What is Philosophy Chapter 2
by Richard Thompson
Logic
(last edited on 18th March 2010)
In this chapter I outline the basics of Logic; there is more about it in Chapters 4
and 5.
I’m not writing a logic text book, so I shall discuss the technicalities of formal logic
only where they are needed to follow the discussion of the comments various
philosophers have made about logic. I learnt the basics of formal logic from W. V. Quine’s
Methods Of Logic, and I have never come upon a better introduction, though of course,
having read one introduction, I’ve never done more than glance at others. William and
Martha Kneale The Development Of Logic is an excellent historical survey, which also
explains the technicalities clearly.
Inferences, Propositions and Entailment
Logic is the study of the validity of inferences. It tells us what follows from what.
Formal Logic gives precise rules that ensure the validity of any inference that satisfies
them.
I’ll start by introducing some terminology. An inference proceeds from a starting
point to an end point. We need a word for the types of entity that can feature in an
inference. The one most commonly used is ‘proposition’. A proposition is some sort of
claim that can be either true of false. Some logicians prefer to talk of sentences, on the
grounds that that gives us a definite subject for discussion. The supporters of ‘proposition’
retort that the meanings of words can change, words can be ambiguous, the same
sentence can mean different things on different occasions, and a variety of different
sentences can be used to make the same claim. Also a sentence belongs to a particular
language, while logic studies ideas that are independent of language. A proposition may
be thought of as what a sentence means on a particular occasion, or what the user is
trying to put across when they use a sentence. W.V. Quine and his supporters, who
preferred to talk of sentences, considered it impossible to define ‘meaning’ or ‘proposition’
independently of a sentence. ‘statement’ has sometimes been adopted as a compromise,
a ‘cowardly’ policy in the opinion of Quine though ‘statement’ has the advantage of
suggesting a particular utterance made by a particular person at a particular place and
time.
I’ve decided to use ‘proposition’. I’ll discuss the arguments for and against the
existence of propositions in Chapter 5, but shall say no more on the matter in this
chapter.
So the central idea in Logic is that of inference from one proposition to another.
The propositions from which an inference begins are called the premisses, and the
proposition at which an inference arrives is called its conclusion. A valid inference is
one in which the truth of the conclusion is guaranteed if the premisses are true. We say
that P entails Q, if P and Q are propositions such that, if P is true, Q must also be true,
so that proceeding from P to Q is a valid inference. The word imply is often used instead
of entail, but as we shall see later Russell and Whitehead confused the issue by using
material implication for a much weaker relation, provoking G. E. Moore to introduce
‘entail’.
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by Richard Thompson
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 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.
What is a good reason for one person to believe something may not be a good
reason for someone else. In particular, if someone believes that P is false ‘P entails Q’ is
not a good reason for him to believe Q. ‘P entails Q’ is said to be a ‘good ’argument,
when P is true, and ‘P entails Q’ is valid.
To identify all good arguments would require us to know the truth of all true
propositions, so that to include the identification of good arguments in Logic would require
that Logic include the whole of knowledge. Aristotle thought that it did, but today there are
few who would agree.
Although the mere validity of an argument does not guarantee the truth of its
conclusion it does not follow that the study of validity is pointless, for an argument that is
not valid is not a good reason for anyone to believe its conclusion.
Aristotelian Logic
The first recorded study of formal logic was by Aristotle who described the logic of
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propositions of four types, namely those that conformed to one the forms: “All S is P”,
“Some S is P”, “No S is P” or “Some S is not P”. The broad outlines of Aristotelian logic
were preserved right into the nineteenth century, but it was elaborated in the middle ages
so I shall discuss that slightly extended form of the logic, and I shall also use some of the
medieval notation.
Aristotle’s logic began with the examination of the logical relations between pairs of
propositions, and then used that as the basis for considering more elaborate arguments
involving larger numbers of propositions.
The medieval logicians referred to the four types of Aristotelian proposition as
A: All S is P, example: all mice are mammals
I: Some S is P, example: some atheists are vegetarians
E: No S is P, example: no razor blades are made of chocolate
O: 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), so we need consider only six distinct propositions.
Pairs of propositions may be related in one or another of several ways.
Contradictories Two propositions are contradictories when the truth of either one is
equivalent to the falsehood of the other that so that one and only one is true.
Contradictory pairs are:
S a P (all S is P) and S o P (some S is not P), eg. ‘all cats like milk’ and ‘some cats
do not like milk’
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’
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 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.
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
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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.
Subcontraries are pairs of propositions that cannot both be false, but might both
be true.
The i and o propositions are subcontraries, since ‘Some S is P’ and ‘Some S is not
P’ might both be true, but cannot both be false. eg. ‘some solicitors are freemasons’ and
‘some solicitors are not freemasons’ could both be true, but they could not both be false,
because if it were false that some solicitors are not free masons, then all solicitors are
freemasons.
Existential Import of Universal Propositions
The forgoing discussion of the four Aristotelian types of proposition assumes that ‘all
implies some’ that is that ‘All S is P’ implies ‘Some S is P’ and ‘All S is not P’ implies
‘Some S is not P’ That assumption is problematical and gave rise to a good deal of
debate among logicians. For sometimes we assert universal generalisations without any
commitment to existence.
For instance if we explained ‘unicorn’ by saying ‘Unicorn’ means ’quadruped
mammal resembling a horse but with a single horn projecting from the middle of its
forehead’ we could confidently assert ‘any unicorn has four legs’ but should not think
ourselves thereby committed to the existence of any unicorns, so we should reject the
inference from ‘all unicorn have four legs’ to ‘some unicorns have four legs‘ since that
involves there actually being some unicorns.
Furthermore, since ‘All S is not P’ is held to be equivalent to ‘All P is not S’ it
appears that S e P not only entails that there are S, but also entails that there are P, so
that ‘No animals are unicorns’ entails ‘No unicorns are animals. If we allow that to entail
‘some unicorns are not animals‘ we should be committed to the existence of unicorns.
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
entailed 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
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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
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.
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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.
Extracting from the third diagram just the information about gardeners and pipe
smokers, we have the last diagram, which corresponds to the conclusion.
Modern Logic
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
Premiss 2
Premiss (1)
subject
enter a
century
of
rapid
growth
into
what is
now
called
‘Mathe
matical
Logic’.
Aristotle’s
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Conclusion
What is Philosophy Chapter 2
by Richard Thompson
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
T
F

P 
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 P&Q
T
T
F
F
T
F
F
F
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
T
T
F
F
Q PVQ
T
T
F
T
T
T
F
F
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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.
The Biconditional , symbol  , P  Q asserts that P and Q have the same truth
value
P
T
T
F
F
Q PQ
T
T
F
F
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
T
F
F
Q PQ
T
T
F
F
T
T
F
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, 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.
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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,
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 represents any proposition at all) the convention is
harmless in the sense that it does not provide a reason to believe an arbitrary conclusion.
For suppose we know P  Q on the basis of P’s being false. For example,
the moon is made of green cheese  Tony Blair is Queen of England.
Although 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,
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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 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
Q PQ
T
T
F
F
T
T
F
T
P
F
F
T
T
P V Q
T
F
T
T
(P  Q) (P V Q)
T
T
T
T
‘P  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.
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 is a tautology A is a
contradiction.
Strict Implication
P  Q, read as ‘P strictly implies Q’ is defined as: ‘P  Q is a tautology’.
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Strict implication has been proposed as an analysis of entailment.
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)
then P V Q from (2) (by I4 and I1)
(1)
(2)
(3)
then Pfrom (1), (by I3)
then Q from (3) and (4), (by I5)
(4)
(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
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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 on other
matters are consistent. Indeed it seems reasonable to suppose that they are generally
inconsistent. For our opinions on matters of fact are at best highly probable, and a set of
propositions all highly probable may well be inconsistent.
Contemplate two successive throws of a standard unbiased die. The probability of
throwing two sixes is 1/36 so we are justified in asserting ‘ Very probably the result will
not be two sixes’.
Altogether there are 36 possible outcomes of the experiment, one corresponding to
each ordered pair of two numbers, either different or equal, selected from {1,2,3,4,5,6}.
Each of those outcomes occurs with probability 1/36, so we are justified in saying of every
one of the outcomes that it is very unlikely. Of course one of those possible results must
occur, but until we have thrown the die, we don’t know which outcome that is.
Now consider the set of 36 propositions of the form:
‘The result will not be (x, y)’ for all values of x, y in {1,2,3,4,5,6}
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.
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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
4
the complete table can be filled in 2 = 16 different ways. In general the truth table for n
n
propositional variables has 2 rows, each of which can be filled with either of the two truth
k
n
values, so there are 2 possible truth functions, where k = 2 .
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
(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
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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)].
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) 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 te
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] )
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‘x’ is called the universal quantifier and is read as ‘for all values of x’
We might also want to say ‘some swans are white, but some swans are not white’
We then employ the existential quantifier, , and our new proposition is written:
E1: (x)([x is a swan]  [x is white]) & (x)([x is a swan]  [x is white])
(x) Is read as ‘there is some x such that…’
notice that ‘but’ is treated as equivalent to ‘and’
The letters x, y... appearing in the quantifiers are called bound variables. They
have the same function as pronouns have in ordinary discourse. If we drop the quantifier
and just write:
E1*
[x is a swan]  [x is white]
the x is then called a free variable and E1* is not a proposition, but something
called an ‘open sentence’, just a sort of blueprint for a proposition. It is roughly
equivalent to ‘it is a swan and it is white’ with no indication of 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’
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 the part of the following expression containing the
instances of ‘x’ to which the quantifier applies. The instances of ‘x’ to which the quantifier
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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 the scope of the first quantifier is ‘[x is a swan]  [x is white]’, while the scope
of the second quantifier is ‘[x is a swan]  [x is white]’
Notice that there is no contradiction in using the same letter ‘x’ both in the assertion
that some swans are white, and in the assertion that some swans are not white, since ‘[x
is white]’ and ‘[x is white]’ fall within the scope 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:
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LG(Arthur) which attributes to Arthur the property of being loved by Gloria, and does
not directly say anything about Gloria.
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, 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 any 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.
(3) Detachment If P and P  Q are both theorems, Q is also a theorem
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(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 in the tautology
(P Q)&(Q R)(P R)
(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.
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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
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 discussed this question a little 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’
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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 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 P a Q and P e Q, not all of the claimed logical relations hold. We
have already noted that if we interpret P a Q as
(x)( P(x)  Q(x)) it does not imply P i Q, which is ( x)( P(x)(Q(x))
However, if we interpret P a Q as ( x)( P(x)  Q(x)) ( x)(P(x)) so that it does
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imply P i Q, it is not the contradictory of P o 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.
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. 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
criterion 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
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clearly rhetorical, assuming the answer ‘No’ . On the other hand when some people ask
the question rhetorically, they expect the answer ‘yes’.
Deductive logic has often been denounced as trivial on the ground that the
conclusions contain no additional information not present in the premises, so we cannot
learn anything new from the deduction. That argument was articulated in the early
seventeenth century by Sir Francis Bacon, in his Novum Organum where he denounced
to what he considered to be the vacuous formal logic of his day, and proposed to replace
it with a fruitful inductive logic. The title was doubtless chosen because when Bacon
wrote formal logic was little different from that expounded by Aristotle in his Organum
The supposition that formal logic is trivial may be reinforced by the examples of valid
argument found in text books on Logic, but that is not a fair test. Examples used to teach
have to be simple enough to be understood by the pupil. When teaching logic one has to
explain the notion that the conclusion of an argument must be true whenever the
premises are true. A clear example is therefore an argument in which it is obvious that
the conclusion will be true whenever the premises are true. Because its validity is
obvious, the argument cited is likely to be trivial. Until the later half of the nineteenth
century the appearance of triviality in formal logic was reinforced by its very limited scope;
it dealt with only a few patterns of argument, and those few were all very simple.
However the supposition that logic is trivial has survived the immense increase in its
scope and power following the work of Russell and Whitehead in the early twentieth
century. Often it is not so much that people are impressed by the perception that logical
argument actually is trivial, but rather that they think it ought to be trivial, because 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
C3: Sophie’s Mother was an only child
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C4: Mary is Alan’s Mother
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 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’
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, so 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.
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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’
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
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explanation of the error.
However many cases are less straightforward and asking the challenger to point to
the error may sometimes be unreasonable; we cannot enumerate all the infinitely many
valid patterns of argument to show that a particular argument does not fit any of them,
while if the argument is valid its proponent could demonstrate that by producing just one
valid form.
Invalid Forms
Someone might suggest that identifying fallacious arguments would be a less hit
and miss affair if, instead of showing that a disputed argument does not fit any valid form
the user of the argument is likely to have in mind, we just show that it conforms to an
invalid form. That is usually impractical because it is only in a few very peculiar cases that
an argument does have a form that guarantees invalidity. In fact I can think of only one.
A valid argument, P therefore Q must be such that the truth of P guarantees the
truth of Q. An invalid form would therefore have to be one such that any argument of that
form must have instances in which the antecedent is true and the conclusion false. The
only way in which the form of the argument can guarantee that, is if the antecedent is
logically necessary and the conclusion a contradiction, producing something like:
(P v P)  (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
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of whom play golf, and suppose nine of those ten drink gin. Then there are eight golfers
of whom five play bridge, and fifteen bridge players of whom nine drink gin, so that most
golfers play bridge, and most bridge players drink gin, yet no golfers drink gin.
However, although such arguments are often useful, finding them is a hit and miss
affair. It is not so much a test for validity, as a strategy that will sometimes detect
invalidity. Although we can sometimes find such reasons for rejecting an argument, there
is no systematic and generally reliable procedure for finding examples in which one
proposition would be true, and another false. The case for placing the onus of proof on
the one who proposes the argument stands.
Notice also that the demonstration that an argument is invalid can be complicated
and may need numerical precision. That is much easier to achieve in writing than in
conversation. I think one of the principle sources of error in logic is conversation; people
insist on talking when they need to write. I’m thinking of installing a white board in my
house to make thinking easier.
An example will show how easily one can stray into confusion. I was once involved
in a discussion that ran roughly as follows:
I: ‘Some researchers have just published results of experiments that show that
listening to very loud music damages people’s hearing’.
X: ‘That’s like saying that looking at big things damages your eyesight’
X thought my argument was of the form :
‘great magnitude in what is perceived damages the organ of perception’ an
argument that is indeed unreasonable, though I was surprised by X’s assumption that
was what I meant as there are two other much more plausible forms either of which fit my
argument. My argument conforms to both the patterns:
(1) ‘Conclusions of published experimental results are likely to be broadly true’,
which was what I actually had in mind, and
(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.
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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
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
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In the Aristotelian logic an important distinction is that between a term that is
distributed in a proposition and one that is not. A term is said to be distributed in a
proposition when the proposition gives information about everything to which the term
applies. Thus in ‘All Fish are vertebrates’ ‘Fish’ is distributed but ‘vertebrates’ is not
because the proposition asserts something of every fish, but not of every vertebrate. In
‘No Mammal is Six Legged’ both ‘Mammal’ and ‘Six Legged’ are distributed because the
proposition tells us something about every mammal - that it is not six legged, and also
tells us something about every six legged creature, namely that it is not a mammal. In
propositions of forms I and O, neither term is distributed.
Syllogisms were divided into four figures:
fig 1
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
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premisses also does so.
Rule (5) is justified by noting that if both premisses are negative, then any one of the
six relations between S and P could be true so that the premisses rule none of them out.
In Formal Logic J. N. Keynes said (P 109) that all the rules of the syllogism could be
deduced from just two:
(a) The middle term is distributed 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 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
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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. Those of the first
figure were assigned letters B, C, D and F which were to be the first letters of their
mnemonic names. 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
PeM
MiS
SoP
No worms have wings
Some winged creatures sting
Some stinging creatures are not worms
The ‘s’s in ‘Fresison represent simple conversion, by applying which to both
premisses - no change is required in the conclusion - we obtain:
MeP
SiM
SoP
No winged creatures are worms
Some stinging creatures have wings
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
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PaM
SoM
SoP
All fish are vertebrates
Some aquatic creatures are not vertebrates
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 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 that of justifying the propositions that
interpret the axioms, and demonstrate that the rules of inference preserve truth.
Mathematicians sometimes regard their collections of formulae as just formal
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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 represents 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))and the law of the excluded middle.
(PvP) Later, other logicians added the law of identity. [(x)(x = x)]
Aristotle considered non contradiction and excluded middle to be self evident,
remarking that any attempt to prove them would therefore be circular. However he went
on to say that the impossibility of proving either law does not rule out all argument in their
favour. The impossibility of a proof rules out any argument calculated to force assent on
someone who is simply disinclined to form an opinion, but if we are confronted with
someone who actually denies either principle there is a good deal we can say, for
powerful ad hominem arguments are available.
Aristotle considered mainly the law of non-contradiction and gave a number of
arguments of uneven quality, but the general drift of his reasoning was that if someone
denies the law, then either he will be so using language that he fails to communicate
anything, or he will be guilty of inconsistency in sometimes himself relying on the very
principle he claims to deny.
The extreme case of irrationality, thought Aristotle, would be someone who asserts
a sentence of the form ‘A is not B’ in every case that he asserts a sentence of the form ‘A
is B’. Such a person would apparently be prepared to assert anything at all, so once we
have noticed that, his remarks will tell us nothing; he fails to communicate anything at all.
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’
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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’, for 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 argued that rejection of non contradiction would imply the rejection of the law of the
excluded middle. For suppose we can assert both ‘A is B’ and ‘A is not B’. From ‘A is B’ it
follows that ‘~(A is not B)’, and from ‘A is not B’ it follows that ‘~(A is B)’, implying the
denials of both ‘A is B’ and ‘A is not B’.
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’ 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) : deducible from the other axioms
(Q  R)  ((P Q) P  R
The rules of inference were:
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rule of substitution: from any axiom or theorem another theorem may be obtained
by substituting any well formed formula in place of (all occurrences of) any letter.
rule of detachment from theorems of the forms P and P  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
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
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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)
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.
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(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)
(b)(PvQ) P
(c) P  (Q  P)
(d)~Pv(P&Q)
(e) (PvQ) & (Pv~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,
(P  Q )  ~(P&~Q), (PvQ)  (~(~P&~Q),
(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
(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
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What is Philosophy Chapter 2
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Equestrians.
(12) Most professional footballers are under the age of 30. Most under 30’s watch
television for more than ten hours per week, therefore some professional
footballers watch television for more than ten hours per week.
(13) All monks are men, some men are virgins, therefore some monks are virgins.
(14) If the red light on the printer comes on then either the printer is out of paper, or the
paper has jammed, but the printer is not out of paper and the paper has not
jammed, so the red light is not on.
(15) To join the Drones Club one must have no paid employment and live either within
one mile of Buckingham Palace, or in a country house North of the Thames with
grounds of at least 60 acres. To join the Country Club one must live in a country
house with grounds of at least 200 acres.
Does it follow that someone who belongs to the Country Club but is not eligible for
membership of the Drones must have paid employment?
(16) Use quantifiers to represent:
(a) There is a man who is either a hypnotist or a vampire
(b) Either there is a man who is a hypnotist, or there is a man who is a vampire
(c) The only topologist living in Rutland is a keen gardener.
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