by Mark English
It is often said that the foundations of logic are the conventions that we (as users of logic) agree upon. I see problems with this view.
For a start, the word ‘logic’ is used in different ways. It is often used to refer to formal logic and applications and developments thereof, but it is also used to refer to the principles of sound reasoning in ordinary informal contexts.
In the case of formal or symbolic logic, agreed-upon sets of axioms do indeed determine how each system operates. In this sense, logical systems are relative and malleable. They can be seen as tools, which may be applied to various subjects and activities. Different applications call for different systems or modifications of the system (or tool) being used.
The problem is that such a view seems to suggest that logic is radically relative, and this could be seen – as it often is, in fact – to justify a broader kind of relativism that would encompass science and possibly validate non-rational modes of thinking. I would want to argue against this view and say that the foundations of logic run deep.
Certainly, it is not satisfactory merely to say that the foundations are conventions we consciously decide on. One reason we know this is because certain logical principles are built into natural language such that an intuitive grasp of something very like what we call logic is a prerequisite for communicating linguistically. In other words, this intuitive understanding of what we might choose to call logical principles precedes any conscious decision we might make about adopting principles, much less sets of axioms. A young child – to the extent that the child uses language – gets it.
‘Not’. ‘Or’. ‘And’. ‘If … then’ Such words and idioms – and perhaps the pre-linguistic intuitions that lie behind them and their equivalents in other languages – are arguably the source of (and are closely related to) the various formal operators. 
Of course, there are big differences between natural and formal languages. For example, the meanings of the words and idioms of natural languages vary and are context-dependent, whereas the meanings of logical constants (which can be seen as being defined simply in terms of how they are used within a system) do not vary within a particular system and are not context-dependent in the way natural language expressions are.
Take the logical connective ‘or’, which is used in the propositional calculus to represent the truth-functional operator of inclusive disjunction. Its meaning/use is quite clear and explicit. When this operator is used, the set of operands is true if and only if one or more of the operands is true. So “Paris is the capital of France or Berlin is the capital of Germany” is true if (a) one of the two statements (operands) is true and (b) if both of the statements are true. It is only false if both are false.
The English word ‘or’ is one of a number of coordinating conjunctions and normally doesn’t work like its counterpart in formal logic. It usually represents exclusive rather than inclusive disjunction. If someone says they wished they had become a doctor or a lawyer, they generally mean either/or, not either/or or both. The slightly awkward expression ‘and/or’ is often used to specify inclusive disjunction in English.
My broader point is that logical structures may often be complex and difficult to discern, but they are generally decipherable and are an integral part of any kind of symbolic processing. Some constraints always apply. It is never a case of ‘anything goes’.
There are many ways you could approach these sorts of questions. You could focus on the relationship between natural and formal languages. Or you could probe into the logical basis of digital computing. But I thought it might be worthwhile here to present – mainly through selected extracts from a section of his book, Objective Knowledge – some of Karl Popper’s ideas on logic and truth. 
I disagree with many of Popper’s views (on the mind, for example) and I also have reservations about his apparent aversion to dealing with questions relating to natural language, questions which may well be pertinent to some of his claims. Nonetheless, I think his views on logic and truth are well worth taking seriously. Driven by a commitment to commonsense and scientific realism, he shies away from conventionalism and relativism.
Popper explicitly rejects the idea of logic being merely a set of more or less arbitrary conventions (whether consciously agreed upon or not). “I am opposed,” he wrote, “to looking upon logic as a kind of game. I know about so-called alternative systems of logic […] but alternative systems of logic can be discussed from very different points of view. One might think that it is a matter of choice or convention which logic one adopts. I disagree with this view.”
Popper sees logic as a theory of deduction or of derivability.
Derivability or deduction involves, essentially, the transmission of truth and the retransmission of falsity: in a valid inference truth is transmitted from the premises to the conclusion. This can be used especially in so-called ‘proofs’. But falsity is also retransmitted from the conclusion to (at least) one of the premises, and this is used in disproofs or refutations, and especially in critical discussions.
We have premises and a conclusion; and if we show that the conclusion is false, and assume that the inference is valid, we know that at least one of our premises must be false. This is how logic is constantly used in critical discussion, for in a critical discussion we attempt to show that something is not in order with some assertion. We attempt to show it; and we may not succeed: criticism may be validly answered by counter-criticism.
Popper is probably most famous for proposing to replace the verification principle of logical positivism with a falsification principle. This was his response to the problem of induction as described by David Hume. Popper’s “backwards” argument from false conclusion to a false premise is a logical notion (cf. modus tollens), but it also relates to science. Indeed, logic is central to the practice of science.
Popper thinks that criticism is a crucial methodological device for the building up of a sound body of knowledge about the world and that effective criticism would not be possible if we could answer criticism by rejecting the logical framework of the critic; by saying, in effect, “Your logic may be all right for you, but I prefer a different logic, and according to my logic this criticism is not valid.”
He argues that strong logics (like classical logic) are to be preferred in critical contexts, “for we want our criticism to be severe [and in order] that the criticism should be severe we must use the full apparatus; we must use all the guns we have. Every shot is important. It doesn’t matter if we are over-critical: if we are, we shall be answered by counter-criticism. Thus we should (in the empirical sciences) use the full or classical or two-valued logic.”
What lies behind this concern is – in part – his rejection of subjectivist interpretations of quantum mechanics. “If we […] retreat into the use of some weaker logic – say, the intuitionist logic, or some three-valued logic (as Reichenbach suggested in connection with quantum theory) – then, I assert, we are not critical enough; it is a sign that something is rotten in the state of Denmark…” (His little joke about the Copenhagen interpretation.)
He recognizes, of course, the importance of these “weaker” logics in proof theory.
[I]f one can prove mathematical theorems with methods weaker than the full battery of classical logic, then this is extremely interesting from a mathematical point of view. Thus in proof theory we are interested in weakening if possible our classical logic, and we can, for example, introduce intuitionist logic […] and investigate how far we can get without using the whole battery. […] So if you wish to prove, or to establish something, you should use weak means. But for disestablishing it – that is to say, for criticising it – we may use strong means.
Alfred Tarski, he claims, introduced two important ideas into logic which are very compatible with scientific realism. The first (partly anticipated by Bolzano) is that logical consequence is truth transmission. The second, according to Popper, is a rehabilitation of the idea that truth is simply correspondence with the facts.
Of the three main theories of truth, the oldest [is] the correspondence theory, the theory that truth is correspondence with the facts, or to put it more precisely, that a statement is true if (and only if) it corresponds to the facts, or if it adequately describes the facts. This is the theory which I think Tarski has rehabilitated.
The second theory Popper discusses is the so-called coherence theory. It comes in many forms but in general terms holds that a statement is regarded as true to the extent that it coheres with the rest of our knowledge. The third theory is that truth is pragmatic utility or pragmatic usefulness: it says in effect that “we should accept a physical theory as true if it turns out in tests, and other applications, to be pragmatically useful, or successful.”
Our problem can be sharply formulated only by pointing out that the opponents of the correspondence theories all made an assertion. They all asserted that there cannot be such a thing as the correspondence between a statement and a fact. This is their central assertion. They say that this concept is meaningless (or that it is undefinable …). In other words, the whole problem arises because of doubts, or scepticism, concerning correspondence: whether there is such a thing as a correspondence between a statement and a fact. […] It is also quite clear that, but for these doubts, the upholders of the coherence theory and of the theory of pragmatic usefulness would really have nothing to argue against. Nobody denies that pragmatic usefulness and such matters as predictive power are important. But should there exist something like the correspondence of a theory to the facts, then this would obviously be more important than mere self-consistency, and certainly also much more important than coherence with any earlier knowledge (or ‘belief’); for if a theory corresponds to the facts but does not cohere with some earlier knowledge, then this earlier knowledge should be discarded.
Moreover, if there exists something like a correspondence of theories to facts, then it is clear that a theory that corresponds to the facts will be preferable to a theory that doesn’t. Or, as Popper puts it, “the pragmatist position will be superseded by a realist position if we can meaningfully say that a statement, or a theory, may or may not correspond to the facts.”
It is important to realize also that, while the coherence and pragmatist theories are generally taken to assert the impossibility or meaninglessness of a correspondence theory, the correspondence theory does not involve denying the importance of coherence or of pragmatic factors.
Popper sees the central issue with defending the correspondence theory as having nothing to do with defining ‘truth’ (he claims to have little interest in definitions), and remains focused on the question of whether or not a statement or a theory can plausibly be said to correspond (or not) to the facts.
According to Popper, Tarski rehabilitated the correspondence theory of truth by providing a simple but precise method of coordinating statements and facts. He did this by deploying the distinction between language and metalanguage in a particular way. A metalanguage is a language in which we talk about some other language (called the object language). Popper gives the example of a grammar of the German language, written in English, which uses English as a metalanguage in order to talk about German (the object language). But you can also use any natural language to talk about, i.e. make metalinguistic assertions about, itself, so that at certain points the language is operating metalinguistically and at other points not.
Here is the key part of Popper’s account of how Tarski’s theory of truth enables us to see how statements can be coherently seen to coordinate – or correspond – with facts about the world.
The characteristic thing about a metalanguage is that it contains (metalinguistic) names of words and of statements of the object language, and also (metalinguistic) predicates, such as ‘noun (of the object language)’ or ‘verb (of the object language)’ or ‘statement (of the object language)’. If a metalanguage is to suffice for our purpose it must also, as Tarski points out, contain the usual means necessary to speak about at least all those facts about which the object language can speak. All this is the case if we use English as our metalanguage in order to speak about German [or English, for that matter] as the object language under investigation.
For example, we shall be able to say in the English metalanguage such things as: The German words ‘Das Gras ist grün’ form a statement of the German language.
On the other hand, we shall be able to describe in our (English) metalanguage the fact which the German statement ‘Das Gras ist grün’ describes. We can describe this fact in English simply by saying that grass is green.
We can now make a statement in the metalanguage about the correspondence of a statement of the object language to the facts as follows. We can make the assertion: The German statement ‘Das Gras ist grün’ corresponds to the facts if, and only if, grass is green. (Or: ‘. . . only if it is a fact that grass is green.’) This is very trivial. It is, however, important to realise the following: in our assertion, the words ‘Das Gras ist grün’, put within quotes, function as a metalinguistic (that is, an English) name of a German statement; on the other hand, the English words ‘grass is green’ occur in our assertion above without any quotation marks: they do not function as a name of a statement, but simply as the description of a fact (or alleged fact).
This makes it possible for our assertion to express a relationship between a (German) statement, and a fact. (The fact is neither German nor English, although it is, of course, described or spoken about in our metalanguage, which is English: the fact is non-linguistic, it is a fact of the real world, although we need of course a language if we wish to talk about it.) And what our metalinguistic assertion asserts is that a certain (German) statement corresponds to a certain fact (a non-linguistic fact, a fact of the real world) under conditions which are precisely stated.
Popper was attracted to the correspondence theory of truth because he saw it as a realistic theory:
[I]t makes the distinction, which is a realistic distinction, between a theory and the facts which the theory describes; and it makes it possible to say that a theory is true, or false, or that it corresponds to the facts, thus relating the theory to the facts. It allows us to speak of a reality different from the theory. This is the main thing; it is the main point for the realist. The realist wants to have both a theory and the reality of the facts (don’t call it ‘reality’ if you don’t like it, just call it ‘the facts’) which are different from his theory about these facts, and which he can somehow or other compare with the facts, in order to find out whether or not it corresponds to them. Of course, the comparison is always extremely difficult.
The coherence and pragmatic theories purport to present us with a method of deciding whether or not a given statement is true. But Popper emphasizes that his version of the correspondence theory (based as it is on Tarski’s semantic theory of truth) is not designed or intended to yield a criterion of truth. In fact, Tarski proved that in a sufficiently powerful language (and in every language in which we can formulate mathematical or physical theories) there can be no criterion of truth.
Although we have no criterion of truth and no means of being even quite sure of the falsity of a theory, it is easier to find out that a theory is false than to find out that it is true […]. We have even good reasons to think that most of our theories – even our best theories are, strictly speaking, false; for they oversimplify or idealise the facts. Yet a false conjecture may be nearer or less near to the truth. Thus we arrive at the idea of nearness to the truth, or of a better or less good approximation to the truth; that is, at the idea of ‘verisimilitude’.
Popper wants to incorporate this idea into logic. While his formal definition of verisimilitude (or ‘truthlikeness’ as it is often now called) has been challenged, the concept has proved to be a fruitful one. This is no surprise, really. Given the provisional and incomplete nature of scientific theories at any given point in time and the obvious phenomenon of scientific progress, this concept or something like it is clearly called for.
Seeing scientific progress merely in terms of effectiveness or some such pragmatic notion is possible, but seems unnecessarily restrictive. Whatever else it may be, scientific activity – encompassing both empirical and formal disciplines – also represents an attempt to understand the world of which we are a part. What’s more, within its self-imposed and natural limits, this quest for understanding can be seen to have been remarkably successful.
- Clearly these expressions and the intuitions which underlie them have in turn arisen from our actions and interactions, and so are rooted in some sense in the physical world.
- Objective Knowledge: An Evolutionary Approach. Oxford: Clarendon Press, 1972. I am drawing on the final section (section 4) of the penultimate chapter. The entire chapter is available here.