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What is Philosophy Chapter 2by Richard Thompson

Logic(last edited on 29 January 2017)

In this chapter I outline the basics of Logic; there is more about Logic in Chapters 4 and 5.

I’m not writing a logic text book, so I shall discuss the technicalities of formal logic only where they are needed to follow the discussion of the comments various philosophers have made about logic. I learnt the basics of formal logic from W. V. Quine’s Methods Of Logic, and I have never come upon a better introduction, though of course, having read one introduction, I’ve never done more than glance at others. William and Martha Kneale The Development Of Logic is an excellent historical survey, which also explains the technicalities clearly.

Inferences, Propositions and Entailment

Logic is the study of the validity of inferences. It tells us what follows from what. Formal Logic gives precise rules that ensure the validity of any inference that satisfies them.

I’ll start by introducing some terminology. An inference proceeds from a starting point to an end point. We need a word for the types of entity that can feature in an inference. The one most commonly used is ‘proposition’. A proposition is some sort of claim that can be either true of false. Some logicians prefer to talk of sentences, on the grounds that that gives us a definite subject for discussion. The supporters of ‘proposition’ retort that the meanings of words can change, words can be ambiguous, the same sentence can mean different things on different occasions, and a variety of different sentences can be used to make the same claim. Also a sentence belongs to a particular language, while logic studies ideas that are independent of language. A proposition may be thought of as what a sentence means on a particular occasion, or what the user is trying to put across when they use a sentence. W.V. Quine and his supporters, who preferred to talk of sentences, considered it impossible to define ‘meaning’ or ‘proposition’ independently of a sentence. ‘statement’ has sometimes been adopted as a compromise, a ‘cowardly’ policy in the opinion of Quine though ‘statement’ has the advantage of suggesting a particular utterance made by a particular person at a particular place and time.

I’ve decided to use ‘proposition’. I’ll discuss the arguments for and against the existence of propositions in Chapter 5, but shall say no more on the matter in this chapter.

So the central idea in Logic is that of inference from one proposition to another. The propositions from which an inference begins are called the premisses, and the proposition at which an inference arrives is called its

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conclusion. A valid inference is one in which the truth of the conclusion is guaranteed if the premisses are true. We say that P entails Q, if P and Q are propositions such that, if P is true, Q must also be true, so that proceeding from P to Q is a valid inference. The word imply is often used instead of entail, but as we shall see later Russell and Whitehead confused the issue by using material implication for a much weaker relation, provoking G. E. Moore to introduce ‘entail’.

Formal logic studies patterns of argument such that any argument conforming to the pattern is valid.

A particular argument can usually be fitted into several different patterns, but to establish it’s validity we need only point to one valid pattern, so when we formalise the propositions of an argument to demonstrate its validity we need not try to capture their full complexity. It suffices to capture sufficient of the content to validate the argument. For instance suppose someone argued thus:

All members of the golf club play either tennis or bridge, Some members of the choir play neither tennis nor bridge, Therefore some members of the choir do not belong to the golf club.

That is an example of the pattern:

Every A is either B or C, Some D are neither B nor C, Therefore some D are not A.

(A = member of the golf club, B = tennis player, C = bridge player, D = member of the choir)

However, the argument also fits the simpler pattern: All P is Q, Some R are not Q, therefore some R are not P, which also suffices to establish its validity. (P = member of the golf club, Q = player of either golf or bridge, R = member of the choir)

We should usually prefer to refer to the simpler pattern when defending the inference in such a case.

One misconception must be removed at the outset. Logic concerns valid arguments, not good arguments, in the sense that a good argument is one that gives someone a good reason for believing its conclusion. A good argument (some people prefer ’sound argument‘) should be valid, but a valid argument may not be a good one. ‘P entails Q’ is only a good reason for a person A to believe Q, if A both realises that P entails Q and also has a good reason to believe P.

‘Simon has a pet cat’ entails ‘Simon has a pet’ is valid, but it does not provide a good reason for believing that Simon has a pet if Simon actually

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does not have a cat.

What is a good reason for one person to believe something may not be a good reason for someone else. In particular, if someone believes that P is false ‘P entails Q’ is not a good reason for him to believe Q. ‘P entails Q’ is said to be a ‘good ’argument, when P is true, and ‘P entails Q’ is valid.

To identify all good arguments would require us to know the truth of all true propositions, so that to include the identification of good arguments in Logic would require that Logic include the whole of knowledge. Aristotle thought that it did, but today there are few who would agree.

Although the mere validity of an argument does not guarantee the truth of its conclusion it does not follow that the study of validity is pointless, for an argument that is not valid is not a good reason for anyone to believe its conclusion.

Aristotelian Logic

The first recorded study of formal logic was by Aristotle who described the logic of propositions of four types, namely those that conformed to one the forms: “All S is P”, “Some S is P”, “No S is P” or “Some S is not P”. The broad outlines of Aristotelian logic were preserved right into the nineteenth century, but it was elaborated in the middle ages so I shall discuss that slightly extended form of the logic, and I shall also use some of the medieval notation.

Aristotle’s logic began with the examination of the logical relations between pairs of propositions, and then used that as the basis for considering more elaborate arguments involving larger numbers of propositions.

The medieval logicians referred to the four types of Aristotelian proposition as

A: All S is P, example: all mice are mammalsI: Some S is P, example: some atheists are vegetariansE: No S is P, example: no razor blades are made of chocolateO: Some S is not P: example, some birds cannot fly

A and I were chosen because they are first two vowels in affirmo and E and O because they are the vowels in nego. S and P are called the terms of the propositions.

At first sight this classification allows eight possible propositions involving any two terms S and P, namely S a P, S i P, S e P, S o P, P a S, P e S, P i S, and P o S, however S i P, (Some S is P), is equivalent to P i S, (some P is S), and S e P, (no S is P), is equivalent to P e S, (no P is S), For example ‘some birds candot fly is equivalent to ‘some creatures that cannot fly are birds’ and ‘some

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mice like chocolate’ is equivalent to ‘some creatures that like chocolate are mice. so Therefore we need consider only six distinct propositions.

Pairs of propositions may be related in one or another of several ways.

Contradictories Two propositions are contradictories when the truth of either one is equivalent to the falsehood of the other that so that one and only one is true. Contradictory pairs are:

S a P (all S is P) and S o P (some S is not P), eg. ‘all cats like milk’ and ‘some cats do not like milk’ are mutually contradictory.

S e P (no S is P) and S i P ( some S is P) eg. ‘no toadstool is edible’ and ‘some toadstools are edible’ are mutually contradictory.

Contraries two propositions are contraries when they cannot both be true but can in some circumstances both be false. Contrary pairs are:

S a P and S e P, and contraries and so are P a S and P e S, eg. ‘all members of the Chess Club play Golf’ and ‘no members of the Chess Club play Golf’. The two propositions cannot both be true, but if some members of the Chess Club play golf, but some do not, both the propositions would be false.

People sometimes confuse contraries with contradictories

Subalternation when one proposition entails another, but not vice versa.

S a P entails S i P, e.g. ‘all birds lay eggs’ entails ‘some birds lay eggs’ , but not vice versa.

and S e P entails S o P. e.g. ‘no freemasons are cannibals’ entails ‘some freemasons are not cannibals’ but not vice versa.

In those relations the universal propositions were referred to as the subalternants and the particular propositions they imply as the subalternate or the subaltern.

Both those examples raise a problem about existence, which I discuss below.

Subcontraries are pairs of propositions that cannot both be false, but might both be true.

The i and o propositions are subcontraries, since ‘Some S is P’ and ‘Some S is not P’ might both be true, but cannot both be false. eg. ‘some solicitors are freemasons’ and ‘some solicitors are not freemasons’ could both be true,

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but they could not both be false, because if it were false that some solicitors are not free masons, then all solicitors are freemasons.

Existential Import of Universal Propositions The forgoing discussion of the four Aristotelian types of proposition

assumes that ‘all implies some’ that is that ‘All S is P’ implies ‘Some S is P’ and ‘All S is not P’ implies ‘Some S is not P’ That assumption is problematical and gave rise to a good deal of debate among logicians. For sometimes we assert universal generalisations without any commitment to existence.

For instance if we explained ‘unicorn’ by saying ‘Unicorn’ means ’quadruped mammal resembling a horse but with a single horn projecting from the middle of its forehead’ we could confidently assert ‘any unicorn has four legs’ but should not think ourselves thereby committed to the existence of any unicorns, so we should reject the inference from ‘all unicorn have four legs’ to ‘some unicorns have four legs‘ since that involves there actually being some unicorns.

Furthermore, since ‘All S is not P’ is held to be equivalent to ‘All P is not S’ it appears that S e P not only entails that there are S, but also entails that there are P, so that ‘No animals are unicorns’ entails ‘No unicorns are animals. If we allow that to entail ‘some unicorns are not animals‘ we should be committed to the existence of unicorns at the same time we assert that there are none.

The matter is more easily discussed with the help of modern logical notation so I shall defer further discussion, except to say that provided the terms in all the propositions do have references, the traditional logic never leads from true premisses to false conclusions.

Having examined individual propositions and their logical relations, Aristotle turned his attention to syllogisms, in which two propositions entail a third. The first two propositions were called the premisses of the syllogism, and the proposition they jointly entail was called the conclusion.

For example(1) All Mammals are vertebrates, (2) All Elephants are mammals, therefore (3) All Elephants are vertebrates

That is a valid inference because it is of the form “All S is P, All Q are S, therefore All Q are P”

Aristotle and his medieval followers developed an elaborate theory of syllogistic inference, which I discuss in Appendix 1.

It was eventually realised that Aristotelian Logic can be illustrated by Page 5


diagrams. In the eighteenth century Euler introduced diagrams in which each of the classes involved is represented by a circle. He distinguished five cases.

Case 1 has the two classes the same, as would be the case if A = human beings, and B = rational animals.

Case 2 has the two classes entirely distinct, as if A = stars, and B = wheelbarrows.

In Case 3 all A are B but not vice versa, as if A were fish and B were vertebrates.

Case 4 is like Case 3 with the positions of A and B reversedCase 5 has A and B overlapping with neither wholly included in the other,

as if A were tennis players and B were dentists.

In the nineteenth century Venn elaborated this method of representation by requiring that A and B should always be represented by overlapping circles, and the actual relationship be represented by shading any region asserted to be empty, and putting an asterisk in any areas asserted to not to be empty. A region about which there is no information is left blank. In a Venn diagram the circles were enclosed in a rectangle representing the Universe of Discourse - the class of all objects under discussion.

Some Venn Diagrams

Figure 1 is a basic diagram, containing no information.

Figure 2 asserts that there is nothing that is B but not A, and there is something that is both A and B, so that all B’s are A’s; the regions corresponding to A’s that are not B and to individuals that are neither A nor B,

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are both left blank indicating that there may, or may not, be some A’s that are not B, and there may, or may not, be some individuals that are neither A nor B.

Figure 3 asserts that nothing is both A and B, but there is something that is A and not B, and there may, or may not, be individuals that are neither A nor B, and there may, or may not be individuals that are B but not A.

Venn diagrams can be used to distinguish more cases than Euler’s diagrams.

Example of Proof by Diagram

Consider the argument:(1) Some people who buy lawn mowers smoke pipes

(2) All who buy lawnmowers are gardeners,

therefore(3) Some gardeners smoke pipes

let P represent pipe smokers, L represent buyers of lawnmowers, and G represent gardeners.

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Extracting from the third diagram just the information about gardeners and pipe smokers, we have the last diagram, which corresponds to the conclusion.

Does Logic just state the obvious?

Perusal of examples provided by logicians often suggests that their validity is so obvious that it is hardly worth mentioning it. That is partly because a good example needs to be clear, but it is also true that Aristotelian argument, unlike modern mathematical logic, rarely deals with inferences likely to puzzle an alert mind.

The principal point of studying logic is not to learn how to recognise valid arguments, but to spot fallacious arguments that resemble valid ones. If we had a list of patterns of valid argument, we should be able to check any suspect piece of reasoning to make sure it matches one of the valid patterns, and much of the content of elementary books on Logic consists of the examination of invalid reasoning. For this purpose it does not matter if the valid arguments appear obviously valid, because the proposed list of valid forms of argument is a check list to be used to identify invalid arguments.

However, identification of fallacy in that way depends on our having a comprehensive list of patterns of valid reasoning. Aristotelian logic has such lists only for syllogisms, and for direct inferences between two propositions of the four types Aristotle recognised.

Aristotelian Logic had no tools for assessing the validity, or otherwise of the following inferences:

(1) If everyone who plays a game knows someone else who plays another game, it follows that if anyone plays any game, there are at least two games that are played by someone or other.

(2) Most mice like chocolate, Most mice like cheese,Therefore some creatures that like chocolate also like cheese

(3) A is B’s nephew, therefore B is A’s uncle

Modern LogicThe three examples at the end of the last section illustrate some of the

limitations of Aristotelian Logic, yet little was done extend formal Logic until Mathematicians began to take an interest in the subject.

For more that two millennia after Aristotle’s death Logic, despite minor amplifications, remained much as he left it. Only when George Boole (1815-

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1864) made a fresh start by constructing a logical algebra did the subject enter a century of rapid growth into what is now called ‘Mathematical Logic’. Aristotle’s Logic is still discernible as part of modern Logic, but it is not a convenient starting point. Nor is Boole’s work, at least not in its original form. I make a fresh start with what are now called ‘truth functions’

Truth Functions.Truth functional logic takes as its units complete propositions.

Truth Values Suppose P, Q, R, and S, represent propositions. Each of those propositions

must be either true or false. If a proposition is true we say its truth value is true, symbol ‘T’ and if false its truth value is false, symbol ‘F’. Boole used ‘1’ for true and ‘0’ for false; some electronic engineers also use that convention, as do electronic calculators that permit boolean operations.

Truth functional logic studies ways of combining propositions into more complicated propositions in such a way that the truth value of the composite proposition is determined by the truth values of its components.

Negation: NOT not P, symbol ~P, is true when P is false and false when P is true. That can be summarised by a truth table which gives the truth conditions for ~P

P ~P T FF T

To deny P is wrong if P is true, but correct if P is false.

Conjunction: AND both P and Q, symbol P & Q, true when P and Q are both true, and otherwise false. Its truth table is:

P Q P & QT T TT F FF T FF F F

P & Q is true only when both P and Q are individually true. P & Q is false in all other circumstances

all other truth functions can be defined in terms of ~ and &, but it is convenient to provide independent definitions for several others.

Disjunction, OR

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In English ‘or’ can have either an inclusive sense, meaning one or the other or both, or an exclusive sense meaning one or the other but not both. In Latin there are different words for the two senses, vel for inclusive ‘or’ and aut for exclusive ‘or’. It is the inclusive ‘or’ that is most often needed in logic, so Russell and Whitehead suggested using the symbol V, from vel. The truth table is:

P Q P V QT T TT F TF T TF F F

The inclusive or is true in all cases but one; it is false only if P and Q are both false

The exclusive or which is used in the theory of computer circuits can be defined as P XOR Q = (P&~Q)V(~P&Q), it is true when just one of P and Q is true, and the other is false.

P Q P XOR Q T T FT F TF T TF F F

The Biconditional, symbol º , P º Q asserts that P and Q have the same truth value

P Q P º QT T TT F FF T FF F T

Material Implication ‘if P then Q’ cannot be accurately represented by any truth function, but the best approximation is something that Russell and Whitehead called ‘material implication’ , symbol É , a symbol originally chosen by Peano, of whom much more later. Many other logicians prefer to call it ‘material conditional’ , and some prefer to represent it by an arrow. The truth table is:

P Q P É QT T TT F FF T TF F T

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If we stipulated that P É Q must be false in all cases where Q does not follow from P, it would always be false, since the truth values of P and Q are not sufficient to establish that one proposition is relevant to the other. We therefore follow the most permissive rule possible, and specify that the conditional must be true in all cases where Q does follow from P. Working through the possibilities:

(T, T): a true proposition may entail another true proposition, example ‘p>3 therefore 10p>30’, so T É T must be counted true

(F, F): a false proposition may entail a false proposition, example‘Prince Charles is the Queen’s father therefore he is older than

the Queen’ is valid even though both premiss and conclusion are false.so F É F must be counted true(F, T): a false proposition may entail a true proposition, example

‘The Queen is Prince Charles’ Father, therefore she is older than he’

so F É T must be counted true(T, F): The only combination of truth values we can rule out is that of a

true proposition entailing a false one, for example ‘the earth is larger than the moon, therefore the moon is larger than the sun’ so T É F must be counted false

According to this definition P É Q is equivalent to ~(P & ~Q), it is true in all cases except that in which P is true and Q is false.

That definition has the paradoxical consequence that P É Q is always true when P is false ‘A false proposition implies anything’.

Of course it doesn’t really, though there are various sayings of the form ‘If P, then I’ll eat my hat’ which are picturesque ways of asserting ~P.

Shakespeare wrote: “Let me not to the marriage of true mindsAdmit impediments. Love is not loveWhich alters when it alteration finds,Or bends with the remover to remove:O, no! it is an ever fixed mark,That looks on tempests and is never shaken;It is the star to every wand'ring bark,Whose worth's unknown, although his height be taken.Love's not Time's fool, though rosy lips and cheeksWithin his bending sickle's compass come;Love alters not with his brief hours and weeks,But bears it out even to the edge of doom:-If this be error and upon me proved,I never writ, nor no man ever loved.

It is said that a colleague once asked G. H. Hardy ‘Can you really prove that 2 + 2 = 3, implies that Bertrand Russell is Pope ?’. Hardy is said to have replied:

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“suppose 2 + 2 = 3, subtracting 2 from both sides gives: 2 = 1Russell and the Pope are two, therefore Russell and the Pope are one.”

Although it may seem strange that F É P should always be true, (I’m using ‘F’ to represent any false proposition, and P to represent any proposition at all) the convention is harmless in the sense that it does not provide a reason to believe an arbitrary conclusion. For if we know P É Q only on the basis of P’s being false that cannot be the basis for a good argument for believing Q.

For example,‘the moon is made of green cheese É Tony Blair is Queen of England‘ is true.

But even though we count that conditional as true, we cannot use it to deduce that Tony Blair actually is Queen, because that inference could only be made if the moon actually were made of green cheese, while the conditional is known to be true only because the moon is not made of green cheese.

P É Q is best thought of as the minimum condition that must be satisfied for there to be any sort of inference from P to Q. It satisfies three important rules that together encapsulate most of the functionality of ‘if ... then’

(1) It satisfies the rule of detachment according to which ‘If P, then Q’ allows us to

pass from P to Q, because the conjunction of P and P É Q entails Q

(2) It is transitive, [P É Q and Q É R] entails P É R

(3) It satisfies the rule of contraposition so that ‘If P, then Q’ is equivalent to:

‘If ~Q then ~P’

Justifying (1): To see that detachment applies, consider the truth table for P É Q and notice that the only case where P, and P É Q are both true is the case where P and Q are both true.

Justifying (2) Once detachment is established (2) can be justified by first constructing the truth table for [P É Q & Q É R] É [P É R] to show that it is always true, and then applying detachment.

Justifying (3): (3) can be justified by constructing a truth table to show that (P É Q) º (~Q É ~P) is always true

Although P É Q is not equivalent to ‘P implies Q’, [P & (P É Q)] does entail Q as I’ll soon show, so that it is impossible for [P & (P É Q)] to be true unless Q is also true.

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Interest in the material conditional goes back to the Stoic philosophers who considered it as a possible analysis of entailment, but they did not try to construct a more comprehensive truth functional logic, and don’t seem to have influenced subsequent work in Logic.

Tautologies A formula that has truth value T for all possible combinations of truth values of its components is called a tautology The tautologies are the theorems of truth functional logic. They are the simplest examples of logically true propositions, by which we mean propositions that are true because they fit a logical pattern that guarantees truth. Sometimes ‘necessary truth’ is used instead of ‘logical truth’.

For example (P É Q) º (~P V Q) is a tautology,

proof:P Q P É Q ~P ~P V Q (P É Q) º (~P V Q) T T T F T TT F F F F TF T T T T TF F T T T T

‘P É Q’ and ‘~P V Q’ take the same truth value for every combination of the truth values of P and Q so whatever propositions are represented by ‘P’ and ‘Q’ ,

(P É Q) º (~P V Q) must be true.

Equivalent Formulae If F º G is a tautology, where F and G are two truth functional formulae, then F and G have the same truth table and are said to be truth functionally equivalent. If either F or G appears in some more complicated truth functional formula H, it could be replaced there by the other without affecting the truth table of H. The following equivalencies are interesting and can be verified by constructing the appropriate truth tables:

P º ~ ~P , (P É Q ) º ~(P & ~Q), P V Q º ~( ~P & ~Q)

the last two equivalencies show that both V and É can be defined in terms of ~ and &

Contradictions A formula which is false for all assignments of truth values is called a contradiction, because the simplest example is the self contradictory formula P & ~P.

If C is a contradictory formula ~C is a tautology, and if A is a tautology ~A is a contradiction.

Strict Implication P Þ Q, read as ‘P strictly implies Q’ is defined as: ‘P É Q is a tautology’.

Strict implication has been proposed as an analysis of entailment.Page 13


It avoids the paradox of the material conditional. P’s falsehood does not guarantee the truth of P Þ Q, but there still is a paradox.

The so called paradox of strict implication is the theorem that a contradiction strictly implies anything since (P& ~P) É Q is a tautology, (exercise: verify that it is a tautology by constructing its truth table).

Although at first sight odd, that result is not easily avoided since it does not depend on the definition of entailment as strict implication and can be derived without using truth tables by appealing to several properties all of which one would expect entailment to have.

I1 to I5 each describe a property one would expect to be satisfied by any relation of entailment

I1: Detachment: from P and P Þ Q infer QI2: Transitivity: from P Þ Q and Q Þ R , infer P Þ R I3: And: P & Q Þ P and P & Q Þ Q I4: OR: P Þ P V Q, for any QI5: OR: from ~P and (P V Q) infer Q

To prove (P& ~ P) Þ Q suppose (P& ~P) (1)then P (by I3) (2)then P V Q from (2) (by I4 and I1) (3)then ~P from (1), (by I3) (4)then Q from (3) and (4), (by I5) (5)then (P& ~P) Þ Q from (1) to (5) , (using I2 several times)

The ‘Paradoxes of Strict Implication’ have been much discussed and logicians have constructed a variety of formal systems, known as modal logics, to define different relations of entailment, but no one has defined a satisfactory relation that fails to satisfy all of I1 to I5, so the ‘paradox’ seems unavoidable. I say a little about modal logics in chapter 5.

Many formal mathematical systems satisfy what is called the deduction theorem, which states that Q can be deduced from P if and only if P É Q is provable, in other words if and only if P strictly implies Q, so that in those systems strict implication is definitely equivalent to entailment.

I shall henceforth say that P entails Q in all cases where P strictly implies Q.

One source of unease with that interpretation may be that some people interpret ‘P entails Q’ as asserting that P constitutes a good reason for believing Q. As I have already remarked, a good reason and a valid argument

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are rather different. What is a good reason for one person may not be a good reason for another, so there is no exclusively logical relation ‘P is a good reason for believing Q’ , and the sentence should be extended to ‘P is a good reason for A to believe Q’ The conditions of that are:

(1) A believes P with good reason(2) P entails Q(3) A believes that P entails Q

The conditions must include (3) since even if P does entail Q, that wouldn’t provide A with a good reason unless A knew of the entailment.

If A believes (2) because he believes that P is a contradiction, he cannot believe P with good reason, so condition (1) is not satisfied. So although a contradiction does entail any proposition Q, it does not provide a good reason for believing any Q.

The argument would be simpler if we included the truth of P in the conditions, but I do not think that that would be correct. Even if P is false, A might have a good reason for believing P, in which case that could be part of a good argument in favour of Q, though that argument would not be conclusive.

The Perils of Inconsistency

Because a contradiction entails any proposition at all, inconsistency is extremely serious. In the formal systems of Mathematics and Logic things now seem to be more or less under control, but it is hard to find any basis for confidence that our beliefs in other matters are mutually consistent. Indeed it seems reasonable to suppose that they are generally inconsistent. For our opinions on matters of fact are at best highly probable, and a set of propositions all highly probable may well be inconsistent.

Contemplate two successive throws of a standard unbiased die. The probability of throwing two sixes is 1/36 so we are justified in asserting ‘ Very probably the result will not be two sixes’.

Altogether there are 36 possible outcomes of the experiment, one corresponding to each ordered pair of two numbers, either different or equal, selected from {1,2,3,4,5,6}. Each of those outcomes occurs with probability 1/36, so we are justified in saying of every one of the outcomes that it is very unlikely. Of course one of those possible results must occur, but until we have thrown the die, we don’t know which outcome that is.

Now consider the set of 36 propositions of the form: ‘The result will not be (x, y)’ for all values of x, y in {1,2,3,4,5,6}

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Although each of these 36 propositions is highly probable, the conjunction of all 36 is inconsistent. They cannot all be true since the experiment must have some outcome.

It follows that we should be wary of long and involved pieces of reasoning, especially when they involve assumptions from different fields of study which are not often checked against each other for consistency. For such an argument may well have inconsistent premisses from which any conclusion at all could be deduced.

Notice that even if do we detect an inconsistency in our beliefs, it may not always be clear how to resolve it, for the inconsistent beliefs may all be ones which, on the evidence, it is reasonable to hold. In Physics, General Relativity appears to be inconsistent with the Quantum Theory, so at least one of those theories needs to be modified, and perhaps both do, but the inconsistency itself does not tell us what modifications are needed.

A Complete Set of Truth Function

The truth table for two propositional variables, P and Q, has four rows, one corresponding to each of the possible combinations of truth or falsity of P and Q. Each of the four rows of the truth table could be completed either with True, or with False, so the complete table can be filled in 24 = 16 different ways. In general the truth table for n propositional variables has 2n rows, each of which can be filled with either of the two truth values, so there are 2k

possible truth functions, where k = 2n.

I shall show that any truth function of more than two variables can be expressed in terms of functions of just two variables.

So far we have given special symbols to six truth function, the two truth functions of one variable, namely P and ~P, and four truth functions of two variables, Ú, É, º, & However, using only Ú, & and ~, it is possible to obtain a formula corresponding to any way of filling in a truth table, with any number of variables. To do so proceed as follows:

(1) Pick out all the rows of the table in which the function takes value T

(2) For each such row, determine which of the propositional variables are marked T and which are marked F, and form a conjunction containing ‘X’ for any variable X marked true and ‘~X’ for any variable X marked false

(3) Use Ú to form the disjunction of all the formulae obtained according to (2)

For example, consider the truth function ‘*’ defined by the truth table:

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P Q P*QT T FT F TF T FF F T

P*Q is marked ‘T’ on the second and fourth rows. On the second row P is true and Q false, so we form the conjunction P &~Q.

On the fourth row P and Q are both false, represented by the conjunction ~P &~Q

So P*Q is equivalent to (P &~Q) Ú (~P &~Q)

The method described above does not always give the simplest formula. In this example P*Q is equivalent to ~Q

For a truth table with substantially more T’s than F’s a shorter formula can be obtained by constructing a formula corresponding to the rows where the formula is false, and then negating it. Consider, for example, the truth function ‡ defined by the table:

P Q P‡QT T TT F TF T FF F T

P‡Q is false only when P is false and Q is true, corresponding to ~P&Qso P‡Q is equivalent to ~(~P&Q) which is, of course, P É Q

It is not necessary to use all three of ~, Ú, &, to define all truth functions. Ú may be defined in terms of ~ and &, since P Ú Q is equivalent to ~(~P & ~Q) so all truth functions could be defined in terms of ~ and &. Alternatively by defining P & Q as ~(~P Ú ~Q) we could define all truth functions in terms of ~ and Ú.

It is even possible to define all truth functions in terms of a single function. Scheffer suggested using either of the functions P|Q equivalent to ‘not both P and Q’, or NOR, where P NOR Q means ‘neither P nor Q’

‘|’ can be used to define ‘~’ and ‘Ú’. For ~P is equivalent to P|P so P Ú Q is equivalent to not both (not P and Not Q) which is (P|P)|(Q|Q).

Once ‘~’ and ‘Ú’. are defined all the other truth functions can be defined in terms of those two.

In the case of NOR it is easiest to define ‘~’ and ‘&’. ~P is equivalent to P Page 17


NOR P and P & Q is equivalent to [~P NOR ~P] which is equivalent to [(P NOR P) NOR (Q NOR Q)]. The remaining truth functions can then be defined in terms of ‘~’ and ‘&’.

Quantification

Truth functional logic treats propositions as units without considering their internal structure, but the validity of almost all arguments depends on the structure of the propositions involved, so we must take our analysis further.

The first step is to extend our notation to accommodate propositions in the subject predicate form, such as those Aristotle considered. Suppose we wanted to say ‘All swans are white’, we construe that as

‘anything that’s a swan is white’, or ‘for any x, if x is a swan then x is white’

Russell used the symbolism (x)( [x is a swan] É [x is white])

Nowadays it is more common to use the notation:

("x)( [x is a swan] É [x is white] )

‘"x’ is called the universal quantifier and is read as ‘for all values of x’

Note that because we use É we are treating ‘if x is a swan then x is white’ as equivalent to ‘either x is white, or x is not a swan’

We might also want to say ‘some swans are white, but some swans are not white’ We then employ the existential quantifier, $, and our new proposition is written:

E1: ($x)([x is a swan] & [x is white]) & ($x)([x is a swan] & ~[x is white])

($x) Is read as ‘there is some x such that…’

notice that ‘but’ is treated as equivalent to ‘and’

The letters x, y... appearing in the quantifiers are called bound variables. They have the same function as pronouns have in ordinary discourse. If we drop the quantifier and just write:

E1* [x is a swan] & [x is white]

the x is then called a free variable and E1* is not a proposition, but something called an ‘open sentence’, just a sort of blueprint for a proposition. It is roughly equivalent to ‘it is a swan and it is white’ with no indication of

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what ‘it’ might refer to. To turn such a blueprint into a proposition we must either replace the ‘x’ by a name, e.g.. ‘Zeus’, or else prefix the expression by a quantifier.

A formula with no free variables is said to be closed. The closed formula obtained by prefixing an open formula F(x) with a universal quantifier is called the universal closure of F(x); if the existential quantifier is used it gives the existential closure so

the existential closure of:

[x is a swan]&[x is white] is ($x)([x is a swan]&[x is white]) , which means ’there is something that is both a swan and white’, or ‘there is a white swan’

It is only closed formulae that represent actual propositions. We often speak loosely of ‘the proposition F(x) Ú G(x)’ but that should be interpreted as ‘some proposition of the form F(x) Ú G(x)’

Either quantifier may be defined in terms of the other. It is usual to take the existential quantifier as basic and define ("x )(F(x)) as ~($x)( ~F(x)). ‘Everything is so and so’ is equivalent to ‘Nothing is not so and so’ Notice that, so interpreted ("x )(F(x)) does not imply that anything actually is an F. ‘All swans are white’ does not entail that there are actually any swans, white or otherwise,

The letters F, G are said to represent predicates. All the examples we have so far encountered involve one place predicates, so called because they govern just one variable. There are also two place predicates, such as ‘greater than, and three place predicates, such as ‘between’.

The Scope of a Quantifier

The scope of $x or of "x is that part of the following expression containing the instances of ‘x’ to which the quantifier applies. The instances of ‘x’ to which the quantifier applies are said to be bound by the quantifier and to come within its scope.

The choice of letter for the variable is of no significance. ‘("x )(F(x))’ means precisely the same as ‘("y )(F(y))’.

In E1 : ($x)([x is a swan] & [x is white]) & ($x)([x is a swan] & ~[x is white]) , the scope of the first quantifier is ‘[x is a swan] & [x is white]’, while the scope of the second quantifier is ‘[x is a swan] & ~[x is white]’

Notice that there is no contradiction in using the same letter ‘x’ both in the assertion that some swans are white, and in the assertion that some swans are not white, since ‘[x is white]’ and ‘~[x is white]’ fall within the scope

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of different instances of ($x), so there is no assumption that the values of x that justify asserting :

($x)([x is a swan] & [x is white])are the same as the values which justify:($x)([x is a swan] & ~[x is white])

It would be possible to emphasize the difference by using different letters in the two quantifiers and rewriting E1 as:

($x)([x is a swan] & [x is white]) & ($y)([y is a swan] & ~[y is white]) , but it is not necessary to do that.

We now have a logic that can deal, not only with Aristotle’s syllogisms, but with a great deal more too, since it can also handle propositions much more complicated than Aristotle’s, including relational propositions involving several terms, like x<y.

‘<’ is an example of a two place predicate since it makes a comparison between two numbers. It is also possible to have three place predicates such as ‘between’ e.g. ‘Simon is sitting between Alice and Margaret’.

It is possible to enlarge the scope of logic even more by introducing propositions with several quantifiers. ‘Everyone loves somebody’ becomes:

("x)( $y)(x loves y), notice that this is quite different from ($y)("x)(x loves y),

which means there is somebody whom everybody loves. Interchanging the quantifiers changes the meaning.

Neither of those propositions can be accommodated in Aristotelian logic, because they contain ‘loves’ which is a two place predicate. The best the Aristotelian Logic could do with a verb like loves would be to manufacture pseudo predicates like:

LA = ‘loves Arthur’ or LG = ‘is loved by Gloria‘, so that:

‘Gloria loves Arthur’ would be rendered either as:

LA(Gloria) which attributes to Gloria the property of loving Arthur, and does not directly say anything about Arthur

Or as:

LG(Arthur) which attributes to Arthur the property of being loved by Gloria, and does not directly say anything about Gloria.

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Suppose we want to define ‘someone is touched by love’ to mean that person either loves, or is loved.

We can’t express that in Aristotelian logic, but in the quantificational notation is can be expressed as:

( $y)(x loves y) v ( $y)(y loves x),

(in English grammar ‘loves’ is a verb, but for the purposes of logical analysis the important point is that it is used to make statements about two individuals)

Proof In Quantificational Logic

The validity of quantificational formulae that involve only one place predicates can be established by an elaboration of the truth table method for identifying tautologies, though I don‘t give the details here. A mechanical test like that is called a decision procedure. There is no decision procedure for the more general logic that includes two place, three place and even more complicated predicates.

In the absence of a decision procedure the predicate logic is much more challenging than the proportional logic.

There seems to be a choice between systems that are complicated to describe, but easy to use, and systems that can be described quite simply, but in which it is often quite hard to construct proofs. In Methods of Logic Quine describes a system of the former kind, which is the one I use if I want to construct a proof. As ease of use comes at a high cost in complexity, the system I have chosen to describe here is of the opposite sort, easily described but harder to use.

Assuming some method, such as truth tables, for identifying tautologies, a system sufficient for quantificational logic is :

(1) Tautologies : Any tautology is a theorem

(2) Substitution for variables representing propositions

(2a) From any theorem we may derive a new theorem by substituting any closed formula for every occurrence of a particular one of the propositional variables

(2b) We may alternatively substitute any open sentence for a propositional variable, provided that none of the places where the substitution is made falls within the scope of a quantifier using any of the individual variables in the open sentence. So that we may not

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substitute F(x) for P if P comes within the scope of "x or $x, though we could substitute F(y) for P in such a context.

(3) Detachment If P and P É Q are both theorems, Q is also a theorem

(4) Quantifier Elimination From ("x)(F(x)) infer F(a) where a is the name of any individual, or alternatively we may infer the open sentence F(x)

(5) Quantifier Introduction

(5a) If P É Q is a theorem, then so is ("x)P É Q(5b) If P É Q is a theorem, and x is not free in P, then P É ("x)Q is

also a theorem

(5b) is equivalent to the rule that if a proposition can be proved for arbitrary x, it is true for all x.

(6) Definition of the Existential quantifier ($x)(F(x)) = ~("x)(~F(x))

It is then possible to deduce subsidiary rules for the existential quantifier:

(4S) from F(a) where a is the name of any individual, infer ($x)(F(x))

(5aS) if F(x) É Q and x is not free in Q, infer ($x)(F(x)) É Q(5bS) From P É F(x) infer P É ($x)(F(x))

There is also a useful subsidiary rule for the universal quantifier:

(UQS) If F(x) can be proved for arbitrary x, we may infer ("x)(F(x))

A consequence of that rule is that from every tautology we may infer its universal closure. For instance P V ~P is a tautology whatever proposition P may be, so using rule (2b) to replace P by F(x) we may infer F(x) V ~ F(x), from which we may use the rule (UQS) to derive ("x )(F(x) V ~ F(x))

As an example of the use of rule (4), from ‘all squirrels are viviparous quadrupeds’ interpreted as ("x)(x is a squirrel É x is a viviparous quadruped) we may infer of our pet Fido, ‘Fido is a squirrel É Fido is a viviparous quadruped’

As an example of (4S) from ‘Richard Thompson has blue eyes’ we may infer ($x)(x has blue eyes).

Example we can justify one of Aristotle syllogisms by showing how to infer ‘All F are G’ from ‘All F are H’ and ‘All H are G’

(1) ("x)( F(x ) É H(x)) premiss(2) ("x)( H(x ) É G(x)) premiss

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(3) F(x ) É H(x) from (1)(4) H(x ) É G(x) from (2)(5) F(x ) É G(x) from (3) & (4), by substituting F(x) for P, G(x) for Q and

H(x) for R in the tautology (P ÉQ)&(Q ÉR)É(P ÉR) and using detatchment

(6) (1)&(2) É (F(x ) É G(x) ) from (5)(7) (1)&(2) É ("x)(F(x ) É G(x) ) from (6) because x is not free in (1) or in

(2)(8) ("x)(F(x ) É G(x) ) from (1), (2), (7) by detachment.

Proof Using the Deduction Theorem

In a system satisfying the deduction theorem, propositions of the form A É B may be proved by assuming A and showing that it is then possible to deduce B.

Since AÚB is equivalent to ~A É B , the same strategy can also be used to prove AÚB, in that case we either start with the assumption ~A and deduce B, or else start with the assumption ~B and deduce A. That strategy forms part of the system Quine described in Methods Of Logic. I earlier used a similar strategy to prove (P& ~P) Þ Q, though that formula involves entailment, not material implication.

The Basis of Logic

Why should we bother about Logic? A popular answer is to avoid contradiction. The law of non contradiction ~(P & ~P ) is a plausible basis for logic because to accept a contradiction is equivalent to rejecting the distinction between truth and falsehood, and would completely undermine any communication between people.

We must not expect too much from a justification of Logic, for no argument or proof will actually stop anyone who really wants to ignore logic from doing so, but many people would be discouraged by the consequences if they realised what they were. Someone seen to reject the distinction between truth and falsehood is likely to find their utterances ignored by most of their fellow men. If their rejection of the distinction were complete and applied to their own thoughts, rather just to what they say to their fellow men, they wouldn’t really have any organised thoughts - nothing more than a jumble of sensations, feelings and impulses.

Sometimes people make dismissive remarks like ‘Life is larger than logic’, and sometimes they say ‘well, whatever you say, it’s true/valid for me’ I think such remarks are either not thought out, or are disingenuous. Although people may claim that they are concerned only with a private reality - with how things seem to them rather than with any objective external reality, it can be no advantage to someone to be without any coherent view of the world. I

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think the anti-logic brigade are more anxious to indulge a taste for intellectual exhibitionism than concerned to protect the integrity of some profound inner vision.

Logic and Language

Philosophers who followed Wittgenstein, especially the philosophers of the ‘ordinary language’ school that flourished in Oxford in the 1950’ and 60’s, criticised the application of modern formal logic outside Mathematics, on the grounds that it does not accurately represent the logic of everyday discourse. PF Strawson articulated that concern in his Introduction to Logical Theory.

I’ve already touched on this question a when considering the inadequacy of ‘É’ as an analysis of ‘if then’ and also when I remarked briefly on the variety of uses of ‘or’ in English.

Critics such as Strawson have questioned the choice of the inclusive ‘or’ as the sense to be represented by a single symbol, but that does not prevent the exclusive ‘or’ being represented too. In contexts where the exclusive ‘or’ is used frequently it can be given a special symbol of its own, indeed computer engineers do have a special symbol for it: ‘XOR’ but the purposes of ordinary logic are adequately served by presenting it as (P & Q) Ú (~P & ~Q . However, whatever the relative frequencies of the two senses of ’or’ in idiomatic English, no logical system could follow ordinary language in having the same symbol for both senses of the word.

Strawson raised other objections to the application of mathematical logic outside Mathematics, pointing out that in ordinary usage ‘and’ is not commutative. ‘She took arsenic and died’ is not equivalent to ‘She died and took arsenic’ That is a trickier example than Strawson realised, for the assumption that she took the arsenic first may not be actually be asserted by the statement; it may be a deduction we make from it on the basis of our general knowledge about arsenic, its effects, and also about the inability of the dead to ‘take’ poisons. Those of us who enjoy detective stories know that ‘She was hit on the head and died’ does not imply that the hitting came first; it might have been done after death in a vain attempt to confuse us about the cause. Yet although the logicians’ ‘&’ cannot on it own do justice to ‘She took arsenic and then died’ , it can be used as part of a more complicated structure that does capture the meaning as in:

( $t1)( $t2)([she took arsenic at time t1]&[she died at time t2] & [t1<t2])

Strawson was also much concerned about the analysis of universal generalisations. The problem there arises from the existential import of universal generalisations and the use of material conditional for ‘if then’.

In the traditional Aristotelian Logic, and often also in ordinary usage, ‘All P Page 24


is Q’ implies ‘Some P are Q’, but (" x)( P(x) É Q(x)) does not imply ($ x)( P(x) & Q(x)) since the former would be true if nothing satisfied P(x) while the later would not.

That is not always as counter intuitive as Strawson seems to think. There’s a case for saying ‘All unicorns have just a single horn’ even though there are no unicorns, but it would definitely be counter intuitive to say ‘All unicorns have gills’, although on our analysis that also is true. However, the problem was not created by modern logic since even before the introduction of truth functions and quantifiers, some logicians advocated interpreting the Aristotelian ‘All P is Q’ so that it does not imply ‘Some P are Q’.

We have to accept that ‘All P is Q’ is sometimes used so that it presupposes that there are some P’s and is sometimes used without that presupposition. Both can be represented in the symbolism of quantification. ("x)( P(x) É Q(x)) represents the interpretation of the universal proposition which does not imply the existence of any P‘s, while ("x)( P(x) É Q(x)) & ($x)(P(x)) represents the sense of ‘All P is Q’ in which is does imply that there are some P. Similarly there are two possible interpretations for ‘No P is Q’.

However there seems to be a special difficulty in Aristotelian Logic, for in whichever senses we interpret ‘All P are Q’ and ‘No P is Q‘, not all of the claimed logical relations hold. We have already noted that if we interpret ‘All P are Q’ as

("x)( P(x) É Q(x)) it does not imply ‘Some P are Q‘, which is ($ x)( P(x)&(Q(x))

However, if we interpret ‘All P are Q’ as (" x)( P(x) É Q(x)) & ($ x)(P(x)) so that it does imply ‘Some P are Q‘, it is not the contradictory of ‘Some P are not Q, for the latter is ($ x)( P(x)&(~Q(x)) so that its contradictory is

("x)( P(x) É Q(x)) which involves no existential commitment.

Alternatively we might interpret every proposed Aristotelian inference A Þ B where A and B are propositions involving the terms P and Q, as assuming the existence of objects of all the kinds referred to so that A Þ B would mean:

[($ x)(P(x))&($ x)(Q(x)] Þ [A É B ]

It would then be possible to represent the basic Aristotelian propositional types as:

P a Q: ("x)( P(x) É Q(x)) P e Q: ("x)( P(x) É ~Q(x)) P i Q: ($ x)( P(x)&(Q(x)) P o Q: ($ x)( P(x)&(~Q(x))

and precede problematic implications with an assertion of existence, thus preserving in a somewhat convoluted form the logical relations Aristotelian logicians traced between them.

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Thus P a Q implies P i Q would be rendered:

($ x)( P(x)) É [("x)( P(x) É Q(x)) É ($ x)( P(x)&(Q(x))]

That would still be a tiresome complication, as it would prevent the useful application of the logic to cases involving terms like ‘unicorn’ that have no reference, but it seems the best that can be done., and may well be close to Aristotle’s own train of thought.

Strawson’s citing of the problem of existential import as a weakness peculiar to mathematical logic is thus entirely misconceived. The problem was already present, and acknowledged, in Aristotelian Logic, and the contribution of mathematical logic has been to provide a notation that makes it easier to discuss the problem.

In everyday discourse ‘All P is Q’ may often be used with the implication that P(x) is, or if it were ever true would be, a reason for believing Q(x); that does set apart the two propositions about unicorns. ‘All unicorns have a single horn’ would be a reason for attributing a single horn to any unicorn that might turn up because the appearance of a unicorn would not undermine its truth, but ‘All unicorns have gills’ would cease to be true as soon as a unicorn appeared and so could never license any inference.

Such an interpretation accords with Aristotle’s view that logic draws out the consequences of the essential properties of the natural kinds that make up the world. However the presence of a logical connection between P and Q is not a necessary condition for asserting ‘All P is Q’ either in everyday conversation or in Aristotelian logic, for that logic allows the assertion in all cases where there are P’s and all of them are Q, and disallows it in all cases where there are no P’s even if there is reason to believe that if there were any P’s they would be Q’s.

The fact that certain words of the English language are not represented in the logical notation by single symbols does not imply that formal logic cannot represent the propositions those English words are used to express, although as English words may be used in a variety of senses, different occurrences of the same word sometimes need to be represented differently. No formal system could contain a symbol that simultaneously represented all the shades of meaning of ‘and’ or of ‘or’, because sentences that use the words differently represent different propositions with different implications. One of the advantages of formalising logic is to expose such distinctions which are sometimes concealed by ordinary usage.

Is Logic Trivial?

I tried to think of a different title for this section, because when I ask the question it is clearly rhetorical, assuming the answer ‘No’ . On the other hand

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when some people ask the question rhetorically, they expect the answer ‘yes’.

The word ‘trivial’ is derived from ‘Trivium’ the first stage in medieval higher education, which followed a pattern derived from Roman education, which had centred on the seven liberal arts, which comprised the four mathematical studies recommenced by Plato, Number, Geometry, Astronomy and Harmony, and also Grammar, Rhetoric and Dialectic. The first four subjects later came to be called the Quadrivium and the latter three, which were taught first were called the Trivium. So in the original sense of the word, it is a truism that Logic is trivial.

Deductive logic has often been denounced as trivial in the modern sense on the ground that the conclusions contain no additional information not present in the premises, so we cannot learn anything new from the deduction. That argument was articulated in the early seventeenth century by Sir Francis Bacon, in his Novum Organum where he denounced to what he considered to be the vacuous formal logic of his day, and proposed to replace it with a fruitful inductive logic. The title was doubtless chosen because when Bacon wrote formal logic was little different from that expounded by Aristotle in his Organum

The supposition that formal logic is trivial may be reinforced by the examples of valid argument found in text books on Logic, but that is not a fair test. Examples used to teach have to be simple enough to be understood by the pupil. When teaching logic one has to explain the notion that the conclusion of an argument must be true whenever the premises are true. A clear example is therefore an argument in which it is obvious that the conclusion will be true whenever the premises are true. Because its validity is obvious, the argument cited is likely to be trivial. Until the later half of the nineteenth century the appearance of triviality in formal logic was reinforced by its very limited scope; it dealt with only a few patterns of argument, and those few were all very simple.

However the supposition that logic is trivial has survived the immense increase in its scope and power following the work of Russell and Whitehead in the early twentieth century. Often it is not so much that people are impressed by the perception that logical argument actually is trivial, but rather that they think it ought to be trivial, because they think it can never do more than tell us what we really knew all along.

The underlying assumption is that any valid inference should be obvious. In the background there may be a picture of knowledge being made up of little atoms of truth, so that every true proposition is a conjunction of various atoms. An argument would then be of the form:

Premisses: t1&t2&.....t20Conclusion t3&t7

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t3 and t7 are both included in the premisses, so the argument is valid

We need do no more than describe that picture to see that it is does not apply to most inferences.

The following example shows how the validity or otherwise of a proposed inference may not be obvious. I give two premisses, and a set of possible conclusions.

Premisses:P1: Alan’s only uncles are Simon and Reginald, who are twins, and

he has no aunts.P2: Sophie’s only aunt is Mary and Simon is her only uncle.

Possible conclusions:

C1: Alan’s Mother has no BrothersC2: Sophie’s Father has no sistersC3: Sophie’s Mother was an only childC4: Mary is Alan’s Mother

‘Aunt’ is to be interpreted as ‘sister of a parent’ and ‘uncle’ as ‘brother of a parent‘. Brothers in law and sisters in law of parents do not count as aunts or uncles, nor do half brothers or half sisters.

There are eight possible arguments to be considered here.P1 & P2 Þ C1, P1 & P2 Þ ~ C1 and so on. Perhaps in one or two cases the

logical status of the argument is obvious, but I don’t think that is so for them all. Deciding which are valid and which invalid is left as an exercise for the reader. Those for whom logic is trivial should of course already know all the answers.

Logical Form and the detection of Fallacy

Today logic is studied mainly as a branch of mathematics but the formal mathematical operations don’t include informal logic - trying to find a logical pattern in an informal argument and trying to clarify ideas when conceptual confusion seems to impede our thought. Proving a theorem in a formal system is one thing, but deciding whether or not a formula fits a particular piece of reasoning is quite another.

People sometimes assume that any proposition and any piece of reasoning has a unique logical form, so that there is only one way to represent it in formal logic. Were that true determining the validity or otherwise of a piece of reasoning would involve no more than testing the resulting sequence of formulae for validity. Even that would not always be as straightforward as the more optimistic are inclined to suppose, but it would still be easier than

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evaluating the intriguing tangles of ambiguous rhetoric we often encounter in everyday conversation.

Things are even more complicated than that because propositions and arguments do not have a unique logical form. The same piece of reasoning may instantiate several patterns. An argument is valid provided at least one of those patterns is valid, it is invalid if none of the corresponding patterns is valid. It is not possible to check all the patterns someone who propounds an argument might have in mind, because arguments often make assumptions that are taken for granted and not explicitly stated.

Therefore, outside Mathematics, detecting invalid reasoning is often much harder than people imagine. An argument is valid if it is an instance of some valid form (pattern), and some people suppose that invalid arguments may be identified as conforming to some invalid form, but that is not so. In some pieces of supposed reasoning the premises have no relevance to the conclusion, so there is no logical structure to analyse. For example suppose I say:

“‘I must be older than he is because my name is ‘Richard’ and his is ‘Peter’ “

The only obvious way to represent that formally is “A therefore B”, but any argument at all, whether valid or not, can be represented in that form.

Someone who wanted to challenge the argument about the ages can’t say much more than ‘That’s fallacious’ or what amounts to the same thing, ‘Names don’t tell us ages’ . If I retort ‘tell me what’s wrong with the argument’ you could legitimately reply ‘So far as I can see there’s nothing right about it. Why do you think it’s valid?’

It is possible to imagine a society in which people’s names incorporated their days of birth - a society of robots might use commissioning date as part of a numerical name. It is possible to imagine a society in which ’Richard’ was reserved for the first born son on any family, and in which ‘Peter’ might only be given to a second born son. If the Richard and the Peter in the story were brothers, the inference from their names to their relative ages would then be valid. Thus it would be incorrect to say that name cannot indicate age, what is wrong with the example I just gave is that there is no reason to suppose there is such a relation in that case.

A common error is to generalise from a single instance, or from too few instances, as happens when someone says ‘Those vegetarians are all communists - like that teacher at the Primary school who stood as a communist in the council election’. Although much of our knowledge is obtained by some sort of generalisation, all generalisation is deductively invalid, so I delay a full discussion till later in Chapter 3, on knowledge, and

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chapter 6, on Science. However it is customary to group reckless generalisation with logical fallacies, so I’ll say a little about the matter here.

Generalisation from a single instance is not always wrong. Consider:

‘This creepy-crawly has eight legs, so creepy crawlies generally have eight legs’

Sometimes (x)(F(x)) may be deduced from F(a) in conjunction with the suppressed premiss: (x)(F(x) É G(x)) v (x)(F(x) É ~G(x)), either all F’s are G, or no F’s are G. Of many sorts of animals it is believed that all in the same species have the same number of legs, so, if F(x) means ‘x is a creepy crawly’ and G(x) means ‘x has eight legs’ we assume that either all creepy crawlies have eight legs, or none do. The discovery of just one eight legged creepy crawly refutes that latter proposition, leaving ‘all creepy crawlies have eight legs’ as the only alternative.

Sometimes a person may appear to be generalising from a single instance when they are actually just giving an example. For example:

‘Hens stop laying when they are frightened. After the next door neighbours garden shed exploded, our Lucy didn’t lay an egg for a whole week, and she usually lays at least every other day’

The speaker may have encountered other cases of frightened hens and have read the generalisation ‘frightened hens don’t lay’ in a book about the care of livestock. He may not have been trying to prove his generalisation, because he assumed it to be already well established; perhaps he was just giving an example to show what it’s like in a particular case.

When the validity of a piece of reasoning is challenged the challenger is often expected to say what is wrong with it. That is all right when the invalid argument is very close to a valid form. It may then suffice for the challenger to point out the difference between the argument in question and a closely similar but valid argument.

For instance an argument of the form: All S is P, All Q is P therefore all S is Q is most likely an unsuccessful attempt to construct an argument of the form All S is P, all P is Q, therefore all S is Q. The particular error of confusing those two patterns is sufficiently common to have a name; it is called the fallacy of the undistributed middle term. (P is the middle term and it is not distributed in either premiss - see appendix I on the syllogistic logic). (Things can be more complicated when the available data does justify the weaker conclusion ‘most S are Q’)

For example someone might say ‘Henry must be a Nazi, because he loves Wagner, and the Nazi’s were great Wagner enthusiasts’. The

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linguistically similar argument ‘Henry is a Nazi, Nazi’s like Wagner, therefore Henry likes Wagner’ is valid, and pointing out how the fallacious argument differs from the valid one should be a sufficient explanation of the error.

However many cases are less straightforward and asking the challenger to point to the error may sometimes be unreasonable; we cannot enumerate all the infinitely many valid patterns of argument to show that a particular argument does not fit any of them, while if the argument is valid its proponent could demonstrate that by producing just one valid form.

Invalid Forms

Someone might suggest that identifying fallacious arguments would be a less hit and miss affair if, instead of showing that a disputed argument does not fit any valid form the user of the argument is likely to have in mind, we just show that it conforms to an invalid form. That is usually impractical because it is only in a few very peculiar cases that an argument does have a form that guarantees invalidity. In fact I can think of only one.

A valid argument, P therefore Q must be such that the truth of P guarantees the truth of Q. An invalid form would therefore have to be one such that any argument of that form must have instances in which the antecedent is true and the conclusion false. The only way in which the form of the argument can guarantee that, is if the antecedent is logically necessary and the conclusion a contradiction, producing something like:

(P v ~P) É (P & ~P)

Although any argument of that form would indeed be invalid, we do not in practice meet such arguments, so that criterion of invalidity would rarely if at all be useful.

Forms of unsound argument are a little more common; any argument with a logically false premiss is unsound, even when it is valid, but even allowing for such cases, it is rare to meet an argument that can be condemned as unsound simply on the basis of its being of an unsound form.

Counter Examples

Unless the person propounding an argument states what its logical form is supposed to be, the best way to show that a putative argument cannot be valid is to make up a story to show that the premisses could be true and yet the conclusion could still be false. That strategy is a little haphazard; on the spur of the moment we may be unable to think of a suitable example, but it avoids the need to determine the intended form of the argument.

For example suppose someone argues thus:Page 31


‘Most golf players also play bridge. Most bridge players drink gin, therefore at least some golf players drink gin.’

That argument is invalid since it is possible for both the premisses to be true while the conclusion is false. We can show that by making up a story. Suppose there are eight golf players of whom five play bridge and none of whom drink gin. Suppose there are fifteen bridge players - the five bridge playing golfers, and ten other bridge players none of whom play golf, and suppose nine of those ten drink gin. Then there are eight golfers of whom five play bridge, and fifteen bridge players of whom nine drink gin, so that most golfers play bridge, and most bridge players drink gin, yet no golfers drink gin.

However, although such arguments are often useful, finding them is a hit and miss affair. It is not so much a test for validity, as a strategy that will sometimes detect invalidity. Although we can sometimes find such reasons for rejecting an argument, there is no systematic and generally reliable procedure for finding examples in which one proposition would be true, and another false. The case for placing the onus of proof on the one who proposes the argument stands.

Notice also that the demonstration that an argument is invalid can be complicated and may need numerical precision. That is much easier to achieve in writing than in conversation. I think one of the principle sources of error in logic is conversation; people insist on talking when they need to write. I’m thinking of installing a white board in my house to make thinking easier.

An example will show how easily one can stray into confusion. I was once involved in a discussion that ran roughly as follows:

I: ‘Some researchers have just published results of experiments that show that listening to very loud music damages people’s hearing’.

X: ‘That’s like saying that looking at big things damages your eyesight’

X thought my argument was of the form :

‘great magnitude in what is perceived damages the organ of perception’ an argument that is indeed unreasonable, though I was surprised by X’s assumption that was what I meant as there are two other much more plausible forms either of which fit my argument. My argument conforms to both the patterns:

(1) ‘Conclusions of published experimental results are likely to be broadly true’, which was what I actually had in mind, and

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(2) ‘Stimuli of high intensity are liable to damage sense organs’, which is at least plausible and offers support to the experimental results in question.

So in this case there are at least three patterns that fit the argument. My interlocutor did not ask which I had in mind, but assumed that because he had found an invalid pattern, the argument must be invalid, whereas to show that he’d have needed to show that all patterns that fitted the argument were invalid.

The forgoing discussion is an example of what I call informal logic. Although formal validity is discussed, the point of the discussion is not to show that particular forms are valid, but to decide which forms fit particular arguments. The example shows that even if some argument A can be fitted into a pattern that it shares with a fallacious argument B, that does not show that A is fallacious. There may be some other, valid, form which fits A but not B.

Even in formal logic, people who are familiar with the subject and ought to know better can easily stray when applying their formulae to particular cases. For example a general principle of inference is that if A entails B, then not-B entails not-A so something that entails what is false must itself be false. A true proposition cannot entail a false one. However the converse does not apply. A false proposition may entail a true one. Consider the following example:

let A be 2 = 3 and B be 2 = 2, then A Þ Bfor A Þ 3=2, since x = y Þ y = x

also (2=3 & 3=2) Þ 2 = 2, since (x = y & y = z) Þ x = z

That is all well known to computer programmers, but that did not stop some of them formulating the so called “GIGO” principle, short for “Garbage in, garbage out”, intended to suggest that wrong input guarantees wrong output. They had it back to front; it ought to be the “GOGI” principle. If the output is wrong, there must be something wrong with the input (assuming the program is correct), whereas wrong input just makes it likely that the output will be wrong.

Another example of people failing to apply in practice the general principles they can represent symbolically, is the wholesale condemnation of circular definitions. ‘Nothing can be usefully defined in terms of itself’ people often say. They are probably thinking of ‘x = x’ which is indeed unhelpfully uninformative about x, but what about ‘x = 2x + 4’ in which x is informatively defined in terms of itself ?

Logical Truth

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Although Logic is primarily concerned with the validity of inferences, it is also capable of establishing the truth of propositions. To every valid inference there corresponds at least one logically true proposition, for if Q can validly be inferred from P, P É Q must be true.

Define a logically true proposition as one that is true by virtue of its form. More precisely, as a proposition does not have a unique logical form, a logically true proposition is a proposition that fits at least one form that guarantees truth.

Logical truths do not all correspond to inferences in the simple way that a logical truth of the form P É Q corresponds to the inference of Q from P. Consider for example the set of truths of the form P Ú ~P , like ‘Either I have a DVD recorder or I don’t have a DVD recorder’. All such propositions are logical truths, as are all propositions similarly obtained from any tautology. In general any substitution instance of a tautology is a logical truth.

P Ú ~P is a tautology so, substituting ‘Simon has a pet dog’ for ‘P’, ‘Either Simon has a pet dog or Simon does not have a pet dog’ is a logical truth

The status of logical truths is highly controversial, and I shall return to it several times in later chapters, eventually, or so I hope, tying up the loose ends in chapter 5.

Appendix 1: The Syllogistic Logic

In the Aristotelian logic an important distinction is that between a term that is distributed in a proposition and one that is not. A term is said to be distributed in a proposition when the proposition gives information about everything to which the term applies. Thus in ‘All Fish are vertebrates’ ‘Fish’ is distributed but ‘vertebrates’ is not because the proposition asserts something of every fish, but not of every vertebrate. In ‘No Mammal is Six Legged’ both ‘Mammal’ and ‘Six Legged’ are distributed because the proposition tells us something about every mammal - that it is not six legged, and also tells us something about every six legged creature, namely that it is not a mammal. In propositions of forms I and O, neither term is distributed.

Syllogisms were divided into four figures:

fig 1 fig. 2 fig 3 fig 4M*P P*M M*P P*MS*M S*M M*S M*SS*P S*P S*P S*P

where each '*' is one or another of A, I, E or O. M is the 'middle term' the term that does not appear in the conclusion The term S that is the predicate in the conclusion is called the major term, and the term P that is the subject in

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the conclusion is called the minor term.

The validity of any putative syllogism can be determined by checking that it satisfies the following rules.

(1) There are precisely three terms

(2) There are precisely three propositions

(3) There is no ambiguity, and the middle term is distributed in at least one premiss

(4) No term may be distributed in the conclusion unless it is distributed in a premiss

(5) Nothing may be inferred from two negative premisses

(6) The conclusion is negative if and only if one of the premisses is negative

Rules (1) and (2) are not criteria of validity, they just define the syllogistic form. There are many valid inferences that do not satisfy those two rules, but such inferences are not syllogisms.

Rule (3) excludes the possibility that the instances of M referred to in the major premiss are all distinct from those referred to in the minor premiss - since in that case the two premisses together would imply nothing about the relation of the major and minor terms.

Rule (4) is justified by the consideration that we cannot be justified in drawing a conclusion that asserts something of all instances of one of the terms, unless one of the premisses also does so.

Rule (5) is justified by noting that if both premisses are negative, then any one of the six relations between S and P could be true so that the premisses rule none of them out.

In Formal Logic J. N. Keynes said (P 109) that all the rules of the syllogism could be deduced from just two:

(a) The middle term is distributed in at least once in the premisses, or alternatively no term is distributed in the conclusion if it is not distributed in a premiss.

(b) To prove a negative conclusion one premiss must be negative.

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These rules are satisfied by some invalid syllogisms, but every invalid syllogism that does satisfy them is equivalent to some other invalid syllogism that does not, so (a) and (b) can be used to establish which syllogisms are valid, but are not sufficient to test for validity in a particular case.

The syllogisms of the first figure were often considered primary, because they can be justified by the dictum de omni et nullo, considered by Aristotle to be the basis of syllogistic logic. The dictum asserts that ‘Whatever is predicated, whether affirmatively or negatively, of a term distributed may be predicated in like manner of everything contained under it’ (Keynes op cit p 256). In other words ‘All S is P’ implies that any particular S is P.

Accordingly syllogisms not in the first figure were to be justified by reducing them to syllogisms in the first figure. Reduction involved some combination of the conversion, obversion, and contraposition of one or both of the premisses, and possibly also of the conclusion.

Conversion consisted in interchanging subject and predicate, so that S*M became M*S. If conversion produces a proposition equivalent to the original the process is called simple conversion, thus ‘Some S is M’ is equivalent to ‘Some M is S’ and ‘No P is M’ is equivalent to ‘No M is P’.

However ‘All S is M’ is not equivalent to ‘All M is S’ and in that case conversion produces only ‘Some M is S’, a transformation called conversion per accidens. Note that such a conversion is only valid if ‘all’ implies ‘some’. An O statement cannot be converted at all; from ‘Some S is not P’ there follows no proposition of the form ‘P*S’

Obversion replaces the original predicate of a proposition by its negation. For instance ‘All S is P’ becomes ‘No S is not-P’ and ‘Some S is P’ becomes ‘Some S is not not-P’ Every obverse is equivalent to its original.

Contraposition produces a new proposition in which the predicate is the subject of the original, and the subject is the negation of the predicate of the original. Thus ‘All S is P’ becomes ‘No not-P is S’. The I proposition cannot be contraposed, and the E proposition ‘No S is P’ cannot be contraposed into a universal but only into ‘Some not-P is S’, and even that contraposition is valid only if universal propositions are granted existential import.

When a syllogism can be reduced to one in the first figure by simple conversion of one or more premises, the process is called direct reduction. Otherwise the reduction will be indirect, and will involve showing that the conjunction of one of the premisses with the contradictory of the conclusion implies a contradiction.

To assist in reduction the syllogisms were given mnemonic names. The first figure was considered primary as all other syllogisms were to be justified

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by reduction to one of the first figure. There were four syllogisms of the first figure and they had names beginning with B C, D and F. The name of every other syllogism began with a consonant from {B,C,D,F} the same as the initial letter of the first figure syllogism to which it was to be reduced.

The remainder of the mnemonic name contained three vowels to represent in order the two premisses and the conclusion. Those vowels were mixed with other consonants of which s, p, m, and c [in the middle of a word] indicated the method of reduction; any others were included for euphony.

s : simple conversionp: conversion per accidensm: premises transposedc: indirect reduction, start by omitting the premiss preceding the c

Using those conventions the names were usually given by the following mnemonic verse:

Barbara, Celarent, Darii, Ferioque prioris:Cesare, Camestres, Festino, Baroco, secundae:Tertia, Darapti, Disamis, Datisi, Felapton,Bocardo, Ferison, habet: Quarta insuper additBramantip, Camenes, Dimaris, Fesapo, Fresison

The fourth figure was a medieval addition to Aristotle's original scheme, hence the 'Quarta insuper addit'

To demonstrate the process of reduction, consider the following examples:

(1) A Syllogism in Fresison of the fourth figure

P e M No worms have wingsM i S Some winged creatures stingS o P Some stinging creatures are not worms

The ‘s’s in ‘Fresison represent simple conversion, by applying which to both premisses - no change is required in the conclusion - we obtain:

M e P No winged creatures are wormsS i M Some stinging creatures have wingsS o P Some stinging creatures are not worms

which is a syllogism in Ferio, of the first figure

(2) A Syllogism in Baroco of the second figure

P a M All fish are vertebratesPage 37


S o M Some aquatic creatures are not vertebratesS o P Some aquatic creatures are not fish.

the ‘c’ represents indirect reduction by omitting the minor premiss.

The contradictory of the conclusion is ‘All aquatic creatures are fish’ . Combining that with the original major premiss we have the two premisses of a syllogism in Barbara, of the first figure. Completing the syllogism by adding its conclusion we have:

M a P All fish are vertebratesS a M All aquatic creatures are fishS a P All aquatic creatures are vertebrateswith a conclusion that contradicts the original minor premiss, hence the

falsity of the conclusion of the syllogism in Baroco contradicts the original premisses , so the syllogism is valid.

Appendix 2: Justifying LogicFrom the time of Aristotle logic has been widely regarded as central to

knowledge, regulating the processes of reasoning and justification, so the question has often been raised, how Logic itself might be justified.

Aristotle himself thought logic self evident. He did not claim that every individual logical principle needed to be justified by a separate intuition of self evidence, but that there were a few self evident principles from which the rest of logic followed. He thus helped to prepare the way for the procedure of axiomatisation, in which a whole body of knowledge is derived from a subset of basic propositions known as the axioms, a procedure followed not just by logicians but most enthusiastically by mathematicians, for instance by Euclid who axiomatised the geometry of his day (around 300 BC).

At first axiomatisation was seen as a key to justification, by reducing the problem of justifying a large (usually infinite) collection of propositions to that of justifying only the axioms, but in recent years axiomatisation has come to be thought of mainly as a way of tidying up a theory. In the case of mathematical theories, it also offers a way of determining to which non mathematical problems some body of mathematics may be applied. If the properties of some physical system can be identified with the symbols of a mathematical theory in such a way that the axioms are true and the rules of inference are valid, then, provided the symbols are interpreted in that sense, the theorems must also be true for the subject matter in question.

However that view does not totally divorce axiomatisation from justification. Suppose we check that, under a certain interpretation of the symbols, a system satisfies the axioms of a theory. We are then checking that, under that interpretation, the axioms are true, and if we check that the rules of inference are valid, that check shows that those rules preserve truth.

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In general, if a set of propositions can be presented as an axiomatised theory, all we need do to justify all the propositions in that set is to justify the propositions that interpret the axioms, and demonstrate that the rules of inference preserve truth.

Some Mathematicians, known as ’Formalists’ regard their collections of formulae as just formal systems - sets of formulae manipulated according to certain rules. They then usually deny that the formulae actually assert any propositions. The relation of the formulae to propositions is, Formalists say, only that formulae are available to represent sets of propositions when someone finds a suitable interpretation for the notation. From that point of view there is no question of the truth of a formula. Of the formula itself we may ask only what place it has in the formal system. Truth arises only when we are given an interpretation of the formal system; we may then investigate the truth of whatever proposition the formula represents in that interpretation.

The Formalist approach has also been applied to mathematical Logic. I shall consider that in more detail in Chapter 4. However Formalism is a recent development and for the moment I wish to examine early attempts to justify the principles of Logic, so I shall return to Aristotle.

Aristotle sought to base Logic on two principles that he considered self evident. They were the Law of non-contradiction ( ~(P&~P)), a proposition can’t be both true and not-true, and the law of the excluded middle. (Pv~P), either something is true, or it isn’t. Later, other logicians added the law of identity. [("x)(x = x)], everything is the same as itself

Aristotle considered non contradiction and excluded middle to be self evident, remarking that any attempt to prove them would be circular. However he went on to say that the impossibility of proving either law does not rule out all argument in their favour. The impossibility of a proof rules out any argument calculated to force assent on someone who is simply disinclined to form an opinion, but if we are confronted with someone who actually denies either principle there is a good deal we can say, for powerful ad hominem arguments are available.

Aristotle considered mainly the law of non-contradiction and gave a number of arguments of uneven quality, but the general drift of his reasoning was that if someone denies the law, then either he will be so using language that he fails to communicate anything, or he will be guilty of inconsistency in sometimes himself relying on the very principle he claims to deny.

The extreme case of irrationality, thought Aristotle, would be someone who asserts a sentence of the form ‘A is not B’ in every case that he asserts a sentence of the form ‘A is B’. Such a person would apparently be prepared to assert anything at all, so once we have noticed that, his remarks will tell us

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nothing; he fails to communicate.

On the other hand, someone who asserts both ‘A is B’ and ‘A is not B’ in only some cases appears to be using a limited version of the rule of non-contradiction. That offers us an opportunity to ask him on what basis he refuses to assert both ‘C is D’ and ‘C is not D’, and why his reason for shunning that contradiction does not extend to ‘A is B’ and ‘A is not B’

It often turns out that when someone wants to assert an apparent contradiction, there is an ambiguity so that the two propositions the person wants to assert are only apparently contradictory, using the same word in different senses. For example someone may say ‘It is raining, and it isn’t’ because it is neither raining heavily, nor completely dry. ‘raining’ and ‘not raining’ are then interpreted as contraries, not contradictories, so that ‘It is not raining’ does not mean the same as ‘~(It is raining)’ but ‘It is completely dry’

Aristotle went on to give examples of how tolerance of contradiction would undermine most rational discussion. Discussing the impact on his theory of substance and essence he observed that if some properties of an object are essential to objects of that sort, such objects necessarily possess them and so necessarily cannot not possess them. Aristotle used the example of ‘men are two legged’. If two leggedness is part of the definition of ‘man’, then ‘men are not two legged’ must be rejected.

He further argued that practical judgements as to what to do would be undermined if, every time we asserted ‘A is B’ we also asserted ‘A is not B’, because any reason we might have for any decision would thus be undermined by the assertion of its contradictory.

Although Aristotle devoted most of his argument to the law of non contradiction, he also cited the law of the excluded middle when he argued that rejection of non contradiction would imply the rejection of the law of the excluded middle. For suppose we can assert both ‘A is B’ and ‘A is not B’. From ‘A is B’ it follows that ‘~(A is not B)’, and from ‘A is not B’ it follows that ‘~(A is B)’, implying the denials of both ‘A is B’ and ‘A is not B’, whereas the law of the excluded middle asserts that one of those must be true.

Aristotle thought that both Heraclitus and Protagoras were guilty of denying the law of non-contradiction. Heraclitus had asserted an apparently blatant contradiction when he said that something can at the same time be and not be.

Protagoras had offended less obviously by saying the man is the measure of all things. Aristotle interpreted that as meaning ‘what seems to each man is so’ which implies that if things seem differently to different people, their different views are all correct. Aristotle thought that Protagoras’ error showed the danger of basing knowledge on our perceptions, which risks their

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misleading us if the sense organs are damaged, or if other factors produce sensory illusions.

Axioms for Modern Logic

In the first edition of Principia Mathematica Russell and Whitehead gave five axioms and two rules of inference for the propositional logic. They usually referred to their axioms as ‘primitive propositions’ using the word ‘axiom’ only occasionally, but ‘axiom’ was clearly what they meant so I shall use the word in describing their system.

The only truth functions that appeared in their axioms were Ú and É which was an odd choice, since they took as primitive truth functions V, &, and ~, defining É so that: P É Q was shorthand for ~P Ú Q

Russell and Whitehead wrote ‘P.Q’ instead of ‘P&Q’ and they represented the universal quantifier as (x)f(x) not as ("x)f(x) but I shall use P&Q and ("x) when discussing their system.

The axioms were:(P Ú P) É PQ É (P Ú Q)(P Ú Q) É (Q Ú P)(P Ú (Q Ú R) É (Q Ú (P Ú R) : later shown to be deducible from the other

axioms(Q É R) É ((P Ú Q) É (P Ú R))

The rules of inference were:

rule of substitution: from any axiom or theorem another theorem may be obtained by substituting any well formed formula in place of (all occurrences of) any letter.

rule of detachment from theorems of the forms P and P É Q, deduce Q

With those rules of inference the fourth axiom was eventually shown to be deducible from the others, so it could be omitted without weakening the system.

To extend the system to include quantification, there were further definitions and axioms and one additional rule of inference for quantified formulae.

Substitution into formulae containing quantifiers had to be restricted so that no formula containing a free variable was allowed to be substituted into any part of a formula that falls within the scope of a quantifier applying to the letter used as free variable.

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Thus in a formula ($x)[(F(x) v P) É G(x)] it would not be permitted to substitute H(x) for P, but the substitution of Q, or of H(y) or of ($x)[(H(x)) would all be allowed.

The axioms for quantifiers were:

f(x) É ( $z)f(z)(f(x) Ú f(y)) É ( $z)f(z)(x)f(x) É f(y) where ‘y’ may be any symbol that names a particular

element of which f(y) could meaningfully be asserted.

The additional rule of inference was that if a formula f(y) can be proved for arbitrary y, we may infer (x)f(x)

There were also several definitions specifying how negation, conjunction and disjunction apply to quantifiers, making the system of Principia rather more complicated than most later systems, which defined one of the quantifiers in terms of the other.

Nicod’s Single Axiom System.

I have already remarked that all the truth functions can be defined in terms of ‘not both’ usually symbolised ‘|’ so that ‘P|Q’ means that P and Q are not both true and is equivalent to ~P Ú ~Q The truth table is:

P Q P|QT T FT F TF T TF F T

In 1917 Nicod showed that the whole of truth functional logic could be deduced from the single axiom [P|(Q|R)]|[([T|(T|T)]|[{(S|Q)|([(P|S)|(P|S)]})

with substitution and the rule of inference:

from P and P|(Q|R) infer R

That discovery was applauded by Russell, who seemed to think that once the number of axioms has been reduced to just one, the position of truth functional logic had been strengthened since it rested on only one assumption instead of on several. Yet no intuitive plausibility attaches either to that one axiom, or to the rule of inference. Indeed I am not entirely sure that I have correctly transcribed the axiom from the book where I found it, and even if I have, there is a small but appreciable chance that, with such a complicated formula, the book may contain a typesetting error undetected during proof

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reading.

An important aspect of ‘intuitive’ assumptions may be, not that intuition provides some guarantee that they are true, but that they are simple enough for us to take them in one piece so to speak and so be sure that we have transcribed them correctly.

At best a system with just one axiom may offer a technical advantage by simplifying proofs about the formal system in some metalanguage. Axiomatic systems have come to be seen mainly in that light - as convenient technical devices.

Hilbert and Bernays System

Trying to combine technical convenience and intuitive plausibility Hilbert and Bernays produced (in 1934) a set of fifteen axioms for the propositional logic, using all five of the standard truth functions, and defining none in terms of the others. The first three axioms involve only material implication, and each of the remaining axioms involves material implication and just one other truth function. The axioms were:

P É (Q É P)[P É (P É Q)] É (P É Q)(P É Q) É (Q É R) É ((P É R))P & Q É PP & Q É Q(P É Q) É (P É R) É ((P É Q & R))P É PÚQQ É PÚQ(P É R) É [(Q É R) É (PÚQ É R)]P º Q É (P É Q )P º Q É (Q É P )(P É Q) É (Q É P) É ((P ºQ ))(P É Q) É (~Q É ~P)P É ~~P ~~P É P

The rules of inference are substitution and detachment - that is from P and P É Q deduce Q

The axioms are all independent, though it would be possible to reduce their number by defining some of the connectives in terms of others.

Exercise.

I added this exercise at the request of a friend who had read an earlier draft of these notes. I strongly suspect that neither he nor anyone else has

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attempted to answer any of the questions, but it was fun making them up.

(1) Which of the following are well formed formulae of the propositional calculus?:(a) (P&Q)vR (b) P&~(Q&vR) (c) R ÉÉ S (d) P&(Q&(R&(S&T)))

(2) Construct truth tables for:(a) P&~Q (b) ~(Pv~Q) (c) Pv(Q&R) (d) (PvQ)&R

(3) Construct truth tables for each of the following formulae and identify any formulae that represent tautologies, and any that represent contradictions:

(a) P É (P&Q) (b) (PvQ) É P (c) P É (Q É P) (d)~Pv(P&Q)

(e) (PvQ) & (Pv~Q) (f) P É ~P (g) P v ~P (h) ~~P º P

(4) Construct truth tables for the following and hence find equivalent formulae containing fewer symbols.

(a) ~(P&~Q) (b) P É ~P (c) ~(Pv~Qv~R) (d) Pv(~P&Q)

(5) Show that each of the following is a tautology,

(a) (P É Q ) º ~(P&~Q), (b) (PvQ) º (~(~P&~Q), (c) (PºQ) º [(P&Q)v(~P&~Q)]

and hence show that each of ‘É’, ‘v’, and ‘º’ can be defined in terms of ‘~’ and ‘&’

(6) By considering:

(P É Q ) º (~PvQ), (P&Q) º (~(~Pv~Q)), and (PºQ)º{~(PvQ)}v{~(~Pv~Q)}]

show that ‘É’, ‘&’, and ‘º’ can all be defined in terms of ‘~’ and ‘v’

(7) Show that ‘&’, ‘v’ and ‘º’ can all be defined in terms of ‘~’ and ‘É’

(8) Find a truth functional formula to represent the following:

(a) Either Mary plays tennis and Anne owns the golf club, or Mary pays golf and Anne does not own the Golf club

(b) If William is a hairdresser then either James is a cook or Felicity owns the hairdressing salon.

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In questions (9) to (14) inclusive decide whether the reasoning is valid:

(9) All members of the tennis club are freemasons, some freemasons play chess, therefore some members of the tennis club play chess.

(10) All members of the potholing club are car owners, some members of the potholing club are also mountaineers, therefore some mountaineers are car owners.

(11) Most jockeys belong to the Union of Professional Equestrians, most jockeys play bridge, therefore some bridge players belong to the Union of Professional Equestrians.

(12) Most professional footballers are under the age of 30. Most under 30’s watch television for more than ten hours per week, therefore some professional footballers watch television for more than ten hours per week.

(13) All monks are men, some men are virgins, therefore some monks are virgins.

(14) If the red light on the printer comes on then either the printer is out of paper, or the paper has jammed, but the printer is not out of paper and the paper has not jammed, so the red light is not on.

(15) To join the Drones Club one must have no paid employment and live either within one mile of Buckingham Palace, or in a country house North of the Thames with grounds of at least 60 acres. To join the Country Club one must live in a country house with grounds of at least 200 acres.Does it follow that someone who belongs to the Country Club but is not eligible for membership of the Drones must have paid employment?

(16) Use quantifiers to represent:

(a) There is a man who is either a hypnotist or a vampire

(b) Either there is a man who is a hypnotist, or there is a man who is a vampire

(c) The only topologist living in Rutland is a keen gardener.

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