Philosophy 201: Reasoning

Contents:

Introduction
Section 1: Sorts of reasoning
Section 2: Means of reasoning (sorts of argument)
Section 3: Disvalues of reasoning
Section 4: Ends (values) of reasoning
Section 5: The three basic Laws of Reasoning
Section 6: An analysis of the Principle of Bivalence
Section 7: Argument as the reverse of explanation
Appendix: The Three Groups in this topic

Introduction

In what follows, my use of G1, G2 and G3 refers to my thesis that most philosophical concepts fall naturally into three groups. (See Philosophy 103 and 103a.)

A unit of reasoning is called an argument. An argument consists of a premise (a proposition), or set of premises, followed by a conclusion. The conclusion is supposed to be true if the premises are true. The step from premise(s) to conclusion is termed 'inference', and is usually indicated by an expression such as 'therefore', or 'so', or 'ergo', or 'it follows that'.

Being a piece of language, an argument has content, form and context (see Philosophy 200).

Section 1. Sorts of reasoning

A very general way of sorting arguments is according to what is predicted or prescribed by the conclusion.

G1. Observational

Observational reasoning predicts a possible observation - what will be observable in the circumstances it describes.

G2. Theoretic

Theoretic reasoning prescribes a belief or a candidate for belief.

G3. Practical

Practical reasoning prescribes either (1) an action or omission of action to cause or avoid a non-moral effect, or, (2) either an action or the omission of an action in order to behave morally correctly.

Section 2. Means of reasoning (sorts of arguments)

There are three main sorts of argument: analogy, deduction, and induction.

G1. Analogy

Analogy is reasoning from premises about particular things that are members of a group (because they have things in common) to a conclusion about one of them.

Here's an example:

The Arctic is a cold region
The Antarctic is a cold region
The Arctic is inhabited by polar bears
Therefore, the Antarctic is inhabited by polar bears.

The form of this argument can be represented as:

A is x
B is x
A is y
___
B is y.

The basic idea of analogy is that two things that are alike in one way (or several ways) might be alike in yet another way.

Analogy is very prone to false conclusions, even if the premises are true. Our example has true premises but science has found no evidence that its conclusion is true. Nevertheless, science does use analogy as a means of making discoveries, because sometimes it does yield a true conclusion.

An analogy is strong (but not necessarily true) when the two things it compares are similar in lots of different ways.

G2. Deduction

Deduction is reasoning from premises about general things (groups) and particular things (members of groups) to a conclusion about a particular thing.

Here's an example of deduction:

Every webmaster is a genius
Jim is a webmaster
Therefore, Jim is a genius.

With its content removed, this has the form:

Every A is x
B is an A
___
B is x.

This form is valid, which means it will always yield a true conclusion from true premises. Unfortunately, not all deductions have a valid form (see Section 3).

G3. Induction

Induction is reasoning from premises about particular members of a group to a conclusion about the whole group.

Here is an example of induction:

This raven is black
That raven is black
Each raven yet seen has been black
Therefore, all ravens are black.

This argument's form:

A1 is x
A2 is x
An is x [n = any number]
___
All A is x.

The basic idea of induction is that something that is true of all known members of a group will be true of all members of that group. To put this another way, it is the idea that what has been true of the past will continue to be true.

Obviously, the truth of the premise of an induction does not guarantee the truth of its conclusion; yet induction is a sort of reasoning we all rely upon every day.

Section 3. Disvalues of reasoning

The disvalues of reasoning are called fallacies. There are many ways an argument can be fallacious, but those ways come under three main headings:

G1. Informal fallacies of content.

Fallacies of this kind are due to a fault in the content of the argument (that is, the actual words used). The determining factor in such fallacies is some sort of confusion. For example, two different meanings of a word or phrase might be confused in the same argument:

Arnold is very healthy.
Smoking is not very healthy.
So, Arnold is healthier than smoking.

Because the phrase 'very healthy' has two different meanings in this argument its conclusion is nonsense.

G2. Formal fallacies.

Fallacies of this kind are due to a fault in the form (the basic structure) of an argument. Usually the term applies only to deductive arguments, in which case the form is said to be invalid. The determining factor in such fallacies is some sort of conflict in the structure . The two sorts of conflict are incoherence (conflict within a premise) and incompatibility (conflict between premises). An incoherent (self-contradicting) proposition cannot be true. And two incompatible (mutually contradictory) propositions cannot both be true together.

It is often difficult to tell whether a form is invalid. One easy method that often works is to remove the content from the argument to expose its form and then give that form some very different content. For example:

If Tom likes Jane he might do her homework.
Tom is doing Jane's homework.
So, Tom likes Jane.

The form of this is:

If P then Q.
Q.
So, P.

But look what happens when you give this form different content:

If there is a power blackout the TV won't work.
The TV won't work.
So, there is a power blackout.

If an argument has a valid form and true premises then the truth of its conclusion is logically necessary. If a deduction has true premises but its conclusion is not true then it has an invalid form.

G3. Informal fallacies of context.

Fallacies of this kind are due either to something that is outside the argument and should be in it, or to something that is in it but does not belong there. So the determining factor is a lack of precision - the argument is either incomplete or overcomplete. For example:

John tells us smoking is bad for our health.
John is still a smoker.
So, we should ignore what John tells us about smoking.

In this argument the premise that John is a smoker is irrelevant to the question of whether John's advice is good advice. This argument both lacks premises it needs and includes a premise that does not belong.

Section 4. Ends (values) of reasoning

G1. Truth from true content and similarity (sound analogy).
G2. Truth from true content and valid form (sound deduction).
G3. Truth from true content and invariable confirmation (sound induction).

Section 5. The three basic Laws of Reasoning

All reasoning is based upon three simple but fundamental laws:

G1. The law of Identity: A is A, and not-A is not-A.

Or, to put it in terms of propositions: a true proposition is true, and the negation of a true proposition is false.

Formally: (tp = t) & (-tp = f)

G2. The law of Non-contradiction: nothing is both A and not-A.

In terms of propositions: no proposition is both true and not true.

Formally: - (p & -p) [Read this as: not(true and not-true).]

G3. The law of Excluded Middle: Everything is either A or not-A.

In terms of propositions: all propositions are either true or not true.

Formally: p V -p [Read this as: either true or not-true.]

To abandon any of these laws is to abandon reason (though there has been disagreement, especially about Excluded Middle).

Section 6: An analysis of the Principle of Bivalence

The Principle of Bivalence is usually, and inadequately, stated as: There are exactly two truth values: true and false.

It is important to distinguish this Principle from the Law of Excluded Middle: All propositions are either true or false.

The Law is about propositions, whereas the Principle is about truth values.

An analysis of the Principle:

G1. The qualitative component: 'true' and 'false' are truth values

G2. The relational component: truth values are primarily instantiated in propositions

G3. The quantitative component: there are exactly two truth values.

The Principle of n-valence

Some logicians have experimented with logics in which there are more than two truth values. Thus there is also a Principle of n-valence:

G1: '1/n true' and '1/n false' are truth values,

G2: truth values are primarily instantiated in propositions,

G3: there are exactly n > 1 truth values (where n > 1 = a number greater than one).

Section 7: Argument as the reverse of explanation

Argument is explanation in reverse. Instead of proposition(s) therefore proposition, we have proposition because proposition(s). Here are examples of three sorts of explanation that are the reverse of the three sorts of argument:

G1: Analogical explanation:

Tom's tape-measure gave a wrong measurement (because) Tim's tape-measure gave a wrong measurement (and) Both tape-measures were left next to the fire.

This is the reverse of the following analogical argument:

Tim's tape-measure was left next to the fire.
Tom's tape-measure was left next to the fire.
Tim's tape-measure gave a wrong measurement.
Therefore, Tom's tape-measure will give a wrong measurement.

G2: Deductive explanation:

The measurement was wrong (because) the metal tape-measure was hot (and) metals expand when heated.

G3: Inductive explanation:

All metals will expand when heated (because) invariably, when a metal has been heated it has expanded.

NOTE: These are not the only sorts of explanation. Other sorts are discussed in Philosophy 302: Philosophy of Science.

Appendix: The Three Groups in this topic

Sorts of reasoning:

G1: Observational
G2: Theoretic
G3: Practical

Means of reasoning (sorts of argument):

G1: Analogy
G2: Deduction
G3: Induction

Disvalues (fallacies) of reasoning:

G1: Informal fallacies of content
G2: Formal fallacies
G3: Informal fallacies of context

Ends (values) of reasoning:

G1: Truth from true content and similarity
G2: Truth from true content and valid form
G3: Truth from true content and exclusion of irrelevance

Laws of reasoning:

G1: Identity
G2: Non-contradiction
G3: Excluded middle

Argument as reverse-explanation

G1: Analogical explanation
G2: Deductive explanation
G3: Inductive explanation