantiation (Fallacy) | ∃x (P(x)) → x exists | There is a unicorn that is magical, so a unicorn must exist. | Unicorns don't exist, despite the statement about magical unicorns. |
| Existential Generalisation | P(x) → ∃x (P(x)) | Socrates is a philosopher, so there exists a philosopher. | |
| Universal Instantiation | ∀x (P(x)) → P(c) for some specific c | All humans are mortal, so Socrates (a human) is mortal. | |
| Universal Generalisation (Fallacy) | P(c) → ∀x (P(x)) | Socrates was a human, so all beings are humans. | Not all beings (e.g., animals) are humans. |
The contrapositive of "All ravens are black" can also be falsified if a non-black raven is found but contraposition changes the domain of variables. A different domain means that the existence of non-black non-ravens does not imply the existence of black ravens and for that matter even ravens. This is an example of existential fallacy.
The big takeway is that the source of paradox is the error of neglecting the domain of subjects in a proposition. Non black entities define a different domain compared to the domain defined by ravens.
In first-order logic, the laws of propositional logic (like contraposition) must consider the domain over which quantifiers are applied. Evidence for contrapositives are logically disconnected from evidence for the original statement as the domains differ.