Ethical Realism

May 10, 2015

Predicate Logic 2: Translation

Filed under: philosophy — JW Gray @ 1:44 am

This is part 2. You should see part 1 before reading this. This is also written with the assumption that you already know propositional logic. Translation is the conversion of natural language into the formal language of some type of logical system. Every statement in predicate logic is either symbolized as a single letter (just like propositional logic), or it requires (1) a predicate letter, and (2) an individual.

1. The most simple type of predicate sentence

The most simple statement of predicate logic (that actually uses a predicate) is the following:

Fa

Which means: a is an F. For example, the following sentence can be symbolized as “Fa”: Barak Obama is the President of the USA in 2015. The scheme of abbreviation would be: Fx: x is the president of the USA in 2015. a: Barak Obama. In this case “F” is a predicate of something (x is a variable), and “a” is a constant (an individual.) Other examples of similarly simple statements of predicate logic are the following:

  1. Lassie is a dog. (Symbolization: “Gb.” “Gx” stands for “x is a dog” and “b” stands for “Lassie.”
  2. Two is odd. (Symbolization: “Hc.” “Hx” stands for “x is odd” and “c” stands for “Two.”)
  3. The Earth goes around the Sun. (Symbolization: “Se.” “Sx” is “x goes around the Sun” and “e” is “the Earth.”)

Note that any of the upper case letters between “F” and “Z” are predicates, they refer to something (a variable such as “x” or “y” or “z”; or a constant, such as any lower case letter between “a” and “w”). Any of these letters can be used based on your personal preference, but a letter can’t refer to two different things in a single scheme of abbreviation. For example, you can’t say that “Fx” could either mean “x is the president of the USA in 2015” or “is odd” for a single scheme of abbreviation. It is also possible to combine simple sentences of predicate logic using logical connectives. For example, the statement “Lassie is a dog, and two is odd” could be symbolized as “Gb ∧ Hc.” In this case the letters represent the same predicates and constants as sentence 1 and 2 above.

2. Simple Quantified sentences

The slightly more complex types of statements for predicate logic are of two main types:

  1. Everything is…
  2. There is at least one…

These types of natural language statements correspond to two main types of statements of predicate logic, and each of them has a (1) quantifier, (2) predicate, and (3) variable, as can be seen on the following table:

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)” and “(∃x)” are the only two quantifiers. “(∀x)” represents “For each x…” and “(∃x)” represents “there is at least one x…” Consider the following three examples of ordinary statements translated into similar sentences of predicate logic: Everything is a dog.

(∀x)Fx UD: The set of everything. Fx: x is a dog.

There is at least one human.

(∃y)Gy UD: The set of all living things. Gx: x is a human.

All mammals are animals.

(∀z)Hz UD: The set of all mammals. Hx: x is an animal.

In each scheme of abbreviation, a universe of discourse is given (the UD), which limits what all the variables refer to (but can also be open to assure us that the variables refer to everything). This is mostly only important for the third sentence given (all mammals are animals). Without the universe of discourse, that sentence in predicate logic would say, “everything is an animal.” Also note that the scheme of abbreviation doesn’t have to use the same variable. For example, “Hx” is the same predicate as “Hz,” and the variables in each of these cases refer to anything from the universe of discourse. The fact that the variables are different letters make no difference in these examples. Even so, using different variables can also be important in at least some situations, and that will be discussed further later.

3. Negated quantified statements

There are two main types of negated sentences used with simple quantified statements:

  1. Some… are not…” or “Not all…” (“¬(∀x)Fx” or “(∃x)¬Fx.”)
  2. Nothing is… or “All… are not…” (“¬(∃x)Fx” or “(∀x)¬Fx.”)

There are four different simple types of negated quantified statements in predicate logic that correspond to these two types of sentences (which can be phrased in two different ways each). In other words, there are two different ways to translate each of these types of sentences from natural language, which are presented on this table:

Symbolization Idiomatic English More precise meaning
¬(∀x)Fx Not all things are 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.
(∀x)¬Fx All x are non-F. All x are non-F.
(∃x)¬Fx Some x are not F. There is at least one x that is not an F.

Here are some examples of how to translate two natural language negated sentences: Some mammals are not dogs.

¬(∀x)Tx Another option: (∃x)¬Tx UD: Set of all mammals. Tx: x is a dog.

There are no unicorns.

¬(∃x)Ux Another option: (∀x)¬Ux UD: Set of all living things. Ux: x is a unicorn.

4. Aristotelian style-statements

There are four types of Aristotelian style-statements (that he focused on when developing his own type of formal logic:

  1. All A are B.
  2. Some A are B.
  3. No A are B.
  4. Not all A are B.

These four types of statements can be translated into satements of predicate logic in a way shown on the following table:

Symbolization Idiomatic English More precise meaning
(∀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.

Here are some examples of translating various sentences from natural language into predicate logic: Every dogs is an animal.

(∀x)(Fx → Gx) UD: Set of all living things. Fx: x is a dog. Gx: x is an animal.

Some numbers are odd.

(∃x)(Hx ∧ Ix) UD: Set of all things. Hx: x is a number. I: x is odd.

No positive numbers are negative.

(∀x)(Jx → ¬Kx) UD: Set of all things. Jx: x is a number. Kx: x is negative.

Some dogs are not reptiles.

(∃x)(Lx ∧ ¬Mx) UD: Set of all living things. Lx: x is a dog. Mx: x is a reptile.

5. Two place predicates

So far the only predicates mentioned refer to one individual, but it is possible for a predicate to refer to two, three, or more individuals. A couple examples of statements translated into statements of predicate logic with two or more individuals for each predicate. Sam is taller than Martha.

Tsm Txy: x is taller than y. s: Sam m: Martha

Nevada is between California and Utah.

Fcnu Fxyz: y is between x and z. c: California n: Nevada u: Utah

I will focus on two-placed predicates, and it is important to note that two-placed predicates that have two different variables will also require two different quantifiers. Some main types of statements using two-placed predicates are the following:

Symbolization Idiomatic English More precise meaning
(∀x)(∀y)Fxy All x Fs all y. For each x and each y, x Fs y.
(∀x)(∃y)Fxy All x Fs some y. For each x there is some y, and x Fs y.
(∃x)(∀y)Fxy Some x Fs all y. There is an x such that for each y, x Fs y.
(∃x)(∃y)Fxy Some x Fs some y. For some x and some y, x Fs y.

Here are some examples of English statements translated into these types of predicate statements: Everyone loves everyone.

(∀x)(∀y)Lxy UD: Set of all people. Lxy: x loves y. Another option: (∀x)(∀y)((Px ∧ Py) → Lxy) UD: Set of everything. Px: x is a person. Lxy: x loves y.

Everyone loves themselves.

(∀x)Lxx UD: Set of all people. Lxy: x loves y. Another option: (∀x)(Px → Lxx) UD: Set of everything. Px: x is a person. Lxy: x loves y.

Some people love themselves.

(∃x)Lxx UD: Set of all people. Lxy: x loves y.

Everyone loves someone.

(∀x)(∃y)Lxy UD: Set of all people. Lxy: x loves y.

Someone loves everyone.

(∃x)(∀y)Lxy UD: Set of all people. Lxy: x loves y.

5. More examples

Here are some more examples of how to translate statements of natural language into predicate logic: Every mammal except monotremes give birth to live young.

Paraphrase: For each animal, if it is a mammal and not a monotreme, then it gives birth to live young. (∀x)((Mx ¬Nx) → Lx) UD: Set of all animals. Mx: x is a mammal. Nx: x is a monotreme. Lx: x gives birth to live young.

All lawyers can practice law, but those who aren’t, can’t.

Paraphrase: Each person who is a lawyer can practice law, and each person who isn’t a lawyer can’t practice law. (∀x)(Lx → Px) (∀x)(¬Lx → ¬Px) UD: Set of all people. Lx: x is a lawyer. Px: x can practice law.

The sum of 4 and a prime number greater than 4 is odd.

Paraphrase: There are two numbers such that if (the first number is the total of the second number added to four, the second number is prime, and the second number is greater than four), then the first number is odd. (∀x)(∀y)((Sxya ∧ (Py ∧ Gya) → Ox) UD: Set of all numbers. Sxyz: The sum of x and y is z. Px: x is prime. Gxy: x is greater than y. O: x is odd. a: 4.

A country between Spain and Germany is in Europe.

Paraphrase: There is a country between Spain and Germay, and it is in Europe. (∃x)(Nsxg ∧ Ex) UD: Set of all countries. Nxyz: y is between x and z. Ex: x is in Europe. s: Spain g: Germany

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