Logic - Chapter 4 - Salmon

(Originally written September 10, 2005 in Book 1)

Logic
Wesley C. Salmon
Chapter 4 - Logic & Language

Words are symbols. A word is given meaning by people assigning it to symbolize something tangible or intangible.

An extension of a word is a class of objects that the word correctly applies to. The intension of a word consists of the properties a thing must have to be in the extension of that word.

To define a word through extensions is called the extensional definition, which uses the class of things the word applies to. To define a word through intension is to use the properties of what a thing must have to be an extension.

There are two fundamentally different ways of indicating the extensions of the word:

1) (Non-verbal) ostensive definition: point to the objects in the extension of the word
2) (Verbal) naming the different types of extensions another way

The problem with extensional definitions is that it is impossible to name the different types or name all of the objects within a class. For example, take dogs. One can't in good, solid conversation list every single dog breed and individually named dog that ever existed. Besides, some dogs haven't been born yet and it is impractical to do this even with supernatural foreknowledge of doggy births.

Extensional definitions hope to indicate a few objects of the extension, assuming that other objects of the extension can be recognized by similar characteristics in the pointed out ones.

Intensional definitions are verbal in nature. One important type of intensional definition is the explicit definition. An explicit definition consists of giving a word or a phrase which means the same thing as the word being defined. For example, pentagon means a five-sided plane figure. In this sentence, the 'pentagon' is the definiendum; 'five-sided plane figure' is the definiens.

A definition is circular if the definiendum occurs in the definiens. For example if the pentagon were described as a plane figure having five sides is a pentagon. A definition can also be circular in a more discreet manner as well. For example:

Mendacity means a lack of veracity
Veracity means an absence of prevarication
Prevarication means mendacity

Many words refer to objects, events or properties. These have extensions and intensions. Other words have meaning only as they function in linguistic context. These words have neither extension nor intension. For example, There is no such thing as 'unless'; there is no event as 'unlessing'; there is no property of being 'unless'. The word 'unless' has no meaning in isolation. 'Unless' and other words like it have purely grammatical functionality. The words like 'unless' gain their meaning through showing how they function in a given context. This type of meaning is called 'contextual definition'.

Since logic is concerned with form or structure, many of the most important logical words are defined contextually.

Definitions are not true or false, but their adequacy can be judged in terms of their ability to fulfill certain functions.

Some definitions are designed to characterize the customary usage of the word so that those people who speak the language correctly (not using slang) can converse in understandable terms. Dictionaries use definitions this way.

Sometimes we define a new word because there is no established way of briefly expressing an important meaning.

A word is vague if there are objects which are neither definitely included nor definitely excluded from its extension.

Sometimes we try to find an intensional definition for a word whose extension is quite well known.
It is important when finding an intensional definition that it be neither too broad nor to narrow in its properties so as not to include objects outside of the extension or too narrow that it excludes objects within the extension.

Some words are given definitions to introduce a word which will have theoretical importance and utility. Such definitions are often found in scientific work.

Words can also have emotive force. This force can be either positive or negative. Definitions whose main function is the transfer of emotive forces are called persuasive definitions. We need words with emotive forces to express our feelings, emotions and attitudes to one another.

Definitions are conventions. While it is true that conventions are arbitrary in nature and generally equitable, some conventions function better than others in real terms.

Analytic, Synthetic and Contradictory statements:

For the purpose of logical analysis it is important to distinguish those statements which are logically true or false from those which are factually true or false.

An analytic statement is one whose truth follow from the definitions of the words which occur in it. (A logical truth).

A contradictory statement is one whose falsity follows from the definitions of the words which occur in it. (A logical falsehood).

Synthetic statements are statements whose truth or falsity is not determined solely by the meanings of the words they contain. Synthetic statements are not logical truths or falsehoods; they are factual statements.

Categories and Contradictions:

Contradictories occur when two statements are made in a way that if one is true the other is false. For example, It's raining here. It's not raining here. One is true; one is false. Logically we cannot determine which, but we know that one must be true and the other is therefore false.

The form of contradictories: P or not-P. It is called the law of the excluded middle. It expresses that every statement is either true or false.

Another form of contradictories: Not both P and not-P. This is called the law of contradiction. It expresses that no statement is both true and false.

Two statements can be made in a way that it is impossible for both of them to be true, but both can be false. These are called contraries.

Contraries and Contradictories:

The relation between the two statements is called contrariety. For example, It is cold here; It is hot here. It is impossible for both to be true, but it could be a moderate temperature and thus, make both statements false.

Ambiguity and equivocation:

Sometimes words have multiple meanings and their definitions are derivative of the word's context. If a word is used in a context that we cannot tell which of the word's meanings is intended then the word has been used ambiguously.

The fallacy of equivocation is when a word uses two distinct meanings in an argument. For example:

Only man is rational
No woman is a man.
Therefore, all women are irrational.