The Wandering Blog

The Series So Far

In the previous articles, we introduced logic and discussed some basic concepts, including the informal test for invalidity:

1. Introduction
2. Soundness and Validity
3. Invalidity
4. The Test for Invalidity

In all of these presentations, we have looked at logic from an informal perspective.

Natural-language Arguments vs. Argument-forms

The next section of our Series introduces you to Formal Logic. Let’s start by looking at the following arguments.

1. If it is raining, then it is cloudy. It is raining. Therefore, it is cloudy.
2. If the trade gap narrows, the deficit will rise. The trade gap narrows. Therefore, the deficit will rise.
3. If you recycle your cans, you will feel warm and fuzzy. You recycle your cans. Therefore, you will feel warm and fuzzy.

These are all examples of natural-language arguments. Their content is different, but their form is the same. That is, they all have the same logical structure, which is:

• If A then B. A. Therefore, B.

This logical structure is called an argument-form, as opposed to a natural-language argument. A natural language argument has both content and form. An argument-form abstracts away the content, and preserves the logical structure. It has only form.

Try to figure out the logical argument-form for the following arguments:

1. You either have the coffee or the tea. You don’t have the tea. Therefore, you must have the coffee.
2. The prize is behind door number one or door number two. It isn’t behind door number two. Therefore, it must be behind door number one.
3. He meet us in the alley or on the corner. He doesn’t meet us on the corner. Therefore, he meets us in the alley.

The argument-form for these argument is:

• A or B. Not B. Therefore, A.

Here is a slightly more challenging set of arguments:

1. If you read this book, it opens your mind. Your mind is not open. Therefore, you did not read this book.
2. Provided that Bob likes potatoes, he will like your dinner. He doesn’t like your dinner. Therefore, he doesn’t like potatoes.
3. It rains only when it is cloudy. It is not cloudy. Therefore, it is not raining.

The argument-form of these arguments may be expressed in any of the following three ways:

• If A then B. Not B. Therefore, not A.
• Since A, B. Not B. Therefore, not A.
• A only when B. Not B. Therefore, not A.

All of these answers are equally correct. Logicians often prefer the first formulation, but these argument-forms are all equivalent, in the sense that they all express the same logical pattern.

Natural-language arguments are expressed in a naturally-occurring language (such as English, French, Ojibwe). Partly as a result of the richnesses of natural languages, ambiguities tend to crop up. Consider the following:

• I have a banana and an apple.
• Press the button and an alarm will sound.

In the first statement, the word “and” functions as a conjunction. That is, the statement is true if I have both a banana and an apple. In the second statement, the word “and” functions completely differently. In fact, it is difficult to know what its function is. Perhaps we are looking at a sign that contains a warning (or an enticement!): if the button is pressed, then an alarm will be sounded. This use of “and” is quite different from the first.

Logical Connectives

One of the virtues to which logicians aspire is clarity. In order to achieve a high level of clarity in argument-forms, we eschew natural-language, and instead use an artificial language. In this artificial language, we are only allowed to use the following words/phrases to express our argument-forms (called our “logical connectives”):

• And: I like marshmallows and I like cake.
• Or: You must take math or French to graduate.
• If … then: If this is a triangle, then it has three sides.
• Not: Your claim is not clear.
• If and only if: You may come to the party if and only if you are a member of the club.

Try to express the argument-forms of the following arguments, using only these words (use placeholders for the content, as above).

1. If the government rigs the elections, there will be riots. If there are riots, the government will fall. Therefore, if the government rigs the elections, the government will fall.
2. If Homer did not exist, it follows that the Odyssey was written by a committee or by a woman. But it was not written by a woman. So, it was written by a committee.
3. The players will go back to work if agreement is reached about their salaries. But this will be achieved only if some of them take early retirement. So the players will not go back if some of them do not retire early.

A properly expressed argument-form will have nothing other than placeholders and the words/phrases above, the words/phrases must be intact wholes (“only if” cannot stand by itself; you must have either “if” or “if and only if”), and a placeholder cannot implicitly contain any of the words/phrases above. By convention, we have been using “therefore” at the beginning of the conclusion.