What is Philosophy Chapter 2
by Richard Thompson
Logic
(last edited on 29 January 2017)
In this chapter I outline the basics of Logic; there is more about Logic in Chapters 4 and 5.
I’m not writing a logic text book, so I shall discuss the technicalities of formal logic only where they are needed to follow the discussion of the comments various philosophers have made about logic. I learnt the basics of formal logic from W. V. Quine’s Methods Of Logic, and I have never come upon a better introduction, though of course, having read one introduction, I’ve never done more than glance at others. William and Martha Kneale The Development Of Logic is an excellent historical survey, which also explains the technicalities clearly.
Inferences, Propositions and Entailment
Logic is the study of the validity of inferences. It tells us what follows from what. Formal Logic gives precise rules that ensure the validity of any inference that satisfies them.
I’ll start by introducing some terminology. An inference proceeds from a starting point to an end point. We need a word for the types of entity that can feature in an inference. The one most commonly used is ‘proposition’. A proposition is some sort of claim that can be either true of false. Some logicians prefer to talk of sentences, on the grounds that that gives us a definite subject for discussion. The supporters of ‘proposition’ retort that the meanings of words can change, words can be ambiguous, the same sentence can mean different things on different occasions, and a variety of different sentences can be used to make the same claim. Also a sentence belongs to a particular language, while logic studies ideas that are independent of language. A proposition may be thought of as what a sentence means on a particular occasion, or what the user is trying to put across when they use a sentence. W.V. Quine and his supporters, who preferred to talk of sentences, considered it impossible to define ‘meaning’ or ‘proposition’ independently of a sentence. ‘statement’ has sometimes been adopted as a compromise, a ‘cowardly’ policy in the opinion of Quine though ‘statement’ has the advantage of suggesting a particular utterance made by a particular person at a particular place and time.
I’ve decided to use ‘proposition’. I’ll discuss the arguments for and against the existence of propositions in Chapter 5, but shall say no more on the matter in this chapter.
So the central idea in Logic is that of inference from one proposition to another. The propositions from which an inference begins are called the premisses, and the proposition at which an inference arrives is called its conclusion. A valid inference is one in which the truth of the conclusion is guaranteed if the premisses are true. We say that P entails Q, if P and Q are propositions such that, if P is true, Q must also be true, so that proceeding from P to Q is a valid inference. The word imply is often used instead of entail, but as we shall see later Russell and Whitehead confused the issue by using material implication for a much weaker relation, provoking G. E. Moore to introduce ‘entail’.
Formal logic studies patterns of argument such that any argument conforming to the pattern is valid.
A particular argument can usually be fitted into several different patterns, but to establish it’s validity we need only point to one valid pattern, so when we formalise the propositions of an argument to demonstrate its validity we need not try to capture their full complexity. It suffices to capture sufficient of the content to validate the argument. For instance suppose someone argued thus:
All members of the golf club play either tennis or bridge, Some members of the choir play neither tennis nor bridge, Therefore some members of the choir do not belong to the golf club.
That is an example of the pattern:
Every A is either B or C, Some D are neither B nor C, Therefore some D are not A.
(A = member of the golf club, B = tennis player, C = bridge player, D = member of the choir)
However, the argument also fits the simpler pattern: All P is Q, Some R are not Q, therefore some R are not P, which also suffices to establish its validity. (P = member of the golf club, Q = player of either golf or bridge, R = member of the choir)
We should usually prefer to refer to the simpler pattern when defending the inference in such a case.
One misconception must be removed at the outset. Logic concerns valid arguments, not good arguments, in the sense that a good argument is one that gives someone a good reason for believing its conclusion. A good argument (some people prefer ’sound argument‘) should be valid, but a valid argument may not be a good one. ‘P entails Q’ is only a good reason for a person A to believe Q, if A both realises that P entails Q and also has a good reason to believe P.
‘Simon has a pet cat’ entails ‘Simon has a pet’ is valid, but it does not provide a good reason for believing that Simon has a pet if Simon actually does not have a cat.
What is a good reason for one person to believe something may not be a good reason for someone else. In particular, if someone believes that P is false ‘P entails Q’ is not a good reason for him to believe Q. ‘P entails Q’ is said to be a ‘good ’argument, when P is true, and ‘P entails Q’ is valid.
To identify all good arguments would require us to know the truth of all true propositions, so that to include the identification of good arguments in Logic would require that Logic include the whole of knowledge. Aristotle thought that it did, but today there are few who would agree.
Although the mere validity of an argument does not guarantee the truth of its conclusion it does not follow that the study of validity is pointless, for an argument that is not valid is not a good reason for anyone to believe its conclusion.
Aristotelian Logic
The first recorded study of formal logic was by Aristotle who described the logic of propositions of four types, namely those that conformed to one the forms: “All S is P”, “Some S is P”, “No S is P” or “Some S is not P”. The broad outlines of Aristotelian logic were preserved right into the nineteenth century, but it was elaborated in the middle ages so I shall discuss that slightly extended form of the logic, and I shall also use some of the medieval notation.
Aristotle’s logic began with the examination of the logical relations between pairs of propositions, and then used that as the basis for considering more elaborate arguments involving larger numbers of propositions.
The medieval logicians referred to the four types of Aristotelian proposition as
A: All S is P, example: all mice are 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), For example ‘some birds candot fly is equivalent to ‘some creatures that cannot fly are birds’ and ‘some mice like chocolate’ is equivalent to ‘some creatures that like chocolate are mice. so Therefore we need consider only six distinct propositions.
Pairs of propositions may be related in one or another of several ways.
Contradictories Two propositions are contradictories when the truth of either one is equivalent to the falsehood of the other that so that one and only one is true. Contradictory pairs are:
S a P (all S is P) and S o P (some S is not P), eg. ‘all cats like milk’ and ‘some cats do not like milk’ are mutually contradictory.
S e P (no S is P) and S i P ( some S is P) eg. ‘no toadstool is edible’ and ‘some toadstools are edible’ are mutually contradictory.
Contraries two propositions are contraries when they cannot both be true but can in some circumstances both be false. Contrary pairs are:
S a P and S e P, and contraries and so are P a S and P e S, eg. ‘all members of the Chess Club play Golf’ and ‘no members of the Chess Club play Golf’. The two propositions cannot both be true, but if some members of the Chess Club play golf, but some do not, both the propositions would be false.
People sometimes confuse contraries with contradictories
Subalternation when one proposition entails another, but not vice versa.
S a P entails S i P, e.g. ‘all birds lay eggs’ entails ‘some birds lay eggs’ , but not vice versa.
and S e P entails S o P. e.g. ‘no freemasons are cannibals’ entails ‘some freemasons are not cannibals’ but not vice versa.
In those relations the universal propositions were referred to as the subalternants and the particular propositions they imply as the subalternate or the subaltern.
Both those examples raise a problem about existence, which I discuss below.
Subcontraries are pairs of propositions that cannot both be false, but might both be true.
The i and o propositions are subcontraries, since ‘Some S is P’ and ‘Some S is not P’ might both be true, but cannot both be false. eg. ‘some solicitors are freemasons’ and ‘some solicitors are not freemasons’ could both be true, but they could not both be false, because if it were false that some solicitors are not free masons, then all solicitors are freemasons.
Existential Import of Universal Propositions
The forgoing discussion of the four Aristotelian types of proposition assumes that ‘all implies some’ that is that ‘All S is P’ implies ‘Some S is P’ and ‘All S is not P’ implies ‘Some S is not P’ That assumption is problematical and gave rise to a good deal of debate among logicians. For sometimes we assert universal generalisations without any commitment to existence.
For instance if we explained ‘unicorn’ by saying ‘Unicorn’ means ’quadruped mammal resembling a horse but with a single horn projecting from the middle of its forehead’ we could confidently assert ‘any unicorn has four legs’ but should not think ourselves thereby committed to the existence of any unicorns, so we should reject the inference from ‘all unicorn have four legs’ to ‘some unicorns have four legs‘ since that involves there actually being some unicorns.
Furthermore, since ‘All S is not P’ is held to be equivalent to ‘All P is not S’ it appears that S e P not only entails that there are S, but also entails that there are P, so that ‘No animals are unicorns’ entails ‘No unicorns are animals. If we allow that to entail ‘some unicorns are not animals‘ we should be committed to the existence of unicorns at the same time we assert that there are none.
The matter is more easily discussed with the help of modern logical notation so I shall defer further discussion, except to say that provided the terms in all the propositions do have references, the traditional logic never leads from true premisses to false conclusions.
Having examined individual propositions and their logical relations, Aristotle turned his attention to syllogisms, in which two propositions entail a third. The first two propositions were called the premisses of the syllogism, and the proposition they jointly entail was called the conclusion.
For example
(1) All Mammals are vertebrates,
(2) All Elephants are mammals,
therefore (3) All Elephants are vertebrates
That is a valid inference because it is of the form “All S is P, All Q are S, therefore All Q are P”
Aristotle and his medieval followers developed an elaborate theory of syllogistic inference, which I discuss in Appendix 1.
It was eventually realised that Aristotelian Logic can be illustrated by 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.
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.
Does Logic just state the obvious?
Perusal of examples provided by logicians often suggests that their validity is so obvious that it is hardly worth mentioning it. That is partly because a good example needs to be clear, but it is also true that Aristotelian argument, unlike modern mathematical logic, rarely deals with inferences likely to puzzle an alert mind.
The principal point of studying logic is not to learn how to recognise valid arguments, but to spot fallacious arguments that resemble valid ones. If we had a list of patterns of valid argument, we should be able to check any suspect piece of reasoning to make sure it matches one of the valid patterns, and much of the content of elementary books on Logic consists of the examination of invalid reasoning. For this purpose it does not matter if the valid arguments appear obviously valid, because the proposed list of valid forms of argument is a check list to be used to identify invalid arguments.
However, identification of fallacy in that way depends on our having a comprehensive list of patterns of valid reasoning. Aristotelian logic has such lists only for syllogisms, and for direct inferences between two propositions of the four types Aristotle recognised.
Aristotelian Logic had no tools for assessing the validity, or otherwise of the following inferences:
(1) If everyone who plays a game knows someone else who plays another game, it follows that if anyone plays any game, there are at least two games that are played by someone or other.
(2) Most mice like chocolate, Most mice like cheese,
Therefore some creatures that like chocolate also like cheese
(3) A is B’s nephew, therefore B is A’s uncle
Modern Logic
The three examples at the end of the last section illustrate some of the limitations of Aristotelian Logic, yet little was done extend formal Logic until Mathematicians began to take an interest in the subject.
For more that two millennia after Aristotle’s death Logic, despite minor amplifications, remained much as he left it. Only when George Boole (1815-1864) made a fresh start by constructing a logical algebra did the subject enter a century of rapid growth into what is now called ‘Mathematical Logic’. Aristotle’s Logic is still discernible as part of modern Logic, but it is not a convenient starting point. Nor is Boole’s work, at least not in its original form. I make a fresh start with what are now called ‘truth functions’
Truth Functions.
Truth functional logic takes as its units complete propositions.
Truth Values
Suppose P, Q, R, and S, represent propositions. Each of those propositions must be either true or false. If a proposition is true we say its truth value is true, symbol ‘T’ and if false its truth value is false, symbol ‘F’. Boole used ‘1’ for true and ‘0’ for false; some electronic engineers also use that convention, as do electronic calculators that permit boolean operations.
Truth functional logic studies ways of combining propositions into more complicated propositions in such a way that the truth value of the composite proposition is determined by the truth values of its components.
Negation: NOT not P, symbol P, is true when P is false and false when P is true. That can be summarised by atruth table which gives the truth conditions for P
PP
TF
FT
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:
PQP Q
TTT
TFF
FTF
FFF
P Q is true only when both P and Q are individually true. P Q is false in all other circumstances