Tutorial 1 Introduction, sketch of course, and symbolizing atomic sentences.

Logical System

Skills to be acquired in this tutorial:

To become familiar with the notions of argument, valid, invalid, premise, and conclusion. To learn how to symbolize atomic sentences.

Bergmann[2004] The Logic Book Chapter 1

Tutorial:

The main role of logic is to assess arguments-- to say whether an individual argument is valid or whether it is invalid. In logic, arguments are taken to consist of two components--premises, and a conclusion.

For example,

If it rains, I get wet.
It rains.

Therefore,

I get wet.

is an argument with two premises 'If it rains, I get wet.' and 'It rains.' and the conclusion 'I get wet'.

In an argument the premises are intended to support the conclusion and the best type of support that the premises can give is if their truth guarantees the truth of the conclusion. Arguments whose premises provide this kind of guarantee are said to be valid.

Take care over this. Individual premises are either true or false; the conclusion is either true or false. When considering whether an argument is valid, we are not looking directly at whether the premises are true or whether the conclusion is true; rather we are considering whether if all the premises are true that condition would mean that the conclusion has also to be true.

For example, the argument

3 is an even number.
All even numbers are exactly divisible by 2.

Therefore

3 is exactly divisible by 2.

is a valid argument, with one true premise, one false premise, and a false conclusion-- it is valid because were its premises to be true, their truth would guarantee that the conclusion is also true.

An argument is invalid if and only if it is possible for all its premises to be true and its conclusion false at one and the same time.

Assessing an argument is done by first symbolizing its premises and conclusion. Then two lines of attack are possible. The argument is like a needle, and it is in one of two haystacks-- the invalid haystack, which contains only invalid arguments, and the valid haystack, which contains only valid arguments. We use 'semantics' to search the invalid haystack, and we use 'derivations' to search the valid haystack. Both these techniques are taught in this course. Should we find the argument in one of the haystacks, we may well have proved it to be 'valid' (or 'invalid', as the case may be).

This course follows the above: symbolization, semantics, then derivations.

The appraisal of arguments can be done using instruments of varying degrees of sophistication. We will start with sentential logic, then move on, after Tutorial 10, to the more advanced predicate logic.

Starting on sentential logic ...

Indicative English sentences are either true or false. For example, 'There are 35 State Governors in the U.S.A.'is an indicative sentence (which happens to be false). Not all pieces of English are indicative sentences. For example, the question 'What day is it today?' is not either true or false (although reasonable answers to it will be either true or false); again, the greeting 'Have a nice day!' is not either true or false.

In an argument, the premises and the conclusion are indicative sentences.

Sentences can be atomic or compound. 'There are 35 State Governors in the U.S.A.' expresses an atomic sentence; whereas 'There are 35 State Governors in the U.S.A. and there is one President of the U.S.A. ' expresses a compound sentence composed of two atomic sentences (one false one and one true one).

Symbols are used to stand for sentences. In the systems used here we use the capital letters 'A' to 'Z' to stand for atomic sentences-- a capital letter is used as a shorthand or code for a sentence. You decide which letter you want to stand for each particular atomic sentence; then having formed a convention or Dictionary, you stick to it throughout each argument.

For example, you may decide to let 'F' stand for the sentence expressed by 'Forests are widespread' in which case when you are trying to symbolize, every time you meet 'Forests are widespread' you symbolize it to 'F' (and, if you are trying to translate back from symbols, every time you meet 'F' you translate it back to 'Forests are widespread').

It is usual to tell the world of your conventions-- the codes you use are not intended to be secret ones.

[Alternative terminology: some logic texts use the word 'proposition' to identify what a true or false indicative sentence expresses. Then they talk about premises and conclusions as being propositions, about propositions being atomic or compound, and, in fact, about the present subject of study (sentential logic) as being 'propositional logic'.]

Exercises to accompany Tutorial 1

Exercise 1 (of 2):

Are each of the premises of the following arguments true, or are they false? What is the truth value of the conclusions? Is each argument valid or is it invalid?

Take note of the fact that many combinations are possible. For example, that an argument is valid does not mean that its premises are true (nor does the truth of an argument's premises mean that the argument is valid) ... One combination is impossible--which one is that?

(You should be able to make your mind up about the truth or falsity; judging validity (in the technical logical sense) is more difficult, after all it is the aim of this course to teach you how to do this, and we are only on Tutorial 1; however, if you understand the definitions given in the Tutorial you should be able to make a reasonable attempt.)

a)

Phoenix is in Arizona,
San Franciso is in California.
Therefore

Los Angeles is in California.

b)

Phoenix is in Arizona,
San Franciso is in California.
Therefore
Phoenix is in Arizona and San Franciso is in California.

c)

Phoenix is in California,
San Franciso is in Arizona.
Therefore
Phoenix is in California and San Franciso is in Arizona.

d)

Phoenix is in California and San Franciso is in California.
Therefore
San Franciso is in California.

a)

Phoenix is in Arizona, (True)
San Franciso is in California. (True)
Therefore
Los Angeles is in California. (True)
(Invalid)

b)

Phoenix is in Arizona, (True)
San Franciso is in California. (True)
Therefore
Phoenix is in Arizona and San Franciso is in California. (True )
(Valid argument)

c)

Phoenix is in California, (False)
San Franciso is in Arizona. (False)
Therefore
Phoenix is in California and San Franciso is in Arizona.(False )
(Valid argument)

d)

Phoenix is in California and San Franciso is in California. ( False)
Therefore
San Franciso is in California.( True)
(Valid argument)

(The impossible combination mentioned earlier is all the premises true and the conclusion false in a valid argument (for if all the premises are true and the conclusion false then it is possible for all the premises to be true and the conclusion false at one and the same time and that means that the argument is invalid).)