CHAPTER 1
BASIC LOGICAL CONCEPTS
Section 1.1 Introduction
Logic is the study of correct reasoning.
The study of correct reasoning is the most important study there can be.
Therefore, logic is the most important study there can be.
There. What a fine piece of reasoning. Arenât you glad you chose logic? Of course you are, but should this argument give you a reason for such gladness? That will depend on whether the argument is correct. If the argument is correct, then the conclusion must be true, and indeed you are now engaged in the most important study that human beings can undertake. What glory! And this argument has shown this to you. But, if the argument is not correct, then the argument does not provide a reason for such gladness, and we will have to look elsewhere for such a reason.
There is a question that can be raised about the second premiss of the argument. What kind of importance are we talking about? How do areas of knowledge gain their importance? Clearly, importance is a relation. Things are never just important by themselves. Things are important to someone, and even more, are important to someone for some goal. Things are important only in relation to the goals that people have. This applies also to areas of knowledge. The study of physics is important for the achievement of such goals as flying to the Moon. The study of physics is not important for the baking of pastry cakes. It is clear now that the above argument is not stated as precisely as it should be. The argument should make clear the kind of importance that is intended. We propose that the intended importance is an epistemic importance, for which the goal at issue is the attainment of truth, the acquisition of knowledge. We also propose that the purpose of logic is to be an instrument in the achievement of that goal. What field of study is better suited to the attainment of that goal? What about physics? It is true that physics produces truths about our universe, but it also produces, as a goal, tentative hypotheses, many of which are later rejected by physics. Here is a more telling consideration: one can do logic without doing physics, but one cannot do physics without doing logic. The same is true for any field of study. Claims of knowledge in any field of study depends on correct reasoningâon a knowledge of logic. Logic is a foundation of physics, of all science, of all fields of knowledge. With this new understanding of the second premiss, we can restate the argument as follows, and it should now be apparent that the argument is entirely correct.
- Logic is the study of correct reasoning.
- The study of correct reasoning is the most important study there can be, for the purpose of attaining truth.
- Therefore, logic is the most important study there can be, for the purpose of attaining truth.
A Definition of Logic
Logic is indeed the study of correct reasoning, but this definition can be made more precise. Reasoning is a somewhat wide-ranging activity, and logic does not really deal with everything that is involved in that activity. Logic has a natural focus on a part of reasoning that can be called argumentation, the making of arguments. The advantage of identifying logic with this focus is that we know exactly what arguments are, and what the standards are for their correctness. There is also a second issue regarding the focus of logic: whether this focus will be organized as a formal system, specifying techniques, rules, laws, and applied to reasoning in general, or whether it will not be organized as a formal system, applied always to concrete cases of reasoning. The former kind of study is called formal logic and involves great precision and abstraction, and the latter kind of study is called informal logic and involves a lack of such precision and abstraction, but addresses real cases more effectively. These considerations lead us to the following definitions:
- Logic is the study of the methods and principles of correct argumentation.
- Formal logic is logic organized as a formal system.
- Informal logic is logic not organized as a formal system.
The logic that we will study in this course will be formal logic. That is, we will study the methods and the principles (these are two different things) of correct argumentation, as these methods and principles are part of a formal system. This definition assumes that the reader has some knowledge of what a formal system is, and for now we can leave it at that, except to say that an excellent example of a formal system is one that most of us are already familiar with, namely, the system of Euclidean Geometry. We can even temporarily define a formal system as a system that resembles Euclidean Geometry in its arrangement, apart from its content. Later, we will construct the formal system of logic, slowly, one step at a time.
The Strange Argument
Letâs start our study with a big bang. Letâs start with a complicated argument, and letâs go through this argument step by step, to see whether or not it is any good. We will use methods and rules here that we wonât introduce until later in the course, and you may not understand very much of what is going on. But that is OK. This is only an example to give you some idea of what we will be doing later on.
You may have heard about The Tooth Fairy, the magical creature that collects the lost teeth of little children and gives them money under their pillows while they sleep.
Is this just a fairy tale, or does the Tooth Fairy really exist? Well, hereâs an argument that claims to prove that it is not a fairy tale, but that the Tooth Fairy really exists.
- If John is in the house, then John came in through the front door.
- If John came in through the front door, then someone saw John come in.
- It is not the case that someone saw John come in.
- Yet, John is in the house.
- So, there really is a Tooth Fairy.
The first question that must always be answered is whether the argument before us is correct. One thing is clear: emotional responses, like âOh, I donât believe in Tooth Fairies,â are worthless, since there is an actual argument here that claims to have proved the exact opposite. We have an intellectual obligation to evaluate arguments that affect our views. We have to show either where such arguments go wrong, or how they are correct. We must conduct a test.
We note at the outset that whether the conclusion follows from the premisses does not depend on the content of the sentences but only on the abstract pattern that the argument has. The reason for this is simple: the laws of logic are themselves abstract patterns with a total disregard for particular content.
In logic, it is customary to use capital letters both to abbreviate sentences and also to represent the patterns that sentences have. We can thus represent the argument as follows. (You will soon become experts in generating such symbolic representations, even ones more complicated and more symbolic than this.)
- if H then D
- if D then S
- not S
- H
So, T
We will test this argument pattern. (Later in the course we can return to this argument to confirm that our method is correct, and that it conforms to the rules and procedures of established logical theory. Again, donât worry if you feel a bit confused at this point.)
- if H then D first premiss
- if D then S second premiss
- not S third premiss
- H#160 fourth premiss
- D lines 1 and 4 correctly produce line 5
- S lines 2 and 5 correctly produce line 6
Letâs assess where we are in the test. We were able to derive steps 5 and 6 with great ease. We have âif H then D,â and H is also available; so, this produces D.
And we have âif D then S,â and D is now available; so, this produces S. But how can one derive something T from items that use only H, D, S? Hmm, this does seem to be a problem. How can there be a connection? It seems, then, that we cannot derive the Tooth Fairy conclusion after all, as all of you believed right from the start. Hmm.
Wait a minute. We can at least bring T into the picture. One can always bring anything into any picture. All one has to do is say âor.â For example, you say âGeorge scored 100 percent.â You must now also say âyesâ to the question âDid either George or Queen Elizabeth score 100 percent?â You must say âyesâ because that is the way âorâ works. Of course, this introduction is harmless, because when you agree to an âorâ sentence, you donât have to agree with both choices. So, there is a harmless way of bringing the Tooth Fairy into the picture. Just say âor.â
7. S or T line 6 correctly produces line 7. Amazing! (but harmless)
Where are we now in our test? We have gotten as far as âS or T.â This is harmless, because line 6 asserts S. Well, it seems that we are no closer t...