Ethical Realism

July 2, 2015

Predicate Logic 3: Interpretation

Filed under: philosophy — JW Gray @ 5:53 am
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This is part 3. You should see part 1 and part 2 before reading this. This is also written with the assumption that you already know propositional logic.

Interpretation is the conversion of a sentence in a formal language of a logical system into a natural language, which is primary done by providing a scheme of abbreviation for a symbolic sentence. The purpose of translation is generally to prove a sentence to be indeterminate, an argument to be invalid, a set of sentences to be logically consistent, or a set of sentences to be non-equivalent.

1. How to develop an interpretation

The main way of providing an interpretation of a sentence of predicate logic is to provide a scheme of abbreviation. You can then use that scheme of abbreviation to form a sentence in a natural language (such as English).

Here are some examples:

Example 1

(∀x)(Fx → Gx)

Scheme of abbreviation:

UD: The set of all living things.

Fx: x is a lizard.

Gx: x is a reptile.

In English: All lizards are reptiles.

Example 2

(∀x)(∃y)Hxy

Scheme of abbreviation:

UD: The set of all Humans.

Hxy: x is child of y

In English: All humans are the children of at least one human.

Example 3

(∀x)((Ix ∨ Nx) → Jx)

Scheme of abbreviation:

UD: The set of all animals.

Ix: x is a dog.

Nx: x is a cat.

Jx: x is a mammal.

In English: All dogs and cats are mammals. (Or: All animals that are either dogs or cats are mammals.)

Example 4

(∀x)(Kx → (∃y)Lyx))

Scheme of abbreviation:

UD: The set of all humans.

Kx: x has a living parent.

Lxy: x is the child of y.

In English: Everyone with a living parent is the child of someone.

2. Truth, falsity, and indeterminacy

It is possible to interpret many sentences to be true, and others to be false. Some could be interpreted either way, and are therefore indeterminate (i.e. contingent). Here are some examples:

True

(∃x)(Mx ∨ ¬Mx)

Scheme of abbreviation:

UD: The set of all things.

Mx: x is a mammal.

In English: There is someone who is either a mammal or isn’t a mammal.

Note that it is not possible to interpret this sentence as being false because it is a tautology (i.e. logical truth), which is to say that it is true under every possible interpretation. Even so, you can’t prove a sentence to be a tautology by using interpretations because the best we can do is give several interpretations of a sentence as being true, but there might be an interpretation we missed that would interpret the sentence as being be false. Tautological sentences are true under every interpretation, and there are infinite interpretations.

False

(∃x)(Nx)

Scheme of abbreviation:

UD: The set of all humans.

Nx: x is a dog.

English: There is a human that is a dog.

Indeterminate

(∀x)(Ox → ¬Px)

Here are two schemes of abbreviation:

UD: Set of all humans.

Ox: x is a doctor.

Px: x is a dog.

In English: No doctors are dogs.

UD: Set of all animals.

Ox: x is a dog.

Px: x is a mammal.

In English: No dogs are mammals.

These two interpretations show that the sentence of predicate logic could be interpreted as being true or false, so it is indeterminate. Indeterminate sentences are not tautologous, and they are not self-contradictory.

3. Logically invalid arguments

In order to prove arguments to be logically invalid by using interpretations, every premise of an argument must be true, and the conclusion must be false while using a single scheme of abbreviation. Valid arguments can’t possibly have true premises and a false conclusion while using a single scheme of abbreviation.

An example of a logically valid argument is “All humans are mortal. There is a human. Therefore, there is a mortal.” Premises using this logical form can’t be true with a conclusion that’s false at the same time.

Consider the following argument:

All dogs are mammals.

There is a mammal.

Therefore, there is a dog.

In predicate logic, the argument form is the following:

(∀x)(Fx → Gx)

(∃x)(Gx)

∴ (∃x)(Fx)

We can then give the sentences of predicate logic the following scheme of abbreviation:

UD: The set of all animals.

Fx: x is a dog on the Moon.

Gx: x is a mammal.

In English:

All dogs on the Moon are mammals.

There is a mammal.

Therefore, there is a dog on the Moon.

The premises are all true, but the conclusion is false. Only invalid arguments can have true premises and a false conclusion. Therefore, the argument is invalid.

Note you can’t prove an argument to be valid using interpretations because that would require infinite interpretations. When we try to prove an argument to be valid using interpretations, the best we can do is provide several interpretations using true premises and a true conclusion. Such interpretations would not prove that it’s impossible to find an interpretation with true premises and a false conclusion.

4. Logically consistent sentences

Logically consistent sentences are non-contradictory, which means they can all be true at the same time. We can prove that by interpreting all the sentences as being true by using a single scheme of abbreviation.

An example of logically inconsistent (i.e. contradictory) set of sentences is “all dogs are mammals” and “some dogs are not mammals.”

Consider the following two sentences:

Nevada is between California and Utah.

California is between Nevada and Utah.

In propositional logic, the sentences have following logical form:

Aabc

Abca

The scheme of abbreviation can then be the following:

UD: The set of all numbers.

Axyz: The sum of x and y is z.

In English:

The sum of 1 and 2 is 3.

The sum of 2 and 1 is 3.

In Mathematics:

1 + 2 = 3

2 + 1 = 3

Both of these sentences are true, so the two sentences of propositional logic are consistent.

Note that you can’t prove sentences to be logically inconsistent using interpretations because there are infinite different possible interpretations that could prove a set of sentences to be inconsistent. The best you could do is offer several interpretations where one sentence is true and the other is false, but that would not prove the sentences to be inconsistent.

5. Non-equivalent sentences

A set of logically non-equivalent sentences can have different truth values, and equivalent ones always have the same truth value given a single scheme of abbreviation. We can prove a set of sentences are non-equivalent by interpreting at least one of them to be true and at least one of them to be false.

An example of a logically equivalent set of sentences is “All unicorns are mammals” and “It’s not the case that some unicorns are non-mammals.”

Consider the following set of sentences:

All numbers are are positive.

Some numbers are fractions.

The sentences have the following logical form in predicate logic:

(∀x)(Fx → Gx)

(∃x)(Fx ∧ Hx)

Consider the following scheme of abbreviation for those sentences of predicate logic:

UD: The set of all animals.

Fx: x is a dog.

Gx: x is a mammal.

Hx: x is a reptile.

In English:

All dogs are mammals.

Some dogs are reptiles.

The first sentence is true, but the second sentence is false. So, these sentences aren’t logically equivalent because they can have different truth values given a single scheme of abbreviation.

Note that we can’t prove sentences to be equivalent by giving interpretations because there are infinite different possible interpretations, and equivalent sentences have the same truth value in every possible interpretation. The best we can do is give several interpretations where the sentences have the same truth value, but that would not prove they always have the same truth value.

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