INTRODUCTION TO LOGIC
1. BASIC CONCEPTS, Some Common Forms of Argument, Kinds of Logic
= Copi, Chapter 1, and Chapter 7, 7.7 and 7.8. The order of presentation however is a little bit different.
Logic has to do with the theory of distinguishing between correct and incorrect reasoning. A study of logic usually sets out also to give some practice in the practice of distinguishing between correct and incorrect reasoning. Both of these are important in so far as incorrect reasoning doesn't prove anything.
Logic in and of itself is not concerned with what the reasoning happens to be about, or where it is happening, whether in the natural or social sciences, philosophy or theology or in common life; merely whether it is correct or incorrect.
This introduction sets out to give you some of the basics of what is called 'Formal' logic. Why it is called that you will see by the end. Much of it will be dedicated to introducing the sometimes unusual technical language which logicians and following them technical philosophers sometimes use.
Arguments, propositions or statements, premises, conclusions, validity, soundness
Bits of reasoning, in the terminology of logicians, are commonly called 'arguments'. In logic, this simply denotes a piece of reasoning in favour of or against a position, without implying anger or the existence of a dispute or anything like that.
Logicians usually talk of 'arguments' or pieces of reasoning as having 'premises' and a 'conclusion'. The conclusion is what one is arguing for. The premises are what one is putting forward, the things one is saying, in favour of the conclusion. Arguments are frequently strung together in series, with the conclusion of one frequently forming one of the premises for the next.
Not everything that poses as an argument really is so. Often what is presented is just one statement after another. If there is no way of determining which are the premises and which is the conclusion, it is highly unlikely that an argument is being presented. The piece of language in question might still have persuasive power, it may be valuable as rhetoric, but does not count as a piece of reasoning and Logic has nothing to do with it one way or the other.
The first problem in the practice of Logic, then, is being able to determine whether there is an argument taking place, and if so, which are the premises and what’s the alleged conclusion, and if more than one argument how are they being all strung together.
This is the burden of most of the rest of Copi and Cohen, Chapter 1, 1.3 – 1.6. This is sometimes a lot of hard work. Even though it is, in theoretical terms only the beginning, it can be the most practically important part of logic.
For the rest of our unit, however, we will be dealing only with genuine argument, i.e. bits of language in which one can distinguish premises and conclusion.
An argument is said to be valid if its conclusion really does follow from its premises.
It is important to distinguish validity and another notion used by logicians called soundness. For a logician, 'soundness' is a stronger notion. A sound argument is a valid argument all of whose premises are true. Only sound arguments prove anything. That is to say, in order to prove anything, the premises have to be true and the conclusion must really follow from the premises (which latter = validity).
This takes us to the next step in our practice of logic in the various sciences and philosophy and common life. Of any argument we can now ask two equally important questions: 1) Does the conclusion really follow from those premises? Is the argument valid? And 2) Are the premises true? An argument in order to be a good argument, a correct argument, a sound argument, has to satisfy both questions. Once again, even if it is only a start for the logician, in practice this is absolutely crucial. Whole articles have been written in which all a person does is to distinguish the premises and conclusion of an argument and then ask and try to give an answer to each of these two questions.
Logic, technically speaking, does not have to do with the truth of our premises except where those premises happen to be logical truths or ‘tautologies’ as they are called by logicians. Technically speaking, Formal Logic deals only with the question whether the conclusions that we allege to follow from those premises really do follow. In this sense, logic is the theory of distinguishing between valid and invalid arguments. It does only part of the job in respect of the soundness of our arguments. It will tell us if our conclusions follow from our premises but, unless the premises themselves are logical truths, not very often, not whether our premises themselves are true. To find this out sometimes we will have to go outside and have a look.
= Copi and Cohen 1.2
The bits of language of which arguments are composed are commonly called 'statements' or 'propositions', occasionally 'sentences'. These in turn are either 'compound' = containing other statements as components, or 'simple', = not containing other statements as components. Some examples below.
Not every bit of living language is a statement or something capable of functioning in an argument for something. Statements or propositions are bits of language in which a truth claim is being made up front. As well as statements, language can be used to give commands or make requests, to make promises, to express emotions etc. These are regarded as 'appropriate' or 'inappropriate', 'sincere' or 'insincere' etc. rather than true or false.
Also, different bits of language can be used to make the same statement, i.e. say the same thing, to make the same claim: e.g. 'Mary loves John', 'John is loved by Mary'. And the same bit of language can be used in different contexts to make different statements: e.g. 'The dog chased the bird', when said of Jack Russell and when said of Pluto or Molly. What counts for logicians is the claims being made, no matter the language, and whether the claim made in the 'conclusion' really does follow from the claims made in the 'premises', the things we say to back it up.
Classes of statements/propositions
= Copi and Cohen, 1.2 and Copi and Cohen 7.7 and 7.8.
Logicians commonly classify arguments in terms of the kinds of statements they contain.
There are a number of different types of statement, of which the following are probably the most common:
Hypothetical or Conditional Propositions/Statements: of the form: "If A, (then) B", e.g. if the cat did it, the dog didn't do it. The first statement, after the 'if', is commonly called the ‘antecedent’; the statement after the 'then' is called the 'consequent'.
Disjunctive Propositions/Statements or Disjunctions: of the form, "Either A, or B (or C, or D etc.)", e.g. either the butler did the murder, or the murder was done by the cousin of the wife three times removed. The various statements are called 'disjuncts'.
Conjunctive Propositions, or Conjunctions, sometimes called 'joint assertions': of the form, "A and/but/although/nevertheless/however etc. B, etc.", when you are joining up two or more assertions, with or without any implied contrast of content, which latter logicians don't worry about. E.g. "The logic class is bright and it is also most co-operative and (they are) always kind to each other", "Even though the butler didn't actually do the murder, he bears some responsibility". The various statements are commonly called 'conjuncts'.
Negative Propositions or Negations: of the form, "It is not the case that...” E.g. it is not the case that the butler did it, the butler didn't do it.
These all count as 'COMPOUND' statements or propositions.
SIMPLE statements or propositions are most commonly what are called
Categorical Propositions/Statements, = the name given by logicians to the kind of statement made by the usual, run of the mill subject-verb-object, or subject-predicate sentences, e.g. the dog chased the bird, the cat sat on the mat, Peter has red hair, all lecturers are difficult to understand, some students are protesters.
There are three main varieties of Categorical statement, each of which may appear as affirmative or as negative (making six in all):
SINGULAR propositions: e.g. the dog chased the bird, Socrates was not a handsome man.
Universal Affirmative: of the form, "All S is P", e.g. all students are protesters, all lecturers are difficult to listen to.
Universal Negative: of the form, "No S is P", e.g. No politicians are honest people, No mice are welcome in the student common room.
Particular Affirmative: having the form, "Some S is P", e.g. Some middle-aged men are going bald, some trees have good fruit.
Particular Negative: of the form, "Some S is not P", e.g. Some books are not easy to read, some Pharisees were not bad people.
In order to show the 'logical' relations that
obtain between them, universal and particular categorical propositions have
been put in the form of a square, the so-called
All S is P (contraries) No S is P
Some S is P (sub-contraries) Some S is not P
Contradictory statements: if one is true, the other is false, and vice versa.
Contrary statement: can't both be true, maybe both false.
Sub-Contraries: can't both be false, may both be true.
Subalterns: if the universal statement is true, so is the particular; if the particular statement is false, so is the universal.
Try these out with examples of your own, to see if it works.
This is on the ‘extra’ list at the moment: we’ll come back to all this later when we do the Logic of the Categorical Syllogism. It belongs in Copi and Cohen Chapter 5.
Classes, Types or Forms of Argument
Arguments, for their part, as we’ve said, are typically classified in terms of the kinds or forms or types of statement they contain. Below are some of the more common forms of argument:
DISJUNCTIVE arguments: typically of the form,
Either A or B.
E.g. Either the butler did it, or the wife's cousin three times' removed.
The butler couldn't have done it.
So the wife’s cousin must have done it...
'MIXED' HYPOTHETICAL arguments (mixed: a mixture of hypothetical and categorical): typically of the two following forms, called after the names given to them by the medieval logicians:
Modus Ponens (= the mode of affirming), of the form,
If P then Q.
P (is so).
Therefore Q (is so).
E.g. If the premises are true and the conclusion follows from the premises, then the conclusion must also be true.
The premises are true and the conclusion does follow from these premises.
So the conclusion must also be true.
Modus Tollens (= the mode of denying), of the form,
If P then Q.
Therefore not P.
E.g. If it rained last night, then the footpath would be wet this morning.
The footpath was dry as a bone this morning. [which is to say, it wasn't wet at all!]
So it mustn't have rained last night.
'PURE' HYPOTHETICAL arguments (all statements hypotheticals): of the form,
If P then Q.
If Q then R.
Therefore, if P then R.
E.g. If the butler did it, the cook didn't do it.
If the cook didn't do it, the butler must have been helped by the assistant cook.
So, if the butler did it, he must have been helped by the assistant cook.
[However, the butler and the assistant cook are the worst of enemies.
If the butler and the assistant cook are the worst of enemies, then they couldn't have helped each other.
So the butler was not helped by the assistant cook.
So the butler didn't do it after all.
If the butler didn't do it, then either the cook did it or the house-keeper did it.
So either the cook or the housekeeper did it.]
But the housekeeper has a watertight alibi.
If the housekeeper has a watertight alibi, she couldn't have done it.
So the cook must have done it.
DILEMMAS: of various kinds, all consisting of a conjunction of hypothetical statements, followed by a disjunction, with conclusion either simple (Simple) or another disjunction (making for a Complex Dilemma). If the conclusion is affirmative, then the Dilemma is called 'Constructive', if negative then the Dilemma is called 'Destructive'. The following are the four major kinds of valid dilemma:
Complex Constructive Dilemma:
If A then B and if C then D.
Either A or C.
Therefore either B or D.
Complex Destructive Dilemma:
If A then B and if C then D.
Either not B or not D.
Therefore either not A or not C.
Simple Constructive Dilemma:
If A then B and if C then B.
Either A or C.
Simple Destructive Dilemma:
If A then B and if A then C.
Either not B or not C.
Therefore, not A.
For examples of Dilemmas, see attached.
Constructive dilemmas are complex variants of Modus Ponens, affirming antecedents, Destructive dilemmas are complex variants of Modus Tollens, denying consequents. In all cases one has to be careful not to do the opposite, i.e. affirm consequents and deny antecedents -- see later.
1. If I am rich I must worry about losing my wealth; if I am poor I must worry about making a living. But I must be either rich or poor. Hence I must always worry.
2. If I am rich I don’t have to worry about making a living; if I am poor I don’t have to worry about losing my wealth. But I must be either rich or poor. So I don’t have to worry.
3. If you were intelligent, you would see the worthlessness of your arguments; and if you were honest you would acknowledge yourself wrong. Either you don’t see that your arguments are worthless or you won’t acknowledge that you are wrong. So you are either wanting in intelligence or else you’re dishonest.
4. If I work I earn money; and if I am idle I enjoy myself. Either I work or I am idle. Therefore, either I earn money or I enjoy myself.
5. If I work I don’t enjoy myself; and if I am idle I don’t earn money. Either I work or I am idle. Therefore, either I don’t enjoy myself or I don’t earn money.
6. If students are fond of logic, they’ll need no exam to make them study; and if they dislike logic, an exam won’t do them any good. But any student is either fond of logic or else dislikes it. So an exam is either needless or of no avail.
7. If you’re pro-communist you’ll vote Labor because it’s the only sizable party anywhere near your position. And if you’re anti-communist you’ll vote Labor because they are the only party capable of taking the poor and needy out of the hands of the communists. You must either be pro-communist or anti-communist. Either way you ought to be voting Labor.
8. The gospel accounts must either agree or disagree. If they agree with each other, it’s because they’ve got together to make it up. If they disagree, then obviously they can’t be accepted as true. So the gospel accounts are to be regarded as untrustworthy.
9. The gospel accounts must either agree or disagree. If they agree then that’s probably because it happened that way. If they disagree then at least we can be assured they haven’t got together to make it up. So the gospel accounts probably can be regarded as trustworthy witnesses.
10. The universe is either self-explanatory or not self-explanatory. If the universe is self-explanatory, then the universe must be said to exist in its own right, in which case it must itself be the Absolute. If not self-explanatory, the universe must have an explanation outside itself which itself does not require an explanation. Either way an Absolute exists.
Kinds of Argument: Summary
Formal Fallacies (ie. Invalid)
Either P or Q
False Disjunctive Syllogism
Either P or Q (inclusive 'or')
Therefore not Q
Modus Ponens (mixed hypothetical 1)
If P then Q
Affirming the Consequent
If P then Q
Therefore P (try dogs and 4 legs)
Modus Tollens (mixed hypothetical 2)
If P then Q
Therefore not P.
Denying the Antecedent
If P then Q
Therefore not Q (once again, try dogs and 4 legs)
(Pure) Hypothetical Syllogism
If P then Q
If Q then R
Therefore if P then R
(Fallacy of Ambiguity:
If P then Q
If Q' then R
Therefore if P then R)
Dilemmas x 4
Constructive Dilemmas: (cf. M. P.)
Complex If P then Q and if R then S
Either P or R
Therefore either R or S
Simple If P then Q and also if R then Q
Either P or R.
Destructive Dilemmas: (cf. M.T.)
Complex If P then Q and if R then S
Either not Q or not S.
Therefore either not P or not R.
Simple If P then Q and if P then R.
Either not Q or not R.
Therefore not P.
Complex If P then Q and if R then S
Either Q or S.
Therefore either P or R
Simple If P then Q and if P then R
Either Q or R.
Complex If P then Q and if R then S
Either not P or not R
Therefore either not Q or not S.
Simple If P then Q and if R then Q.
Either not P or not R.
Therefore not Q.
Categorical Syllogisms x 15 or x 24 (out of 256)
Most common fallacy, probably, = 'undistributed middle term'
E.g. All dogs have 4 legs. All cats have 4 legs Therefore all dogs are cats.
All the above forms of argument apart from categorical syllogisms not yet treated contain at least one compound statement or proposition. The branch of logic which deals with this kind of argument, i.e. the logic of compound propositions, is commonly termed 'Propositional Calculus' or sometimes 'Sentential Calculus'. This part of logic deals essentially with the logic of 'connectives' between statements like 'and', 'or', 'not', 'if...then...", 'if and only if...'. Given the truth of the premises, and the meaning of the above connectives as relevant, what follows concerning the conclusion?
One common way of showing a long argument to be valid is to reduce it to a series of shorter arguments, each of which is known to be valid. You can try this for yourself, with the above example of cooks and butlers and assistant cooks. Alternatively, one could show that one or more of the arguments being used in a particular discourse is demonstratively invalid.
The Logic of the Categorical Syllogism/'Predicate Calculus'
There is another whole branch of logic dealing with arguments composed entirely of simple or 'categorical' statements or propositions, called the Logic of the CATEGORICAL SYLLOGISM.
A Syllogism = an argument consisting of two premises and a conclusion. E.g. all the argument forms already listed.
A Categorical Syllogism is a syllogism all of whose propositions are, or can be put into, categorical statements. Strictly speaking, Singular propositions need to be translated into Universal propositions, by such round-about techniques as "all people identical with Socrates are not handsome", "All dogs identical with Jack Russell are friendly". Most of the time, however, we don't worry about this. The branch of logic dealing with this kind of argument is nowadays more commonly called 'Predicate Logic' or 'Predicate Calculus', sometimes 'Quantification Theory'.
All mammals have lungs. of the form: All M is P.
Dogs are mammals. All S is M.
So dogs must have lungs. Therefore all S is P.
There are 255 other varieties or forms of categorical syllogism, only 24 of which are valid. But there is no need to go into this at this stage. We will confine ourselves to look at only some of the more obvious fallacies or mistakes.
[To continue our murder mystery, from above.
However, the housekeeper has a watertight alibi.
People with watertight alibis couldn't have done the deed.
Therefore the housekeeper didn't do it.
So the cook is the culprit.]
Other types of Logic:
In addition to Propositional Logic and Predicate Logic, there are a number of other kinds of logic, including:
The Logic of Relations: this deals with predicates like, …to the left of…, …is the mother of…, …is the sibling of…, etc. This can be looked at as a complication of predicate logic, where the predicates in question are two or three or more placed predicates.
Modal Logic: deals with the logic of 'modal' expressions such as 'necessary', 'possible' and 'impossible' and how these relate to each other. Modal Logic can get very complicated, and there are a lot of different systems.
Logic nowadays is likely to be done as part of pure Mathematics, though attempts to reduce one to the other have proved to be illusory. Indeed, certain limits have been demonstrated even within Logic: there is no decision procedure for relational or polyadic predicate logic, for example and that’s provably the case (called ‘Church’s Theorem). So even Logic can’t quite be done entirely by computers….
Deductive and Inductive Logic:
These three or four kinds of Logic all belong to Deductive Formal Logic.
Side by side with Deductive Logic there is the whole field of what used to be called Inductive Logic, namely the logic for developing general laws (whether 'probabilistic' or 'deterministic' - not just probabilistic), starting with experience of particular instances. To deal adequately with this would take us, however, into complicated discussions in the Philosophy of Science. Inductive Logic, once very much in the province of logicians, is nowadays more likely to be dealt with by mathematicians interested in probability and statistics on the one hand, and philosophers of science on the other, rather than by technical logicians. However, Copi and Cohen do have a go at doing something at least a little bit helpful in this regard (see 1.8, plus the whole of Part III).
Roughly, deductive logic has to do with relations of ideas or what follows given the meaning content specifying the logical operational force of words like "and", "or", "not", "If...then...", "if and only if", also "all", "some", "none"; also, but not so far noted, "is identical with", "is related to symmetrically/transitively/reflexively" ( = the logic of relations) "possible", "necessary", "contingent" ('modal logic'). In so far as whether it follows is a function of the very meaning of the words used or concepts, equivalently how they function logically given the meaning that they have, what follows, follows with absolute certainty. A valid argument in deductive logic is valid in all logically possible worlds.
Inductive logic, on the other hand, has to do with contingent relations between matters of fact and existence in this particular world, as based on experience. In so far as we are always extrapolating beyond experience, otherwise no value for prediction and control, it is always possible that further experience will falsify the general principles and laws and theories we may come up with. Conclusions follow only with a greater or lesser degree of certainty and in theory at least are always open to correction. We can distinguish at least two broad categories of argument:
All A's in our experience have been followed by B's.
Therefore all A's are B's.
In ??% of the cases in our experiences, A's have been followed by B's.
Therefore, the probability of A's being followed by B's is ??% (with a margin of error of !!%, given the size of the sample).
The difference between the two types has to do with the nature of the conclusion, a 'deterministic' versus a 'probabilistic' or statistical law. Either way, if we have any sense we will assent to the conclusion only with a greater or lesser degree of certainty, quite as much for deterministic as for probabilistic. The truth-value of our conclusions is contingent on the kind of world we happen to be in, the kind of knowers we happen to be in our little corner of the universe, and the way the cookie has crumbled in the history of the sciences so far.
There will be a brief, Historical introduction to the Philosophy of Science in Part D of this unit.
A Final Note: Validity depends of form: the use of 'counter-examples'
As is perhaps becoming clear, logicians determine validity by concentrating on the kind of argument being used. Arguments of one form may be valid, e.g. all arguments listed above. Arguments of a different form may be invalid: if one uses an invalid type of argument, one can't rely on one's conclusion.
For example, if a person argued along the lines of:
All P is M.
All S is M.
Therefore all S is P.
That this way of arguing is no good i.e. is invalid, that a person can't argue like this, can be seen by substituting S = Dogs, P = Cats and M = has four legs. In common language: just because two things share a common characteristic, doesn't make them the same thing.
In general: VALIDITY IS A FUNCTION OF FORM, or of kind of argument being used, rather than what the argument happens to be about. In logic, as in ethics, what is good for the goose is good for the gander. And logic is the study of the forms into which our arguments must be capable of being put in order to be taken as valid – not necessarily how we actually do think, how we actually do get from A to B, which is rather a province for Psychology and sometimes the Philosophy of Science.
The technique just used is an example of probably the most common technique in practical logic for showing that an argument is invalid, that is, the use of a counter-example. "You may as well say, All cats have 4 legs, all dogs have 4 legs, therefore all dogs are cats." This is just as legitimate and rhetorically much more powerful than saying, "you have just committed the fallacy of undistributed middle term". Technically, a counter-example is the use of an obviously invalid argument, of the same kind/having the same form/of the same variety as the argument whose invalidity you may wish to allege. E.g., in our case, Christians oppose the military regime, Communists oppose the military regime, therefore all Christians are Communists. To repeat: in logic, as in ethics, the principle of universalizability obtains, what is good for the goose has to be good for the gander.
This is enough by way of introduction to the study of 'Formal' logic. Enough to give you the idea of what logic is about and the kinds of things that logicians go on with. Rather than look any more at this point at good type arguments, we will concentrate in the next section on the more common bad types, common 'fallacies' or ways of going wrong.
(I) Various grounds for contesting arguments:
Arguments or bits of reasoning, our own and others, may be refuted or critiqued or otherwise regarded as incorrect on a number of grounds, including the following.
1. Unsoundness, because not having true premises.
As already noted, logicians usually talk of 'arguments' or pieces of reasoning as having 'premises' and a 'conclusion'. The conclusion is what one is arguing for. The premises are what you are putting forward in favour of the conclusion. Arguments are frequently strung together in series, with the conclusion of one frequently forming one of the premises for the next.
To repeat, a sound argument is a valid argument all of whose premises are true. Only sound arguments prove anything. That is to say, in order to prove anything, the premises have to be true and the conclusion must really follow from the premises (which latter = validity).
Alleging unsoundness in common parlance usually amounts to claiming that one or more of the premises is actually false. This may be the case whether or not the conclusion follows from the premises. But, as the word is used by logicians, an argument also counts as unsound if the conclusion does not in fact follow, even if all the premises are true.
One may also propose that the truth of the premises has not yet been substantiated, whether or not the conclusion follows from these premises. In such a case, one is alleging that the soundness of the argument has yet to be established. Until the premises are established, the conclusion hasn't yet been proved, no matter how valid the argument. This is not sufficient to show that the conclusion itself is false: to allege that at this stage would itself be a fallacy.
2. Invalidity: your/my argument is not valid. Whether or not the premises are true, the conclusion does not follow from the premises, so once again I/you or your opponent is not proving anything.
As already noted, sometimes the most expeditious and certainly the most rhetorically effective method of pointing out invalidity is to utilize a 'counter-example', which is to say an obviously invalid argument which has the same form, is the same species of argument as the argument you are attacking. For example, if someone were to say,
The communist party opposes apartheid.
One could refute this by something like: it doesn't necessarily follow. Just because two things share a common characteristic doesn't make them the same: you may as well say,
Dogs have four legs.
Cats have four legs.
Therefore dogs are cats.
This is a good, general method if you don't know the name of the actual fallacy or fallacies being committed, and even if you do it is a good way of ramming the point home. A 'fallacy' = a mistake in reasoning. Some of the more psychologically persuasive mistakes have been given names, for the sake of easier detection and avoidance. See below, last item on the agenda, for a short list of 'formal fallacies', i.e. fallacies whose psychological persuasiveness derives from an apparent likeness to valid form of argument. The fallacy committed in the above is the fallacy of 'Undistributed Middle Term'.
Fallacies can be divided into two broad classes,
Formal fallacies: mistakes whose psychological persuasiveness derives from similarity to valid forms of argument; and
Informal fallacies: mistakes in reasoning that have some other source. Informal fallacies themselves come in two main kinds:
Fallacies of Ambiguity: where the mistake in reasoning arises from some ambiguity of terms or phrasing in the course of the argument; and
Fallacies of Relevance/Irrelevance: where the argument is, strictly speaking irrelevant to the truth of the conclusion.
There are a number of special cases of all of the above, sufficiently common to be worth a look at. But there are also a number of more subtle grounds for rejecting or criticizing arguments. For the rest of this section we will concentrate on some of the more subtle grounds, returning to a list of informal and formal fallacies in the next section.
3. Incoherence or Inconsistency or Self-Contradiction: I am/ you are contradicting yourself.
Contradictory statements cannot both be true. It cannot be the case that a person who makes contradictory claims in the course of an argument is mounting a sound argument: there is no way, in such a case, that all the premises are or could be true. All one has to do to show unsoundness in such a case is to point out the fact of the contradiction. You yourself don't have to choose which statement is mistaken.
(For certain technical reasons, arguments with contradictory premises count as valid: if it is impossible for all the premises to be true, it is impossible for all the premises to be true and the conclusion false. But in so far as it is impossible for all the premises to be true, such an argument never actually proves anything. Such an argument maybe valid but it cannot be sound. So this strange quirk of logic does not matter.)
If contradictions occur not within one argument but between premises or conclusion of one argument in one part of the text and premises or conclusion of another argument by the same author: the author has to choose which argument he or she wants to keep in his or her repertoire. One or the other argument has to be acknowledged as either invalid or unsound.
Either way, if you can point out a contradiction, you have the person in serious trouble. The same goes if you yourself are involved in a self-contradiction.
Claims that are so opposed that they cannot both be true may occur within a few pages of text or sometimes within a paragraph or two, in which case they are easy to detect. Except that sometimes the contradiction results only when the logical implications of what has been said are drawn out.
Sometimes the contradiction may occur, however, across a wide section of text: the author appears to have forgotten claims that s/he made in a previous chapter or a few chapters ago. His or her own thought might have moved on in the course of the research, without this being reflected in text written at different times. This is also something you/I have to worry about yourself/myself, particularly when writing a long essay or a thesis or dissertation. It helps to have someone else fairly sharp read ones text for content, therefore, and not only as a proofreader.
4. Reductio Ad Absurdam: showing that the argument, when pushed, reduces to absurdities. In so far as it leads to absurdities, your argument is that there has to be something wrong with it. Alternatively, you yourself or I myself could be inadvertently committed to absurdities which you/I wouldn't entertain in a thousand moons.
For example: "Arguing along such lines (as Descartes in his First Meditation for example) you would soon end up believing you were the only person in the world. But if that is the case, why bother arguing at all, and where do you get the language from in order to mount the argument?"
For example: "If you want God to be directly responsible for absolutely everything, then you are going to have God directly responsible for all the evils in the world. In which case you would need to posit God as a criminal or a heartless puppet master. But if that is so, why believe in God at all?"
The strongest form of the 'reductio ad absurdam' is where we show that accepting a certain point of view will lead to an out and out contradiction. That point of view must therefore be mistaken. This way of argument. As you probably realize from elsewhere, can also be put to positive use: showing the contradictory of the position you want to espouse leads to self-contradictions. If that is false, then your point of view must be true. This is often used in mathematics and sometimes makes for a much shorter proof than proving something directly. It is an 'indirect proof, by reductio ad absurdam'.
5. Begging the Question/Arguing in a Circle (Petitio Principii): surreptitiously assuming what you are endeavouring to show or prove. This is sometimes disguised by a different manner of phrasing. Such arguments are trivially valid, but incapable of advancing knowledge.
(Listed in Copi and Cohen under the heading, Fallacies of Presumption.)
Examples: I.Q. tests measure intelligence. What is intelligence? Well, it is what I.Q. tests measure!
The Bible is inspired. How do I know? Because the Bible tells me so. How do I know that? Because the Bible is inspired.
What is Physics about? It is about Matter-Energy. What is Matter-Energy? It is what Physics is about.
What is Matter? It is what takes up space. But what is it that takes up space? Oh, that's Matter.
A particular form of this is the so-called Complex Question, supposedly well beloved of lawyers in court. E.g. "When did you stop beating your wife?" "When did you start taking drugs?" The first question assumes that the accused has been beating his wife, and the second that the person has been taking drugs. The way to answer is to 'split the question': I've never beaten my wife, so the question of when I stopped does not arise, I've never taken drugs, so the question of when I started taking them does not arise. Such an issue may arise in academia also, as when someone starts by asking, "Why is this position so obviously false?"
6. Self-Reflexive Invalidity: you/I are undermining yourself/myself.
Formally stated, the charge here is that the argument is self-undermining: when applied to itself, it leads to its own self-destruction. The claim made by the argument does not apply to itself: technically, it is not "self-reflexively valid".
Example: Logical Positivism. In the 1930's a group of philosophers called Logical Positivists argued that the only meaningful statements are statements which are either empirically verifiable or analytically true, i.e. ones which can be demonstrated by either science or pure logic. The positivists were embarrassed when it was asked of them whether their claim, in itself, was to be classified as being empirical or analytic truth. According to its own criteria, the positivist claim is itself meaningless: it is not a truth of logic, but neither is it something capable of empirical verification. This was indeed quite embarrassing.
Example: Trying to doubt your own existence. If I doubt that I am, there is still the doubting. I doubt, therefore I think, I think, therefore I am.
Example: The claim that there is no truth. If it is true, then of course there is a truth and so it is not true. If it is not true, then once again there is truth. The making of the claim undermines the claim that is made.
Some varieties of the 'hermeneutic of suspicion' have to watch themselves on this point. If 'everything is ideology', then what of the statement that 'everything is ideology'?
7. Unfalsifiability, and Conspiracy Theory: your/my statement is unfalsifiable, you've stacked the decks, you/I am putting up what amounts to a conspiracy theory. This one we owe to the work of Karl Popper in Philosophy of Science.
This is what happens when a supposedly empirically based theory builds up a protective wall around itself. It may do this either positively and aggressively by absorbing all attempts at refutation as but further demonstrations that the theory is true ('conspiracy theory'), or by having a way of construing the phenomena alleged so that they are always at least consistent with the theory ('unfalsifiability). This is the case with some versions of Psychoanalysis and some versions of Marxism and other such all embracing ideologies and closed systems, including some kinds of fundamentalism and perhaps some kinds of contemporary 'economic rationalism'??. No matter what one says, the advocate of the theory always has a way of construing the phenomena in his or her own terms, in a way with conforms to their system, and oftentimes even supports it.
Example: all behaviour is a way of asserting power.
reply: that is just not true!
reply to reply: ah, there you go again, asserting your power over against mine.
If a theory is going to be right, no matter what the facts, then the facts can't reasonably be taken as giving support to the theory either. The game has been rigged in advance: heads I win, tails you lose.
Claims in theology need to be falsifiable also, though not necessarily in a direct and straightforward manner. If no experience whatsoever could be claimed as telling against our faith, then our experience can't be claimed as telling in favour of it either. No risk, no gain.
8. Infinite Regress: I/you are involved in an infinite regress.
A claim or argument can be such that instead of leading towards a satisfactory conclusion, it simply leads us down the garden path to more and more questions and problems with no hope of any answers forthcoming. You explain nothing, if what you rely on requires just as much explanation as what you were setting out to explain in the first place.
A very ancient example: the world rests on the back of a very large elephant/a very large turtle. What does the elephant or turtle rest on? An even larger elephant/turtle!
A slightly less ancient example (Aristotle): if everything which is changing requires an external cause, then God can't be subject to change. Otherwise you would be involved in an infinite regress.
9. Irrelevance or Triviality: theoretical and practical. What I/you claim is irrelevant in the sense of not really dealing with the issue, or else irrelevant for all practical purposes.
Here one is alleging that the conclusions arrived at when looked at closely have little or nothing to do with the original questions asked. Or that the solution put forward doesn't in fact solve the problem. (Theoretical triviality, often called 'Ignoratio Elenchi', ignoring the question.) Example: you have proved that such and such, e.g. pleasure, is 'desirable' in the sense of capable of being desired. You have not yet proved that it is 'desirable' in the sense of 'ought to be desired'. There are lots of things that people are capable of desiring but which positively are not desirable in the latter sense.
Alternatively, one might be alleging that the claims made, no matter what their theoretical value, are not such as can be or will be acted upon in practical everyday life. This is sometimes said about the theory of universal determinism. One can be a universal determinist/double predestinationist in ones philosophical or theological closet. However, when put in a situation of choice, one is going to act as if you were free and as if the situation were truly open. Similarly with theories about the unreality or illusory character of the external world. A person who professes this ought to be as prepared to leave his or her room via the window, as via the door, they are both equally unreal or illusory after all. But in fact of course you leave via the door. As the philosopher David Hume says about certain kinds of philosophers, when they leave their philosophical closets they mingle with the rest of mankind in the exploded opinions of the vulgar. (T I, IV, IV, p. 216.) This can sometimes happen with sophisticated theologians also.
Having looked at these rather special ways of going wrong, we now turn our attention to some of the more common fallacies, formal and informal.
2. COMMON WAYS OF GOING WRONG and what to do about them (cont'd)
(II) Some common logical 'fallacies', informal and formal, to guard against:
Once again, these are things to be avoided oneself, as well as things sometimes found in the argumentation of other people.
(A) Formal Fallacies: psychologically persuasive mistakes in reasoning which get their persuasive character via specious resemblance to valid forms of argument. Some of these also have been given names:
False Disjunctive Syllogism:
Either A or B (where ‘or’ is meant inclusively, i.e. not in fact ruling out the possibility that both happen to be true)
Therefore not B.
The Fallacy of Affirming the Consequent:
If it's a dog then it has four legs.
It has four legs.
Therefore it's a dog.
The Fallacy of Denying the Antecedent:
If it's a dog then it has four legs.
It's not a dog.
Therefore it doesn't have four legs.
'Antecedent' = the first part of an "if...then...", what comes before the 'then'; 'consequent' = what comes after the 'then'.
These fallacies get their persuasive character via specious resemblance to the perfectly valid forms of argument commonly known as Modus Ponens: If 'p' then 'q'. 'p' is so. Therefore 'q'; and Modus Tollens: If 'p' then 'q'. 'q' is not so. Therefore 'p' is not so.
Corresponding to these two fallacies are various kinds of false dilemmas, specifically:
False Constructive Dilemma: = affirming consequents:
If A then B and if C then D. If A then B and if A then C.
Either B or D. Either B or C.
Therefore either A or C. Therefore A.
False Destructive Dilemmas: = denying antecedents:
If A then B and if C then D. If A then B and if C then B.
Either not A or not C. Not a or not C.
Therefore either not B or not D. Therefore not B.
Another very common formal fallacy has already been mentioned, the Fallacy of an Undistributed Middle Term:
The Catholic Church opposes the G.S.T.
The Socialists oppose the G.S.T.
Therefore the Catholic Church is Socialist.
The 'middle term' is the term by means of which you link the other two terms. It is, equivalently, the term which occurs in each of the premises but not in the conclusion. In order to be sure of a link, in order to verify a genuine connection, at least one of the premises has to be talking about the whole of that class: thus 'distributed' versus 'undistributed' = talking only about part of the class. The problem with the above argument is that the Catholic Church and the Socialists could be two entirely unrelated parts of the large class of people who oppose the G.S.T.
All dogs have legs.
All cats have legs.
Therefore all dogs are cats.
(B) Informal Fallacies: Fallacies of Relevance: (or of Irrelevance)
Argumentum Ad Hominem: playing the person, not the ball.
It is not at all uncommon to try to reject someone's argument, not by showing its weakness or the falsity of its premises, but by ignoring it altogether and instead mounting a personal attack on the argument's propounder. Suppose, for example, that I rejected a strongly feminist conclusion just because it came from a committed feminist. "She is a feminist after all. What do you expect?" Then my rejection would be fallacious, as I have not shown anything at all to be wrong with the arguments themselves. Also, "what do you expect of a Catholic/Protestant/Jew?" etc. The fallacy of argumentum ad hominem is that of arguing against the person instead of his/her arguments. In common parlance, playing the person, not the ball.
The Genetic Fallacy: a more sophisticated variety of playing the person, not the ball.
Here we reject a person's belief, not because we find their reasons inadequate, but because we note that they are in some way caused to hold that belief. It is observed, for example, that the particular brand of religious belief that an individual has is nearly always that which s/he was brought up with. Catholic children rarely end up as Presbyterian adults, and Buddhists and Moslems tend to stay Buddhist and Moslem all their lives. To hold that a person's belief can be dismissed simply because s/he was caused to believe it, however, is a fallacy. It is called the genetic fallacy since it confuses the genesis or origin of a person's belief with the reasonableness of that belief in itself.
(What used to be called) The Straw Man Fallacy
By this is usually meant: attacking a caricature of another position rather than taking on one of its stronger forms or the more nuanced form actually proposed by its better proponents. A person sets up a 'straw man' and then knocks it over and pretend they've done great things. Possible response: what you attack is a caricature of my position. My position is much more nuanced than that and is not susceptible to the objections you raise against the caricature.
Various other fallacies of (lack of) relevance or irrelevance, many of which have been given names for the sake of easier detection and avoidance:
Argumentum ad Baculum (appeal to the stick): an appeal to force or the implied threat of force, in order to get a conclusion accepted. This is frequently extended to include any threat of unhappy consequences being used to have a conclusion accepted.
Argumentum ad Misericordiam (appeal to pity): often used in court. It may be a legitimate move when the question arises as to sentence to be imposed on someone found guilty, but is strictly speaking not relevant to the question as to the guilt or otherwise of the alleged perpetrator.
Argumentum ad Verecundiam (appeal to authority): that is, an appeal to authority outside the scope of their expertise, or in a situation or at a level of scholarly debate where you should be arguing the matter for yourself. According to the scholastics, an appeal to authority is the weakest form of argument, even within the person's field of expertise. Of course, it does lend weight to your/my position, to have other bright people agreeing with one, provided they are acknowledged authorities in your field, as long as one is also arguing it for oneself.
Argumentum ad Crumenam: an appeal to the hip-pocket nerve, when used in order to gain acceptance of a claim or all too convenient theory of some kind.
Argumentum ad Fidem: appeals to believing for no reason at all. Making a virtue of the irrationality of the position. You just have to believe this. Trust me.
Argumentum ad Populum: an "appeal to the people", usually used for appeals to mass emotions, as in using pretty/handsome persons in advertisements promoting cars or tennis shoes or toothpaste; or snob appeal.
Argumentum ad Quietem: an appeal to 'leave things as they are', not to rock the boat.
These are all ways of avoiding the trouble of actually having to mount a logical argument or to collect empirical evidence or data or whatever.
Once again, see Copi for more.
C. Informal Fallacies: The Fallacy/Fallacies of Ambiguity
This refers to a class of cases where an argument seems plausible only because an ambiguity of some sort is being traded on. Once the ambiguity is realized, the argument is seen to be a sham. For example, showing that something is desired in actual fact; that therefore it is 'desirable' in the sense of 'capable of being desired'. And then going on to pretend that you have shown that it is 'desirable' in the sense of 'ought to be desired'. For example: everyone desires happiness. So we ought all to desire the greatest happiness of the greatest number.
Sometimes a distinction is made here between mistakes arising from
Equivocation or Ambiguity of terms: where there is a subtle shift from one sense of a term to another sense in the course of an argument; and mistakes arising from
Amphiboly or Ambiguity of Phrasing: e.g. the duke yet lives, that Henry shall depose, you should not speak evil of your friends etc.
See Copi for more on this.
D. Some other miscellaneous fallacies
Fallacy of Rejecting the Conclusion of a Bad Argument
This rather subtle fallacy involves rejecting a conclusion solely on the grounds that a particular argument being used to support it is a poor one. For all you or I know, the conclusion might still be true, and be susceptible of proof by much better arguments. In order to be able to claim it as actually false one has to mount a positive argument against it, or alternatively a good argument in favour of a contrary, incompatible position.
On the other hand, if a person does manage to exhaustively and conclusively destroy all plausible arguments in favour of a position, one may be able to claim this position as not such as reasonable people should any longer accept in the sense of regarding it as positively true - without forcing them to acknowledge that it is actually false.
Argumentum ad Ignorantiam
This fallacy (the Appeal to Ignorance) is a more generalized version of the previous one. It involves claiming a position as false because it hasn't been proved true, or true because it hasn't been proved false. If there is at present insufficient evidence to decide in either direction, all that one can claim is that the matter has yet to be decided, one way or the other. The status of paranormal phenomena, telepathy, telekenesis, etc. might perhaps be in this class??
The Fallacy of Requiring Unattainable Standards in Argument
This is a fallacy beloved of tobacco companies. In mathematics the demand for full rigorous proof is reasonable. Rarely outside. In the words of Joseph Butler, "probability is the very guide of life". If a person is arguing on the causes of lung cancer or how to reduce inflation or increase employment, the best one can do is provide some measure of support, well below conclusive, for your views. It is a fallacy to dismiss an argument for not being quite conclusive if, by the nature of the subject being discussed, it is not reasonable to expect conclusive proof. Doubt based upon the lack of conclusiveness is in many cases unreasonable doubt, and commits the fallacy of requiring unattainable standards. From Aristotle: "it is a mark of the trained mind never to expect more precision in the treatment of any subject than the nature of that subject permits." (Nichomachean Ethics, Book I, 3.)
Confusion in respect of Necessary and Sufficient Conditions and Relevant Factors:
'Necessary condition', 'sufficient condition', 'necessary and sufficient conditions' are terms well beloved of certain kinds of philosophers, though sometimes they make the mistake of thinking that because a condition is neither necessary nor sufficient it is in no way relevant. Conditions can be necessary but not sufficient, sufficient but not necessary, both necessary and sufficient or neither necessary nor sufficient but still relevant. Of course, factors or conditions can also be entirely irrelevant, in no way connected.
Necessary condition (conditio sine qua non) but not sufficient: e.g. clouds for rain. You need clouds in order to have rain (thus 'necessary') but just because you have clouds doesn't mean you will have rain. In order for a person to be elected to parliament, it is necessary for that person to be an Australian citizen; but only a very small number of Australian citizens are elected to parliament, so this is obviously not a sufficient condition.
Sufficient condition (but not necessary): being shot three times in the head (usually) suffices for you to be dead, but there are lots of other ways of getting dead. In order for an Australian citizen over the age of 21 who is not a convicted criminal in jail, to be elected to parliament, it suffices that they should get a majority of first preference votes in a general election. However, they might also get into parliament on the preferences of minor parties, even if they do not have a majority of first preference votes. So it is not actually necessary to get a majority of first preference votes.
Necessary and Sufficient Condition(s): no brain activity with respect to death. All the conditions specified above taken together for getting into parliament, including both ways of getting elected, plus the possibility of being appointed by state parliaments in case of the death or retirement of a sitting senator, plus the possibility of a by-election.
Neither necessary nor sufficient but still Relevant: smoking with respect to cancer: people can get cancer without smoking and can smoke without getting cancer, but smoking it would appear does still dispose one to get cancer, it is one of the relevant factors. Another example: having the same name with respect to being the same person. A person can have the same name as someone else and yet be a different person; you can be the same person and yet have a different name, e.g. if you changed your name. Even so, whether or not the name is the same is one of the criteria we use in deciding in particular cases whether it is the same person.
Hasty Generalization or the Inductive Fallacy
Induction is argument from the particular to the general, e.g. from particular experienced cases to general or universal principles or statements about a whole set of phenomena. It is a fallacy to infer too much from too little. For example, I cannot infer anything substantial about Australian religious education out of a case study of just one local school's religious education program. Though this might well give me some clues as to hypotheses to explore with further case studies.
Hastily inferring Causation out of Correlation
A statistical correlation between two factors x and y does not by itself prove that x caused y. The causal relation could well go the other way. Indeed, it is quite possible, and quite often the case, that x and y are both caused by a third, more basic factor, z. For example, a person might find a statistical correlation between the level of alcoholism and the level of poverty among indigenous peoples. A well-to-do person might jump to the convenient conclusion that alcoholism causes the poverty, thereby absolving him or her of blame; whereas the truth of the matter is that both the alcoholism and the poverty are caused by the dispossession of the indigenous people, the destruction of their culture, religion and language and of the economic base of their pre-colonial existence. Enough to drive anyone to drink! Compare Rousseau's attack on the justification for slavery on the grounds that slaves have a servile nature. Slaves have a servile nature because they have been made and are treated as slaves: this is one of the effects of enslavement. You ought not to regard an effect on people of an evil treatment as a reason for justifying that treatment. (Jean-Jacques Rousseau, The Social Contract, Book I, Chapter II.) Which is to say, there are usually good logical and empirical reasons for not blaming victims, as well as good ethical reasons.
Following this look at some of the things that can go wrong, we turn now to a particular kind of Formal Logic, namely the Formal Logic of the Categorical Syllogism. This was invented by Aristotle and perfected in the Middle Ages. Together with the Geometry of Euclid, it counts as one of the great achievements of ancient thought. It was the dominant system of logic in its domain until late in the 19th Century, and is frequently called the Traditional Logic of the Categorical Syllogism. It still remains valid within its limited but valuable scope of application, provided certain assumptions made be noted.