# Defining logic

What is logic?

What is the difference between inductive and deductive?

Logic is the study of validity, where validity is the necessary transmission of truth. In more detail, we have: a sentence is a grammatically correct structure of words, a statement is a sentence that is either true or false (not all sentences are statements: e.g. questions, requests, orders), a proposition is the meaning of a statement, and an argument is a structure of propositions (written or stated as a structure of statements) which is either valid or invalid. An argument consists of one or more propositions called its premises and a final proposition called its conclusion; and an argument may have intermediate propositions between premises and conclusion. An argument is valid if, given that the premises are all true, then necessarily the conclusion must be true also; if the conclusion could be, or is, false, given true premises, then the argument is invalid. (Note that when the words are used correctly, statements are true or false and arguments are valid or invalid: in everyday discourse these concepts are often confused.)

Propositions may be simple or compound: compound propositions are one or more simple propositions joined together by connectives: such words as ‘not’ (meaning ‘it is false that…’), ‘and,’ ‘or,’ ‘if…then…,’ and ‘if…and only if…’. For example, if we represent propositions by upper case letters P, Q, R, etc. then we might have Not-P, P or Q, Q and R, If P then Q, and P if and only if Q. Finally, there are simple forms of argument, easily seen to be valid, which are standard argument forms.

This is all a bit much to digest in one go, but some examples will help. Suppose we have the propositions ‘Pat is pregnant’ and ‘Pat is female’ and connect them in the form ‘If Pat is pregnant then Pat is female,’ which is true. In symbols this could be ‘If P then Q.’ Now we can ask what happens if P is true or if P is false. Obviously, if P is true then Q is necessarily true, so the argument form ‘If P then Q, and P, therefore Q’ is valid (written if full this is ‘ If Pat is pregnant then Pat is female, and Pat is pregnant, therefore pat is female); it is called affirmation of the antecedent. (Here ‘P’ stands for ‘P is true’; if we wanted to say that P is false we would write it as ‘Not-P’). On the other hand, if not-P then obviously Q could be true but also could be false (i.e. Pat could be male) so this argument form is invalid; it is called the fallacy of denial of the antecedent. We can also consider what happens if Q is true or Q is false. If Q is true then Pat is female but that does not mean that she is pregnant, so P could be true and could be false. This argument form, ‘If P then Q, and Q, therefore P,’ is called the fallacy of affirmation of the antecedent. But if Q is false we have ‘If Pat is pregnant then Pat is female, and Pat is not female, therefore Pat is not pregnant’ is clearly valid; this form is called denial of the consequent.

So there you are. A very brief introduction to logic. There are more advanced branches of logic. For example, quantificational logic uses concepts such as all and some, and modal logic uses concepts such as necessarily and possibly. There is also traditional logic, due to Aristotle. All of this is available in any good logic textbook.

Deductive logic is the logic that I have sketched in answering your previous question. It is valid. Inductive logic is not really logic at all, since it is invalid. The concept is due to Aristotle, who wanted to be able to derive general statements, such as ‘All men are mortal’ from less general statements, such as ‘Some mem are mortal’. Since all men who have ever lived and died so far have died, the latter statement is true; but the former one is not necessarily true. So to go from some to all is invalid. But the problem is that sometimes we want to make this move and claim that the conclusion is true. This occurs particularly is science, where a formula is repeatedly shown to be true, and is therefore generalised into a scientific law. How to justify this is called the problem of induction. Induction is in fact only a fancy name for generalization, and some generalisations are very bad. Superstition is generalisation from a coincidence to a supposedly universal truth: breaking a mirror or walking under a ladder bring bad luck, and touching wood or crossing your fingers averts bad luck. Stereotypical thinking is also bad generalisation, from a few examples. In fact, generalisation is animal thinking: animals that can learn, such as cats and dogs, do so by generalisation; a dog quickly learns that if you pick up its leash then it is going for a walk.

In my view the problem of induction cannot be solved in isolation. What happens in science is that generalisations of empirical observations lead to scientific laws and theoretical scientists then try to explain these laws by inventing theories that describing their underlying causes. It is a most important feature of theories in the mathematical sciences that they are able to predict empirical things never previously observed. For example, Maxwell’ equations predicted radio, and Einstein’s special relativity predicted nuclear energy. The only way they can do this is by being true, and if they are true then so are the scientific laws that they are explaining, in which case the inductions to those laws is justified.

Logic can be defined as that branch of philosophy that reflects upon the nature of thinking itself. Since all of philosophy employs thinking, logic is perhaps the most fundamental branch of philosophy.

Philosophers have traditionally divided logic into two branches: (ii) deductive logic and (ii) inductive logic. Both are concerned with the rules of correct reasoning – or correct argumentation. Deductive logic deals with reasoning that attempts to establish conclusive inferences. That is, it attempts to show that if the reasons given are true, then it will be impossible for the inference based upon these reasons to be false. Such reasoning is called valid reasoning. Deductive logic, then, is concerned with for determining when an argument is valid.

Deductive logic

Reasoning which involves arranging knowledge, or information, into some kind of system is called logic. Philosophers divide logic into two branches: deductive and inductive. The simplest form of deductive logic, or reasoning, is called a syllogism. A syllogism, which means ‘linking together’, is an argument which contains two premises and an inference. The two premises, or propositions, must not be equivalent, so that neither of them can be derived from the other. That is, that an inference can be drawn from the first by the second. The third proposition, which is derived from the two preceding premises, is the inference. Thus, every syllogism must contain exactly three propositions.

By use of the syllogism, deductive logic attempts to establish conclusive inferences. An inference is conclusive when reasons given make it impossible for the inference, based on those reasons, to be false. Such reasoning is called ‘valid’ reasoning. Deductive reasoning, then, is a system which is concerned with determining when an argument is valid. Syllogistic form of deduction is that which takes a general rule, or principle, with which we start and the reasoning consists in combining a related fact with the general rule through some characteristic which actually connects it with the rule.

Let us consider an example of deductive reasoning, by use of the syllogism, in which the general rule is stated in the form of a universal affirmative proposition – that is a universally held principle.

A. All men are mortal (general rule)
B. Socrates is a man (related fact)
C. Socrates is mortal (inference)

From the above syllogism we see that A and B are the premises, while C is the conclusion. We also see that deductive reasoning moves from a general rule to a particular rule. This kind of reasoning can be described in the following way:

A. All M is P (Major Premise)
B. S is M (Minor Premise)
C. S is P (Conclusion)

The first and second propositions are the premises, while the third is the conclusion.

It should be pointed out that each proposition contains four parts: a quantifier, a subject term, a predicate term, and a copula. For example, in the first proposition of the above syllogism ‘all’ is the quantifier, ‘humans’ is the subject term, ‘fallible’ is the predicate term, and ‘are is the copula. The copula is the word that connects the subject term to the predicate – it is usually the verb ‘to be.

In this example, in the major premise, the subject, M, is used as a symbol of the common term which connects the two premises, and for this reason it is called the Middle Term (M for Middle): the same term appears in both premises. The predicate in the Major Premise, P (for predicate), is called the Major Term, and the Subject of the Minor Premise, S (for subject), is called the Minor Term. The conclusion consists of the Minor Term, S, of the Minor Premise and the Major Term, P, of the Major Premise. The two Middle Terms effectively cancel each other out. So the inference is always the subject term of the minor premise and the predicate of the major premise.

Propositions, then, are made up, primarily, of a subject term and a predicate term. The predicate tells us something about the subject. For example, in the sentence ‘Socrates is mortal’, the predicate tells us that even a great man like Socrates cannot live forever. There are two kinds of predicate: analytic and synthetic. An analytic predicate does not tell us anything about the subject that we cannot conclude by analysis of the subject term itself. A synthetic predicate, on the other hand, does tell us something that is not contained in the subject term. ‘Socrates is a man’ is an example of an analytic predicate, in that we can deduce the truth of the predicate by analysis of the subject term, while ‘John is Irish’ is an example of the synthetic predicate, since we cannot draw the inference from the predicate.

Validity and Truth

Inferences are true or false: they are valid or invalid. It is possible for an inference to be valid, while the conclusion is false. Take the syllogism:

A. All students are poor
B. Some wealthy people are students
C. Some wealthy people are poor

The conclusion is clearly false, yet, on the basis of the premises, the inference appears to be valid. It is also possible for the premises and conclusion to be true, while the inference is invalid. For example,

A. Some Englishmen are politicians
B. David Cameron is a politician
C. David Cameron is an Englishman

Here we have the case where the conclusion is actually true, yet the inference is not valid. That is, the conclusion does not follow logically from the premises, or, to put it another way, the premises do not necessarily imply the conclusion. However, it is also possible for the premises and the conclusion to be false while the inference is valid.

Take the example:

A. All planks of wood are more intelligent than teachers
B. Fido the dog is a plank of wood
C. Fido is more intelligent than teachers

In this case, although the conclusion follows logically from the premises, the premises are false. Thus, factual truth depends on whether the proposition corresponds to the facts. Validity does not. Hence, a conclusion is true if it agrees with the facts to which it refers. A conclusion is valid if the premises necessarily imply it.

The issue of validity concerns the relationship of implication between the premises and the conclusions. That is, whether the relationship ‘necessarily holds’. Therefore, the issue is whether the premises necessarily lead to a conclusion. Where they do, the inference is valid, where they do not, the inference is invalid. The validity of an inference requires the identification and analysis of logical forms. In short, its validity is determined by the fact that the argument contains a firm, deductive structure. The argument is valid because, were the premises true, the conclusion would be true. On the other hand, to determine the factual truth of individual proposition, it is necessary to identify and observe the proposition’s content. To do this we must first establish the truth of the premises. This is not a logical task. Then we must establish that the conclusion follows necessarily from the premises; that is, we must determine whether or not the inference is valid. This is a logical task. If it is the case that both of these conditions are met, we must conclude that the truth of the conclusion is logically demonstrated. Thus we can say that the logically ideal rational case consists of two distinct steps: (1) you must prove the truth of the premises, and (2) you must show that the inference is valid. In a rational discussion a conclusion may be rejected on any one of the following three grounds:

1. The premises are false
2. The inference is invalid
3. The conclusion on the independent factual investigation can be shown to be false.

There is another form of deductive reasoning or logic that should be mentioned at this juncture. That is modus tollens (from tollere: to remove): an inference from a conditional statement. This has the form of:

If P, then Q.
Q is false.
Therefore, P is false.

For example, I can say ‘If John is sick his mother will be at home. John’s mother is not at home, therefore he is not sick’. Thus we have the pattern of an inference from a conditional statement, together with the denial of the consequent, to the denial of the antecedent.

In conclusion, it can be said that deductive logic is a study of deductive ‘valid’ reasoning: in a valid argument the premises provide ‘conclusive’ reasons for a conclusion and it is quite impossible for the premises to be true and the conclusion to be false. However, most of the reasoning we engage in falls short of this ideal. In both science and in everyday life arguments we use do not provide us with conclusive reasons for their conclusions. They may offer good reason to believe their conclusions, but they do not compel us in the same way as deductive arguments. The form of logic that deals with those arguments which do not allow us to arrive at conclusive conclusions is called inductive reasoning.

Inductive logic

Inductive logic is not concerned with rules for correct reasoning in the sense of ‘valid’ reasoning, but with the soundness of the inferences for which evidence is not conclusive. That is, while deductive logic is concerned with drawing particular inferences from general assertions, inductive logic is concerned with inferences from the particular to the general – and the inference of a general proposition from particular assertions can never be conclusive. For example, if we want to establish the truth of the proposition that ‘All Irishmen are mortal’, we can do this in two ways: either by deductive reasoning or inductive reasoning. Let us take deductive reasoning first. To do this we will turn to one of the most well known and well-used forms of deductive logic: the syllogism. A syllogism is defined as an argument which contains two premises and a conclusion – or, to put it another way – every syllogism must be composed of three propositions. The proposition which one is trying to prove is the conclusion of the argument. The other two propositions provide reasons for asserting that the conclusion is true. These propositions are called the premises of the argument. Now let us return to the example “All Irishmen are mortal”. To prove whether the premise is valid by deductive logic we set up a syllogism thus:

A. All Irishmen are human beings
B. All human beings are mortal
C. All Irishmen are mortal

Here we have an inference from the assertion about all human beings to an assertion about some human beings. That is, from the general to the particular, which is, of course, deductive reasoning.

On the other hand, one may not accept that ‘All Irishmen are mortal’ is true. It may be the case, you might argue, that we cannot know this for certain until after every Irishman is dead. So, in order to establish whether ‘All Irishmen are mortal’ we need to take a different approach. That is, we need to argue from the particular to the general. If we accept that (a) ‘Every Irishman born before 1830 is dead’, and (b) that ‘Irishmen are still dying’, we can conclude that the proposition ‘All Irishmen are mortal’ is true. However, unlike the reasons provided by the deductive argument, the truth of these reasons does not establish with certainty that ‘All Irishmen are mortal’. It is still possible that (a) and (b) are true, and yet that there is some Irishman alive today or one born in the future, that may be immortal. Inductive logic, then, is not concerned with valid inferences, but with inferences which are probable, given as evidence the truth of certain propositions upon which they are based. Inductive logic has one of its most important uses in connection with science. The scientist employs deductive methods, and even intuitive guesses, to investigate the world, but it is inductive logic that it the scientist’s most important tool. For example, Galileo described the rate at which a particular body accelerated when he dropped it. He then dropped other bodies and in each case measured their rate of acceleration. If we reconstruct this activity, we might say that he arrived at a number of true individual statements describing the rate at which particular bodies accelerated. He noticed, for instance, that body A, when dropped, accelerated at a rate of 32 feet per second per second; body B fell with the same acceleration, so did body C, and so on. Thus, from the truth of particular propositions he inferred a general truth of nature, sometimes called a ‘law of nature’. He inferred that all bodies dropped would fall with an acceleration of 32 feet per second.

In short, we can say that inductive logic is a theory about what reasons provide evidence for a given conclusion’s truth when the reasons in question are not conclusive ones.

The difference between inductive and deductive is this: Inductive refers to the real world, where you have experiences and collect facts. Over time you will notice similarities between some experiences and facts, and then you may wish to group them together. This is called inductive inference. Actually what you are doing is to make a deduction from the occurrence of similar facts, but usually we keep the word deductive out of it, because it fits formal logic better.

A really good example of induction is this: we have been observing ducks, geese and swans forever, and we noticed that swans are always white. So we group all swans together and state, inductively, and make inference such as ‘all swans are white’ .

There are two issues with this. The first is that we cannot be sure that we have seen all swans in existence. We could be wrong with our definition. Then later black swans were discovered in Australia which shows that inferences from inductions are not 100% truth. On the other hand the importance of induction is that we can discover things, acquire knowledge and make classifications as to what belongs together. In this way we organize the facts of the world for the benefit of our knowledge and convenience, for survival and exploitation.

As I said inference and deduction are much the same. But when we use logic exercises we use a very formal way of arriving at a conclusion. We start with a premise that every one agrees to e.g. all humans are mortal. Then we can deduce from the fact that Socrates is human that he must be mortal.

Another example is this:

Premise: ‘all nations occupy a land mass’

Minor premise: ‘the moon surface is a land mass’

Conclusion: ‘therefore all nations occupy the moon’

From these two examples you can see that deduction does not refer to the world, but is a mechanism of truth finding down to a conclusion. If the deduction is done correctly the result must be true. So why the nonsense conclusion in second example?

It is because deduction relies on the premises. If any premise is false than the answer will necessarily be rubbish, so deduction by itself cannot help us with the truth. As it is a formal device, and strictly regulated, deduction can only give you the truth if the truth is already in the premises. (In science it is the best way of making sure that experiment and research are free of self contradiction.)

Let me give you another example:

The fundamental doctrine of the Christian’s church is, (premise) that all believers will be saved, (minor premise Cathy is a believer,) (conclusion) therefore Cathy will be saved.

This is a valid conclusion for Christians. It is a truth resulting from a truth premise. However the idea of being saved is not a fact in the world, it is an idea in the minds of people. So this proof is absolute evidence for Christians, but worthless to those who don’t believe in salvation.

So to conclude: Induction refers to the world of facts. It consist in classifying those facts by inferences based on sameness and difference. Deduction comes after and uses formal logic to test the truth of the inferences.

Both methods are vulnerable. Inductive inference is limited to the results of human experiences, and deductive logic is absolutely dependent on the prior truth of its premises.

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