Download Chapter 2, Logic

What is Philosophy Chapter 2
by Richard Thompson
Logic
(last edited on 1st August 2011)
In this chapter I outline the basics of Logic; there is more about Logic
4 and 5.
in Chapters
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 conclusion. A valid
Page
What is Philosophy Chapter 2
by Richard Thompson
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 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.
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 does have a cat.
‘Simon
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,
Page
What is Philosophy Chapter 2
by Richard Thompson
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:
I:
E:
O:
All S is P, example: all mice are mammals
Some S is P, example: some atheists are vegetarians
No S is P, example: no razor blades are made of chocolate
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 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.
Page
What is Philosophy Chapter 2
by Richard Thompson
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
both be true.
are pairs of propositions that cannot both be false, but might
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, 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
Page
What is Philosophy Chapter 2
by Richard Thompson
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) Mammals have four chamber hearts,
(2) Elephants are mammals,
therefore (3) Elephants have four chamber hearts”
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 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 reversed
Page
What is Philosophy Chapter 2
by Richard Thompson
Case 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, 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.
Page
What is Philosophy Chapter 2
by Richard Thompson
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 have
a list of patterns of valid argument, we can 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 it 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 Logic
Page
What is Philosophy Chapter 2
by Richard Thompson
The 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 (1815-1864) 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.
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
F
F
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
T
T
F
F
Q
T
F
T
F
P&Q
T
F
F
F
P & Q is true only when both P and Q are individually true. P & Q is false in all other
circumstances
Page
What is Philosophy Chapter 2
by Richard Thompson
all other truth functions can be defined in terms of and , but it is convenient
to provide independent definitions for several others.
Disjunction, OR 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
T
T
F
F
F
T
F
PVQ
T
T
T
T
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
T
T
F
F
F
T
F
Q
T
T
T
F
P XOR Q
F
The Biconditional, symbol  , P  Q asserts that P and Q have the same truth
value
P
T
T
F
F
F
T
F
Q PQ
T
T
F
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 , 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
T
Q PQ
T
T
Page
What is Philosophy Chapter 2
by Richard Thompson
T
F
F
F
T
F
F
T
T
If we stipulated that P  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 others. 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 ‘>3
therefore 10>30’, so T  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  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  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  F must be counted false
According to this definition P  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  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 minds
Admit impediments. Love is not love
Which 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 cheeks
Within 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:
“suppose
2 + 2 = 3,
Page
What is Philosophy Chapter 2
by Richard Thompson
subtracting 2 from both sides gives: 2 = 1
Russell and the Pope are two, therefore Russell and the Pope are one.”
Although it may seem strange that F  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 we know P  Q 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  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  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  Q entails Q
(2) It is transitive, [P  Q and Q  R] entails P  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  Q and notice that the only case where P, and P  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  Q & Q  R]  [P  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  Q) (Q  ~P) is always true
Although P  Q is not equivalent to ‘P implies Q’, [P & P Q)] does entail Q as I’ll
soon show, so that it is impossible for [P & P Q)] to be true unless Q is also true.
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
Page
What is Philosophy Chapter 2
by Richard Thompson
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  Q) (P V Q) is a tautology,
proof:
P
T
T
F
F
T
F
T
F
Q PQ
T
F
F
F
T
T
T
T
P
T
F
T
T
P V Q
(P  Q) (P V Q)
T
T
T
T
 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  Q) (P V Q) must be true.
‘P
Equivalent Formulae If F 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 P , (P  Q ) (P & Q),
P V Q ( P & Q)
the last two equivalencies show that both V and 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 contradiction.
A is a tautology A is
Strict Implication
P  Q, read as ‘P strictly implies Q’ is defined as: ‘P  Q is a tautology’.
Strict implication has been proposed as an analysis of entailment.
Page
What is Philosophy Chapter 2
by Richard Thompson
It avoids the paradox of the material conditional. P’s falsehood does not guarantee
the truth of P  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)  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  Q infer Q
I2: Transitivity: from P  Q and Q  R , infer P  R
I3: And: P & Q  P and P & Q  Q
I4: OR: P  P V Q, for any Q
I5: OR: from Pand(P V Q) infer Q
To prove (P& P)  Q
suppose (P& P)
then P
(by I3)
(1)
(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)  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  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
Page
What is Philosophy Chapter 2
by Richard Thompson
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 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.
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.
Page
What is Philosophy Chapter 2
by Richard Thompson
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}
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,
 However, using only  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
Page
What is Philosophy Chapter 2
by Richard Thompson
(3) Use  to form the disjunction of all the formulae obtained according to (2)
For example, consider the truth function ‘*’ defined by the truth table:
P
T
T
F
F
Q
T
F
T
F
P*Q
F
T
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)  (~P ~Q)
The method described above does not always give the simplest formula. 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‡Q
T
T
T
T
F
T
F
T
F
F
F
T
P‡Q is false only when P is false and Q is true, corresponding to ~PQ
so P‡Q is equivalent to ~(~PQ) which is, of course, P  Q
It is not necessary to use all three of ~, , to define all truth functions.
may be defined in terms of ~ and , since P  Q is equivalent to ~(~P ~Q)
so all truth functions could be defined in terms of ~ and . Alternatively by
defining P  Q as ~(~P ~Q) we could define all truth functions in terms of ~
and .
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 ‘’. For ~P is equivalent to P|P so P  Q is equivalent
to not both (not P and Not Q) which is (P|P)|(Q|Q).
‘|’
Page
What is Philosophy Chapter 2
by Richard Thompson
Once ‘~’ and ‘’. 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 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]  [x is white])
Nowadays it is more common to use the notation:
(x)( [x is a swan]  [x is white] )
‘x’
is called the universal quantifier and is read as ‘for all values of x’
Note that because we use  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
Page
What is Philosophy Chapter 2
by Richard Thompson
equivalent to ‘it is a swan and it is white’ with no indication of 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)  G(x)’ but that should be interpreted as ‘some
proposition of the form F(x)  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
Page
What is Philosophy Chapter 2
by Richard Thompson
white, since ‘[x is white]’ and ‘[x is white]’ fall within the scope 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.
Page
What is Philosophy Chapter 2
by Richard Thompson
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 procedures 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 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.
Page
What is Philosophy Chapter 2
by Richard Thompson
(3) Detachment If P and P  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  Q is a theorem, then so is x)P  Q
(5b) If P  Q is a theorem, and x is not free in P, then P  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)  Q and x is not free in Q, infer (x)(F(x))  Q
(5bS) From P  F(x) infer P  (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  x is a viviparous quadruped we may infer of
our pet Fido, ‘Fido is a squirrel  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 )  H(x)) premiss
(2) (x)( H(x )  G(x)) premiss
(3) F(x )  H(x) from (1)
(4) H(x )  G(x) from (2)
(5) F(x )  G(x) from (3) & (4), by substituting F(x) for P, G(x) for Q and H(x)
Page
What is Philosophy Chapter 2
by Richard Thompson
for R in the tautology
(P Q)&(Q R)(P R) and using detatchment
(6) (1)&(2)  (F(x )  G(x) ) from (5)
(7) (1)&(2)  (x)(F(x )  G(x) ) from (6) because x is not free in (1) or in (2)
(8) (x)(F(x )  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  B may
be proved by assuming A and showing that it is then possible to deduce B.
Since AB is equivalent to ~A  B , the same strategy can also be used to prove
AB, 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)
 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 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 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
Page
What is Philosophy Chapter 2
by Richard Thompson
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 ‘’ 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)  (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 like to watch ‘Morse’
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 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 is
Q’ implies ‘Some P are Q’, but ( x)( P(x)  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
Page
What is Philosophy Chapter 2
by Richard Thompson
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)  Q(x)) represents the
interpretation of the universal proposition which does not imply the existence of
any P‘s, while (x)( P(x)  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)  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)  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)  Q(x)) which involves no existential commitment.
Alternatively we might interpret every proposed Aristotelian inference A  B where
A and B are propositions involving the terms P and Q, as assuming the existence
objects of all the kinds referred to so that A  B would mean:
[( x)(P(x))&( x)(Q(x)]  [A  B ]
It would then be possible to represent the basic Aristotelian propositional types
as:
P a Q: (x)( P(x)  Q(x))
P i Q: ( x)( P(x)(Q(x))
P e Q: (x)( P(x)  ~Q(x))
P o Q: ( x)( P(x)(~Q(x))
while preserving in a somewhat convoluted form the logical relations Aristotelian
logicians traced between them.
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.
Page
What is Philosophy Chapter 2
by Richard Thompson
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’, for 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 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 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 Quadriviumand 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
Page
What is Philosophy Chapter 2
by Richard Thompson
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&.....t20
Conclusion t3&t7
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 Brothers
C2: Sophie’s Father has no sisters
Page
What is Philosophy Chapter 2
by Richard Thompson
C3: Sophie’s Mother was an only child
C4: Mary is Alan’s Mother
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.
‘Aunt’
There are eight possible arguments to be considered here.
P1 & P2 C1, P1 & P2  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 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 pattern 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’
Page
What is Philosophy Chapter 2
by Richard Thompson
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 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 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)  G(x)) v (x)(F(x)  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’
Page
What is Philosophy Chapter 2
by Richard Thompson
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 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
Page
What is Philosophy Chapter 2
by Richard Thompson
is logically necessary and the conclusion a contradiction, producing something like:
(P v P)  (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 antecedent 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:
‘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
Page
What is Philosophy Chapter 2
by Richard Thompson
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
(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  B
for A  3=2, since x = y  y = x
also (2=3 & 3=2)  2 = 2, since (x = y & y = z)  x = z
Page
What is Philosophy Chapter 2
by Richard Thompson
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
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  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  Q corresponds to the inference of Q from P. Consider for example
the set of truths of the form P  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  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
Page
What is Philosophy Chapter 2
by Richard Thompson
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
M*P
S*M
S*P
fig. 2
P*M
S*M
S*P
fig 3
M*P
M*S
S*P
fig 4
P*M
M*S
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 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 premiss 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.
Page
What is Philosophy Chapter 2
by Richard Thompson
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.
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
Page
What is Philosophy Chapter 2
by Richard Thompson
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 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 conversion
p: conversion per accidens
m: premises transposed
c: 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 addit
Bramantip, 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 wings
M i S Some winged creatures sting
S o P Some stinging creatures are not worms
Page
What is Philosophy Chapter 2
by Richard Thompson
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 worms
S i M Some stinging creatures have wings
S 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 vertebrates
S o M Some aquatic creatures are not vertebrates
S 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 vertebrates
S a M All aquatic creatures are fish
S a P All aquatic creatures are vertebrates
with a conclusion that contradicts the original minor premiss, hence the falsity
of the conclusion of the premiss in Baroco contradicts its premisses , so the
syllogism is valid.
Appendix 2: Justifying Logic
From 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
Page
What is Philosophy Chapter 2
by Richard Thompson
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.
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
Page
What is Philosophy Chapter 2
by Richard Thompson
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 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
Page
What is Philosophy Chapter 2
by Richard Thompson
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 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  and  which was an
odd choice, since they took as primitive truth functions V, , and ~, defining 
so that: P  Q was shorthand for ~P Q
Russell and Whitehead wrote ‘P.Q’ instead of ‘P&Q’ and they represented the universal
quantifier as (x)(x) not as (x)(x) but I shall use P&Q and (x) to discuss their
system.
The axioms were:
(P  P)  P
Q  (P Q)
(P Q)  (Q P)
(P (Q R)  (Q (P  R) : later shown to be deducible from the other axioms
(Q  R)  ((P Q) P  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.
Page
What is Philosophy Chapter 2
by Richard Thompson
rule of detachment from theorems of the forms P and P  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.
Thus in a formula (x)[(F(x) v P)  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:
(x)  ( z)(z)
((x) (y ( z)(z)
(x)(x)  (y) where ‘y’ may be any symbol that names a particular element of which
(y) could meaningfully be asserted.
The additional rule of inference was that if a formula (y can be proved for arbitrary
y, we may infer (x)(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  ~Q The truth table is:
P
T
T
F
F
Q
T
F
T
F
P|Q
F
T
T
T
In 1917 Nicod showed that the whole of truth functional logic could be deduced from
Page
What is Philosophy Chapter 2
by Richard Thompson
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 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  (Q  P)
[P  (P  Q)]  (P  Q)
(P  Q)  (Q  R)  ((P  R))
P  Q  P
P  Q  Q
(P  Q)  (P  R)  ((P  Q  R))
P  PQ
Q  PQ
(P  R)  [(Q  R)  (PQ  R)]
P  Q  (P  Q 
P  Q  (Q  P 
(P  Q)  (Q  P)  ((P Q ))
(P  Q)  (~Q  ~P)
Page
What is Philosophy Chapter 2
by Richard Thompson
P  ~~P
~~P  P
The rules of inference are substitution and detachment - that is from P and P 
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.
(1) Which of the following are well formed formulae of the propositional calculus?:
(a) (P&Q)vR
(b) P&~(Q&vR)
(c) R  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  (P&Q)
(e) (PvQ) & (Pv~Q)
(b) (PvQ) P
(c) P  (Q  P)
(d)~Pv(P&Q)
(f) P  ~P (g) P v ~P
(h) ~~P  P
(4) Construct truth tables for the following and hence find equivalent formulae
containing fewer symbols.
(a) ~(P&~Q)
(b) P  ~P (c) ~(Pv~Qv~R)
(d) Pv(~P&Q)
(5) Show that each of the following is a tautology,
(a) (P  Q )  ~(P&~Q), (b) (PvQ)  (~(~P&~Q), (c) (PQ)  [(P&Q)v(~P&~Q)]
and hence show that each of ‘’, ‘v’, and ‘’ can be defined in terms of ‘~’ and ‘&’
(6) By considering:
(P  Q )  (~PvQ),
(P&Q)  (~(~Pv~Q)), and (PQ){~(PvQ)}v{~(~Pv~Q)}]
show that ‘’, ‘&’, and ‘’ can all be defined in terms of ‘~’ and ‘v’
(7) Show that ‘&’, ‘v’ and ‘’ can all be defined in terms of ‘~’ and ‘’
(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
Page
What is Philosophy Chapter 2
by Richard Thompson
(b) If William is a hairdresser then either James is a cook or Felicity owns the
hairdressing salon.
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.
Page