I have discussed a logical system called “propositional logic.” I will now discuss predicate logic, a system that is a bit more complex than propositional logic because it introduces predicates, quantifiers, constants, variables, and the universe of discourse. It also uses the elements of propositional logic (propositional letters and connectives). I recommend you learn about propositional logic before learning about predicate logic.
Predicate logic is a deductive symbolic logical system that allows us to determine valid reasoning and consistency between propositions. Unlike propositional logic, it allows us to capture the form of propositions, which allows us to correctly determine that certain arguments are valid that propositional logic would fail to determine to be valid. For example, consider the following argument:
- All dogs are mammals.
- All mammals are animals.
- All dogs are animals.
Propositional logic would symbolize this argument as the following:
According to propositional logic, the argument is invalid. However, predicate logic would find the argument to be valid, and would symbolize it in the following way:
(∀x)(Fx → Gx)
(∀x)(Gx → Hx)
(∀x)(Fx → Gx)
In order to understand predicate logic, the first step is to know a little about the different elements of symbolization.
Predicate letters and connectives
Any proposition can be represented with a capital letter between A and O.
Logical connectives are relations between propositions and the negation of a proposition.
|¬A||Negation||It is not the case that A.|
|A ∧ B||Conjunction||Both A and B.|
|A ∨ B||Disjunction||Either A or B.|
|A → B||Conditional||If A, then B.|
|A ↔ B||Biconditional||A if and only if B.|
Lower case letters between a and v represent constants, which are specific individuals, which can be identified by name (such as George Washington) or a definite description (such as the first president of the United States). Specific numbers can also be used, such as 1, 2, or 3.
We can define constants using a scheme of abbreviation. For example:
g: George Washington
Upper case letters between A and O represent predicates, which are properties and relations, such as “is a dog” or “…loves…”
Predicates and constants can be used to construct propositions. For example, “George Washington is the first President of the United States can be symbolized as “Fg.”
The scheme of abbreviation would be the following:
Fx: x is the First President of the United States.
g: George Washington.
Fx is a one place predicate because it is only about one thing. However there can be predicates that are about two, three, or more things. For example, consider the sentence, “Aaron Burr shot Alexander Hamilton.” That proposition can be symbolized as “Gbh.”
The scheme of abbreviation is the following:
Gxy: x shot y.
b: Aaron Burr
h: Alexander Hamilton
An example of a three place predicate is between. Consider the sentence “Nevada is between California and Utah.” We can symbolize this proposition with “Bncu.”
The scheme of abbreviation is the following:
Bxyz: x is between y and z.
x, y, and z are variables. They range over various possible constants without stating which constant they are about. For example, “Fx” is the predicate used above for “is the first president of the United States.” What exactly does the “x” stand for? Anything we want. We can use this predicate say Barak Obama is the first president of the United States and symbolize the proposition as “Fb.” In this case “b” stands for “Barak Obama.”
Note that the formula “Fx” is not true or false because it doesn’t say who or what “x” is. It is a variable. “Fx” is not actually a formula that we would use in an argument in isolation because it’s not true or false. Finally, note that variables in schemes of abbreviation stand for any variable—they are metavariables. So, “Fx” and “Fy” say exactly the same thing in a scheme of abbreviation.
There are two quantifiers: The universal quantifier (∀x), which is about every x, and existential quantifier (∃x), which states that there is an x. Both of the quantifiers require the use of a predicate and range over that predicate. For example, if we use the above scheme of abbreviation, “(∀x)Fx” says that “everything is the first president of the United States,” and “(∃x)Fx” says “there is a first president of the United States.”
Main types of propositions using quantifiers are the following:
|Symbolization||Idiomatic English||More precise meaning|
|(∀x)Fx||Everything is an F.||For all x, it is an F.|
|(∃x)Fx||Something is an F.||There is an x that is an F.|
|(∀x)(Fx → Gx)||All F are G.||For all x, if it is an F, then it is a G.|
|(∃x)(Fx ∧ Gx)||Some F are G.||There is an x that is an F and a G.|
|(∀x)(Fx → ¬Gx)||No F are G.||For all x, if it is an F, then it is not a G.|
|(∃x)(Fx ∧ ¬Gx)||Some F are not G||There is an x that is an F and not a G.|
|¬(∀x)Fx||Some things are not F.||It’s not the case that all x are F.|
|¬(∃x)Fx||Nothing is an F.||It’s not the case that an x is an F.|
Note that each quantifier refers to a variable, there must be a predicate that is bound by that variable, and there must not be any variable that is unbound. In the proposition “(∀y)Fy,” “y” is the variable, and “Fy” is bound by the quantifier. In “(∀z)(Fz → Gz),” Fz and Gz are both bound by the quantifier. A variable is bound by a quantifier if either of either of these criteria are met:
- If there is a single predicate next to the quantifier, and the predicate has the same variable as the quantifier. For example, “(∀y)Fy.”
- If there is a predicate within parentheses next to the quantifier, and the predicate has the same variable as the quantifier. For example, “(∀z)(Fz → Gz).”
“(∀y)Fx” is not a proposition because “Fx” has to be bound by “(∀y),” but it has the wrong variable. Similarly, “(∀x)Fa” is not a proposition in predicate logic because “a” is a constant, but Fa is not supposed to be bound by a quantifier at all.
Also, “(∃x)Fx ∧ Gx” actually fails to be a proposition of predicate logic because Gx must be bound by a quantifier, but it’s not bound by a quantifier. This symbolization is ambiguous. Two possible propositions it could refer to are the following:
|(∃x)(Fx ∧ Gx)||Some F is a G||There is an x that is an F and a G.|
|(∃x)Fx ∧ (∃x)Gx||Something is an F and something is a G.||There is an x that is an F, and there is an x that is a G.|
Finally, it is important that each variable of the predicates are only bound by one quantifier each. “(∀x)(∃x)(Fx ∧ Gx)” is not a proposition because it’s not clear which quantifier is supposed to bind each variable.
Universe of discourse
One element of schemes of abbreviation is the universe of discourse, which is the range over what variables can be about. For example, consider the following symbolization and scheme of abbreviation:
UD: The set of all humans.
Mx: x is a mammal.
This proposition says “All humans are mammals” because the universe of discourse is restricted to humans. If the universe of discourse was “the set of all crows,” then the proposition would say, “All crows are mammals.”
Learning more about predicate logic
Fully understanding predicate logic requires that we learn to translate ordinary language into predicate logic, interpret predicate logic into ordinary language, and use natural deduction to validly derive conclusions from premises. I plan on writing about how to do each of these things.