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LOGIC

SECTION 1

LOGIC

Introduction
Principles
Subject, Predicate, and Context
Logic and Rhetoric 
Inferences and Proofs

SECTION 2
 FALLACIES

Argumentation
Non-sequitur 
Category Errors 
Fallacies of distraction 
Appeal to motives 
Changing the subject 
Inductive fallacies 
Statistical syllogisms 
Causal fallacies 
Missing the point 
Ambiguity 
Syllogistic fallacies 
Fallacious explanation 
Fallacies of definition 
 


 

 

INTRODUCTION

Logic is the study of the patterns of coherent or consistent speech. Its most important applications are the search for inconsistencies in stories or reports and the identification of valid and invalid forms of reasoning or argumentation.

Logic rests on the fact that there are statements that necessarily are true and therefore cannot be falsified no matter what is or is not the case. Such statements are called tautologies. Here are some simple examples of tautological statements:

• It rains or it does not rain.
• Boys are boys.
• No circle is a rectangle.

Because tautologies are true no matter what is or is not the case it simply is impossible to find, construct or even imagine a counterexample (a situation in which the tautology would not be true). For the same reason, the negation of a tautology necessarily is false and therefore cannot be verified no matter what is or is not the case. Negations of tautologies are called contradictions. It is impossible to find, construct or even imagine an example (a situation in which the contradiction would be a true statement). Here are the contradictory statements that are the negations of the tautologies listed above:

• It rains and it does not rain.
• Some boy is not a boy.
• Some circle is a rectangle.

Incoherent speech involves the speaker in a contradiction, which may be more or less obvious to his audience or so well-hidden in his arguments that only diligent logical analysis will bring it to light.

PRINCIPLES 

In the following table, we list some basic principles of logic. Each one of them is a tautology.

For many purposes, logicians will develop formalisations that are more sophisticated than this one is. For other purposes, no formalisation is necessary.

To speak or write logically, one should not contradict explicitly or implicitly any of the principles listed in the table.

For example:

It is illogical to say of yourself

• that you are no thing (which violates ‘Existence’)

• that you are not you (which violates ‘Identity’)

• that you are me (which violates ‘Uniqueness’).

It is illogical to say of your cat

• that it has no properties (which violates ‘Specificity’)

• that it is dead and not dead (which violates ‘Non-Contradiction’)

• that it is neither in good health nor not in good health (which violates ‘Excluded middle’).

SUBJECT, PREDICATE AND CONTEXT 

The word ‘thing’, which occurs in each one of the principles of logic, refers to anything about which you may want to say something. Thus, an object (the Eiffel tower, your computer) is a thing. So is an animal (your cat), a person (me, you, your father), or a fictional character (Mickey Mouse). An historical or fictional event (the Second Gulf War, the Big Bang, your birth, your neighbour's marriage, the death of Sherlock Holmes) is a thing. Other things are a letter of the alphabet, a word, a sentence, an argument; and so on. In short, a thing is anything that is or can be the subject of something one says.

What one says about a thing is called its predicate -- it is what one predicates of it. For example, you might predicate of a subject that it has, or does not have, a certain property; or that it stands, or does not stand, in a certain relation to some thing(s).

Notice that one should always make a clear distinction between a thing and the names or descriptions by means of which one refers to it. The name 'Oliver' is composed of six letters, but the person (if any) to whom the name applies is not composed of letters. The name 'Dracula', as the proper name of a vampire, does not refer to a real thing--thát Dracula does not exist--but obviously the name itself does exist. Consequently, in the context of a description of the real world, the 'axiom of existence' applies only to the name 'Dracula', but not to the non-existent Dracula. Thus, one should not read the axiom of existence as if it said 'for every name, there is a thing to which the name refers'. 

Sometimes, we find that a thing is known by more than one name or description. For example, the names 'the morning star' and 'the evening star' refer to the same planet. However, that fact does not give us a counterexample or an exception to the principle of uniqueness. In other words, it is not the case that we have here a pair of things--the morning star and the evening star--such that the one thing is the other thing: there is only the one planet. Nor is the case that we have here a pair of things--the name 'the morning star' and the name 'the evening star'--such that the one name is identical to the other. 

We should note that the principles of logic refer to a given context. In the Dracula-story, the name 'Dracula' refers to something that is supposed to be really existing. The story would not make sense, if you did not make that supposition. Mickey Mouse does not exist in the real physical world, but he certainly is supposed to exist in the Mickey Mouse stories. Of course, while you know that the story is fictional, in order to enjoy it, you have to separate clearly what it tells you from what you know is true in the real world. Getting mixed up about the context of real life and the context of a particular piece of fiction or imagination is not going to help you make sense of either the one or the other. 

Keeping track of contexts is an essential move in logic. Finding out which statements can, and which statements cannot, refer the same context, is the primary purpose of logic. Your cat may have been alive and well yesterday but ill this morning—and now it may be dead. That statement is not contradictory. However, it cannot be true that your cat is alive and well, ill and dead—all at the same time.  

A statement, imagination or story may not be true, but that does not mean it is illogical. We certainly can check whether a story is illogical or not, regardless of whether it is meant to be true. A novel that in chapter one reports that the butler discovered the body of his employer and in chapter eight states that the butler was already dead by the time his employer died is illogical. It tells a story that cannot possibly be true. On the other hand, a logically consistent or coherent story conceivably could be true even if it is not.

Obviously, checking whether a story is consistent is not the same thing as checking whether it is true. Checking whether one story agrees with another is not the same as checking whether it agrees with what we know of the real world. 

If two people disagree on some point, at least one of them must be saying something that is not true. It also is possible that both are saying something that is false. However, if they were not pretending to discuss the real world or the same fictional story but merely producing stories for the enjoyment of their readers then they presumably would not care about the correspondence of their literary products with the facts of reality or the facts of any story but their own.

While there are statements that are true in one context and false in another, tautologies are true in all contexts and contradictions are false in all contexts. That is just another way of saying that tautologies necessarily are true and therefore cannot be falsified no matter what is or is not the case; and that contradictions necessarily are false and therefore cannot be verified no matter what is or is not the case.

LOGIC AND RHETORIC 

To say something illogical is to say something that, if taken literally, cannot be true. It is to say something that we even cannot imagine being true—and not because of a lack of imaginative power.

If someone says ‘My cat is dead and not dead’ then what he says cannot be true, at least if we take him literally. To make sense of his assertion, we have to assume that he uses the word ‘dead’ in two different senses, for example, ‘My cat is alive but she is so listless that she might as well be dead’. That interpretation removes the contradiction but it does so only by taking him to be saying something else than what he literally said.

When someone says ‘I am not myself today’ then we tend to assume that he means something like ‘I do not know what is wrong with me today but my present behaviour is unusual for me’. However, if he insists that we take his words literally then we cannot make sense of what he says. It could not possibly be true.

When someone deliberately says something that prima facie is illogical, there is a good chance that he does not want his audience to interpret it literally. He probably is speaking rhetorically to make or emphasise a point. There is nothing wrong per se with such rhetorical flourishes but they should be used with care because they increase the risk of misunderstanding. After all, one is saying something that should not be taken literally but one leaves it to the audience to find out what one really wants to say. 
In addition, rhetorical expressions can be misleading. Demagogues and tricksters often use them to divert their audience's attention from relevant facts or to induce them to associate one thing with another when there is no objective basis for the association. The less trained in logic the audience is, the easier it is for the demagogues and tricksters to mislead them. As Bertrand Russell said, 'Logic is the best defense against trickery.'

Often the illogical nature of what a person says is not obviously, or is obviously not, an intended result. It may be that it appears only on closer analysis of what he said or by combining different parts of his message. Alternatively, it may appear only by making explicit what he did not say in so many words but should affirm because it is implied in what he did say explicitly. Sometimes a speaker is not fully aware of all the logical implications of what he says. Sometimes he may be unaware of the existence of factual or theoretical knowledge that applies to what he is saying. Consider the following message:

1. I bought a piece of flat land that is a perfect rectangular triangle.

2. One side is 30 meters long.

3. One side is 40 meters long.

4. The third side is 55 meters long

That looks like a simple description of a piece of land with no hint of rhetorical embellishment or exaggeration. However, an elementary knowledge of geometry (in particular, of the relevant Pythagorean theorem) reveals that no rectangular triangle with the dimensions that the speaker mentions can exist. If what the speaker said were true then the Pythagorean theorem is wrong! On the other hand, if the theorem is true then at least one of his measurements, or his description of the shape of his land, is wrong. Therefore, assuming reasonably that the theorem is true we can infer that the speaker made a mistake or lied about the land he claims to have bought.

INFERENCES AND PROOFS

Suppose that Jane is a student and that her teacher tells you that all the students in Jane’s class passed the examination. Although the teacher is not saying it in so many words, you are entitled to infer that Jane passed the examination. After all, Jane is a student in her class.

Premise 1: All the students in Jane’s class passed the examination.
Premise 2: Jane is a student in Jane’s class.
Conclusion: Jane passed the examination.

This inference is valid. However, it does not prove that Jane passed the examination. After all, the statement that Jane passed the examination is inferred merely from what the teacher said. Did the teacher speak the truth? Suppose that it turns out that Jane did not pass the examination. Then we can prove that what the teacher told Jane was not true. The proof goes as follows:

Fact 1: The teacher said that all the students in Jane’s class passed the examination.
Fact 2: Jane is a student in Jane’s class.
Fact 3: Jane did not pass the examination.
Inference: At least one student in Jane’s class did not pass the examination.
Inference: It is not true that all students in Jane’s class passed the examination.
Conclusion: What the teacher said was not true.

Another proof of the same conclusion would be

Fact 1: The teacher said that all the students in Jane’s class passed the examination.
Fact 2: Jane is a student in Jane’s class.
Inference: If what the teacher said were true then Jane passed the examination.
Fact 3: Jane did not pass the examination.
Conclusion: What the teacher said was not true.

Again, the conclusion is validly inferred from the statements that precede it (the premises of the argument). However, because it is inferred from facts by means of other valid inferences, we can now say that we have a proof that the conclusion is true. A proof is a valid inference starting from facts (which are communicated by means of true statements). However, valid inferences can be made from statements that are not true.

Clearly, a proof is a valid inference but not every valid inference is a proof. Consider

Premise 1: Lions are birds
Premise 2: Birds have wings
Conclusion: Lions have wings

The conclusion is validly inferred from the premises but we should not say we have proven that lions have wings. The conclusion is false—and we logically cannot claim to be able to prove what is false. Consider also

Premise 1: Lions are birds
Premise 2: Birds are animals
Conclusion: Lions are animals

Again, the conclusion is validly inferred from the premises. This time the conclusion is true: lions are animals. However, the inference still is not a proof of the conclusion. One of the premises is false—and we logically cannot claim that a falsehood provides support for a statement.

Obviously, neither inference proves that its conclusion is true. Yet, both of them are valid inferences because each of the following hypothetical statements is a tautology:

• If lions are birds and if birds have wings then lions have wings

• If lions are birds and if birds are animals then lions are animals

In these hypothetical statements, nothing is said about the truth or falsehood of the premises or the conclusions of the inferences. The statements merely affirm that if the premises are true then the conclusion is true.

For example, the inference about Jane’s examination result is valid because the following hypothetical statement is a tautology:

• If all the students in Jane’s class passed the examination and if Jane is a student in Jane’s class then Jane passed the examination

Again, nothing is said about the truth or the falsehood of the premises or the conclusion of the inference. All that is said is that

• If the premises are true then the conclusion is true.

Moreover, because that pattern represents here a tautology, which is true no matter what may or may not be the case, we can say

• If the premises are true then the conclusion must be true

Because the hypothetical statements with which we are dealing here are tautologies, their negations are contradictions. With respect to the inferences we took as our examples, those negations satisfy the pattern

• The premises are true and the conclusion is not true

For example, ‘All the students in Jane’s class passed the examination and Jane is a student in Jane’s class but Jane did not pass the examination’; ‘Lions are birds and birds have wings but some lion has no wings’.

Moreover, because the said pattern represents here the negation of a tautology, it represents a contradiction:

• The statements ‘The premises are true’ and ‘the conclusion is not true’ are contradictory

Thus, if we are dealing with a valid inference, one logically cannot affirm the premises of the inference without also affirming its conclusion. Affirming the premises of a valid inference while refusing to affirm its conclusion involves one in a contradiction—in holding something to be true that simply cannot be true. In other words, it involves one in incoherent speech.

From what we have said so far, it is easy to understand how a logician goes about checking the validity of an inference. He does so by trying to find, construct or imagine a situation in which the premises are true but the conclusion is false. In other words, he tries to come up with a counterexample. If he succeeds in his attempt, he has proven that the inference is not valid.

However, the mere fact that he does not succeed in producing a counterexample gives us no compelling reason to say that he has proven the validity of the inference in question. It may be that his search for a counterexample was not exhaustive—that he did not consider all possibilities.

Unless he can show that his attempt has considered all possibilities and therefore amounts to a proof that the search for a counterexample is futile and hopeless, his negative result is inconclusive. On the other hand, if he can show that he has considered all possibilities and still could not find a counterexample, then he is entitled to say that no counterexample can exist and that, therefore, the inference he is investigating is valid.

Hence, we can also understand that logical thought consists primarily in taking account of all possible cases and contexts.

[ About argumentation in general, see section 2 on Fallacies ]