MPP From: A ⊃ B, A
To infer: B (syntactic definition)
(semantic definition)
An attempt could be made to justify the above argument by either definition. The justification would, however, take the form:
A3 (1) If ‘if A, then B’ and ‘A’ are true, ‘B’ is true
(2) ‘if A, then B’ and ‘A’ are true
∴ ‘B’ is true
This argument, however, takes exactly the same form as the original argument, and so in order to accept it we must accept the validity of MPP. But this was exactly what we were supposed to be proving, and so the so-called justification is circular.
Another demonstration of this comes from a paper by Lewis Carroll (of Alice in Wonderland fame), presented as a short story about the Tortoise and Achilles. The Tortoise starts off with an argument of the form shown above, which he has Achilles write in his notebook. However, before he writes the conclusion, the Tortoise has him imagine a reader who accepts the first two propositions, but refuses to accept the conclusion. To remedy this Achilles adds a third premise, making the argument*:
A4 (1) If A, then B
(2) A
(3) If (1) and (2) are true, ‘B’ must be true
∴ B
Achilles is sure that this renders the argument incontrovertible, but the Tortoise has him imagine another reader who fails to accept the conclusion. Again, another premise is added (“although,” remarks Achilles, “such obtuseness would certainly be phenomenal”):
A5 (1) If A, then B
(2) A
(3) If (1) and (2) are true, ‘B’ must be true
(4) If (1) and (2) and (3) are true, ‘B’ must be true.
∴ B
At this point Achilles thinks he has triumphed, and when the Tortoise suggests that someone could fail to accept the conclusion, he says that “Logic would take you by the throat, and force you to do it”. The Tortoise points out that “whatever Logic is good enough to tell me is worth writing down”, and they add a predictable fifth premise. This continues as expected, and by the time the narrator revisits them they have more than a thousand premises, with “several millions more to come”. The lesson, of course, is that one cannot justify deduction by inserting extra premises ad infinitum; these premises are analogous to restating the principle in question – endless repetition does not make it any more valid. Thomson argues that Achilles should never have conceded the original extra premise, because if the argument is valid to begin with the added premise is unnecessary, and if the argument was not valid to begin with the added premise does not constitute justification for it.
A way by which we could see whether a proposed justification of a deductive inference is useful is through seeing whether the same method would justify an inference we consider to be invalid. If the method will justify both equally well, it cannot be regarded as useful in helping us decide whether the inference is justified or not. To illustrate this, we may introduce the classically invalid rule modus morons (MM):
MM From: A ⊃ B, B
To infer: A
Using this, we can see that analogous arguments to those above can be constructed for MM. For example, compare the following to A3:
A6 (1) If ‘A’ is true, then ‘if A, then B’ and ‘B’ are true
(2) ‘if A, then B’ and ‘B’ are true
∴ ‘A’ is true
If A3 can be counted as justifying MPP, A6 must also justify MM, and it is therefore difficult to see how either argument is useful in justifying one form of logic rather than another. We seem therefore to be unable to demonstrate a method for justifying inferences which does not leave us guilty of circular reasoning, as we end up using the very form of inference we are trying to justify, or leave open the possibility of justifying alternative rules of inference that we do not use. I will return to the topic of alternative logics later in this essay.
3. Civilised Circles
In the previous section I showed that any attempt to justify a deductive inference leads to circularity – you cannot justify deduction without using deduction. We are thus in a position with no candidate at all for a rigorous justification of deduction. However, it is debatable whether this kind of circularity means that such arguments are entirely without merit. To begin with, we must distinguish between the terms ‘circular’ and ‘viciously circular’. The former tends to be used both descriptively and evaluatively, whereas the latter is always in some way evaluative. Following Brown, I shall use the term ‘circular’ as a purely descriptive term, reserving ‘viciously circular’ for arguments which are both circular and defective.
The most obvious form of circularity is when the conclusion of an argument can only be reached by assuming that conclusion. For instance, imagine I had a piece of paper with, among other text, the words “the information on this piece of paper is true”. I could construct a circular argument to allow me to believe all the content on the paper:
A7 (1) The paper says that the information on it is true.
(2) The information on the paper is true.
∴ The information on the paper is true.
The assumption (2) leads me to accept the conclusion, which is in fact the same as the original assumption. This argument is viciously circular, as the circularity removes all merit that the argument could have had – it gives us no reason to accept the conclusion, which is only supported by itself.
Another more subtle form of circularity is known as ‘begging the question’, which occurs when one of the premises of an argument cannot be known without knowing the conclusion. The reason such an argument can be considered defective is that it provides no new reasons for knowing the conclusion. Strictly speaking, however, this principle would eliminate all arguments that utilise disjunctive syllogism: statements such as ‘A or B’. If I know that ‘A’ is false, to know ‘A or B’ is true requires me to know that ‘B’ is true. An argument of this form is therefore defined as begging the question:
A8 (1) ¬A
(2) A ∨ B
∴ B
However, this argument need not be useless, if for example the premises became known in a different order – I might have good reasons for believing the disjunction, but not be sure which of the two disjuncts is true (I could be aware that it is the weekend, but be hung-over and unsure if it is Saturday or Sunday). If I then came into further knowledge that settled the matter, the argument would still be useful. It seems that context is very important in analysing whether an argument is viciously circular. The idea of a gap in knowledge between the premises and conclusion is something that will be returned to later.
Another example of circularity that does not necessarily make its argument defective comes from science. To test a hypothesis, an experiment must be able to give a negative result; one which contradicts the hypothesis. If an experiment is based directly on a theory, it may be unusable as support for that theory. However, there are cases in which an experiment can be devised, with equations produced under the assumption that the theory is correct, than nevertheless can produce answers that disprove the theory.
In a case from contemporary physics, an experiment was carried out using Earth-based telescopes that measure the change in angle of two distant celestial objects A and B. Their relative linear velocity is calculated from the measured red shift (as objects move faster away, their colour shifts measurably towards the red end of the spectrum) using an equation, but Newtonian physics and relativity yield different formulae. Since it is currently believed that relativity is correct, the relativistic formula is used (where β is the recession velocity, S is the red shift):
β = (S2 – 1)(S2 + 1)
Here, as we are using the relativistic formula, β cannot be greater than or equal to 1 (the speed of light in vacuo). However, when this formula is combined with other measurements and calculations to yield a tangential velocity, there is nothing in the procedure to prevent a result greater than the speed of light – if this were to occur it would be evidence against relativity, even though the experiment was, in a way, ‘circular’. The fact that a theory is assumed in producing an argument does not preclude that argument from showing the theory to be incorrect.
Similarly, some ways to justify an inference that are considered circular are still open to challenging that inference. Imagine, for example, a system of logic which permits arguments of this form:
Q From: A or B
To infer: A
If we were to construct an argument using this rule, we could still challenge it:
A9 (1) Q is invalid or B
∴ Q is invalid
It should not be assumed, therefore, that the fact an argument is circular means it will always support a given inference. The fact that an argument can be circular and still self-defeating is important, as it means we are not entirely without criteria in trying to determine whether to use a given deductive system. If that system can be shown to contradict itself, this would be a strong reason not to use it.
Another reason we may have for accepting circular arguments rests in their purpose. Dummett divides arguments into two types: suasive and explanatory. Suasive arguments are intended to convince the reader of something they do not already believe. It is therefore important for a suasive argument that the reader agrees with all the premises, and can follow the inferences to the conclusion, which they must then accept. Vicious circularity in a suasive argument is clearly a problem. An explanatory argument, however, is intended to explain to the reader how something functions, which they already believe in. The epistemic direction in an explanatory argument, in fact, may run counter to the direction of argument, because the aim is to show how the premises are the best explanation for the conclusion which is already accepted – to show how something works, rather than that it works. Circularity in an explanatory argument is therefore less of a problem.
A deductive inference is not something which we are establishing from scratch, and the search for a justification thereof is the search for an explanation of deduction’s role in our use of language, and of how we use deductive inferences to lead to a conclusion. We already use deduction and accept its conclusions, and so it is acceptable to class a justification as explanatory, and to permit deduction to be used in it. With this in mind, it is possible to look deeper at potential ways to justify deduction.
At this point we can return to the soundness and completeness theorems mentioned above. One model for the use of these proofs as justification is that the syntactic definition allows us to prove positive results – to show that such-and-such an argument is valid, whereas the semantic definition allows us to prove negative results – to show that such-and-such an argument is invalid. The soundness theorem shows that if an argument is semantically invalid it cannot be reduced to logical steps, and the completeness theorem shows that if an argument cannot be reduced to logical steps, it is not semantically valid. Together they provide a way to formalise how deductive arguments, and the rules they are made of, can be guaranteed to be truth-preserving. In addition to this they show that a set of deductive rules will not lead to a contradiction: they are self-consistent. This would not be enough for a suasive argument, but as an explanation of how a given deductive system works they are sufficient.
So far we have examined the possibility of justifying the rules we use in our deductive reasoning. We have come up with a way that soundness and completeness proofs can produce an explanatory justification for deductive reasoning, but because the justification is inevitably circular, in that it will involve the reasoning it is trying to justify, we do not have a justification for using one particular type of reasoning over another. This question will be returned to below, but I would first like to examine a slightly deeper question regarding deductive reasoning.
There are three ‘levels’ to which a justification of deduction could be addressed: the first is the unproblematic level of a specific argument, and the question of what rules are followed to make that argument valid, the next is the level of the correctness of a particular type of inference, which is what has been discussed thus far, and the third is the level of the existence of deduction as a whole – the fact that it is possible to make deductive arguments at all. To try to justify this is a problem closely related to the philosophy of meaning. There is a tension between two aspects of deductive reasoning: the way that the conclusion is ‘contained within’ the premises, but that a deductive argument is nevertheless useful. We can question the reason that we have an indirect means for obtaining the truth of a statement.
Quine’s model of language is as a ‘web of knowledge’, a network of interconnected sentences. Those at the periphery are those which depend on empirical observation for their truth, and those towards the middle depend on those further outside. The inner sentences cannot be confirmed by observation, but by reasoning based on other sentences. Thus far, there is nothing puzzling about sentences whose meaning requires deductive reasoning based on other sentences for their truth to be established.
However, deductive reasoning also allows us to reason outwards on this web, and this is much more interesting. The meaning of a sentence on the periphery gives a way for it to be established – by observation. How is it then that we can establish it by a completely different method, perhaps without reference to that observation at all? To answer this we must consider the way in which sentences have meaning. There are two general models for language in this respect: holism and molecularism. Under the holistic view, language can be understood only as a totality, and the meaning of a sentence cannot be given without reference to all other sentences. The meaning of a given sentence, then, already includes all the ways in which its truth could be established, whether those are empirical or logical.
Under the molecularist view, it is possible to represent the meaning of a sentence independent of the whole of the rest of the language. In this view the form of a sentence becomes important, and we are back to semantics again. The value of soundness and completeness proofs becomes something external to logic, resting on whether two-valued truth values and conditionals are an accurate way of describing a sentence’s meaning. The question of how we can justify a deductive inference rests on how we define meaning in our language, and deductive reasoning will only work to the extent that the assumptions made in the deductive system match up to the way our language functions.
4. Dissident Logics
Thus far the discussion has focused on classical logic; this is the ‘standard’ logic we use regularly, formalised into simple rules. However, there are also other forms of logic which some philosophers and logicians advocate, and their existence has important implications for the question of whether any deductive reasoning can be justified. One such logic is intuitionist logic. This proceeds by moving from the idea of there being an absolute truth-value for all statements, to focusing on the provability of statements: we can assert ‘A’ only if we can prove it. We can assert ‘A ∧ B’ if we can prove both ‘A’ and ‘B’. Following from this, we lose some of the most important characteristics of classical logic: in particular bivalence – a statement always being true or false. For instance, in classical logic, ‘A ∨ ¬A’ is always true. In intuitionist logic, this is not the case, because ‘A’ may be such that we do not have a proof for or against it.
The reason this is important is that thus far we have an explanatory justification for a given deductive system, but not a justification for why we should pick any one system over another. Provided a system can be shown to be self-consistent, there seem to be no reasons why classical logic should be ‘better’ than intuitionist logic, or any other. For example, another non-classical logic is modal logic, which in concerned with the expressions “it is necessary that” and “it is possible that”. These can be formalised in various ways to give a system which can be used and analysed just as classical logic can. Multi-valued logics reject bivalence in order to represent vagueness (for example, when we are unsure about whether a given predicate applies or not) more accurately. There are many other varieties of logic which have developed, and some philosophers believe that these should have precedence over classical logic; for instance, much of mathematics can be rederived from intuitionist logic, which according to some philosophers puts it on a more solid philosophical footing (holding that classical mathematics makes existence claims about mathematical objects that have no real basis).
One possible response to the question of what justifies any particular deductive system is to say that that logic is empirical. In other words, we should be prepared to change our logical theory if new evidence comes to light that contradicts it. This may seem a strange approach to take – in most cases we take evidence that contradicts logic to mean that the evidence itself is flawed. However, Quine held that logical rules were a part of his web of knowledge just like any other, and accordingly that there could be evidence to which the correct response would be to revise logic itself.
A recent example of this is quantum logic, which arose as a consequence of quantum mechanics. One important difference between quantum and classical logic is that it is no longer the case that ‘A ∧ (B ∨ C)’ implies ‘(A ∧ B) ∨ (A ∧ C)’ (the distributive law). For instance, imagine that ‘A’ means ‘the particle is moving to the right’, ‘B’ means ‘the particle is in the interval [-1,1]’ and ‘C’ means ‘the particle is not in the interval [-1,1]’. While ‘A ∧ (B ∨ C)’ is true, neither ‘(A ∧ B)’ nor ‘(A ∧ C)’ are true, because the uncertainty principle does not allow such strong assertions to be made. The disjunction is false, and so the distributive law fails.
The idea of a revision of logic, however, seems counter-intuitive. Logic is something that seems to be ingrained within us much more deeply than anything else; it would not help someone mystified with quantum mechanics to tell them simply to give up the distributive law. It must be clear what we mean by changing our logic – should quantum logic replace classical logic in all fields, or should we use each in their own context? Moreover, the idea that philosophy could conclude that almost every rational field has been doing logic ‘wrong’ seems very odd; in this instance I would argue that philosophy should be serving a more purely explanatory role about our use of logic, rather than prescribing a method to everyone else.
Another response to the question of which logic to use is the idea that logic is conventional. The results of logical arguments – ‘logical truths’ – are true because of the convention of using a particular system. In an example from mathematics, geometry has its origins in Euclidean geometry, which is based on certain axioms. Non-Euclidean geometry, which developed later, stems from rejecting some of these axioms in favour of others. The theorems within each system could be said to be true within that system, and thus true by convention. Returning to logic, we use one form rather than another simply because that is the conventional logic to use, rather than because of something intrinsically ‘better’ in one form of deductive reasoning.
The final response which I shall examine is a pragmatic one: our use of classical deductive logic is justified because it conforms to our accepted practice. Goodman argues that the system we use arises as a result of bringing linguistic practice and self-consistent rules into line with each other: “a rule is amended if it yields an inference we are unwilling to accept; an inference is rejected if it violates a rule we are unwilling to amend”. Our choice of logic therefore accommodates both our intuition and the formal rigour that we require of a deductive system. This solves the problem of why we use a particular system of deductive reasoning, as well as the deeper problem mentioned above of why our reasoning can correlate to facts in the real world. No one particular consideration leads the other, but rather we come to a compromise which produces a system we are happy to use. The soundness and completeness proofs show that the system will not lead us to contradictions, and the fact that the system conforms to our intuition about arguments means that we do not end up with conclusions we cannot accept.
5. Conclusion
This essay began by explaining why we feel that a justification of our deductive processes is necessary; to back up our use of deductive arguments and give us confidence that such arguments are not going to lead us into false conclusions. Although it is not possible to justify that the deductive reasoning we use is in some way better than any other, it is nevertheless possible to show first that the system we use is not self-contradictory, and provide an explanation for how it works and where it comes from. I believe that this is enough to support our continued use of deductive processes.
I contend that the justification of classical deduction is that it is a convention implicit in our use of language. We may ask ourselves “Would we accept a logic that disagrees with intuition?”. I believe that the answer to this is no, and that any further justification than this is impossible with regards to why we use the deductive systems that we do. Analysing this topic has shown that this is not a weakness of the system we use but rather a necessary part of it.
Bibliography
Brown, H. I., ‘Circular Justifications’, in Philosophy of Science Association, Vol.1 (University of Chicago Press, 1994)
Carroll, L., ‘What the Tortoise said to Achilles’, in Mind, Vol.4, No.14, (Oxford University Press, 1895)
Dummett, M., ‘Is Logic Empirical’, in Truth and Other Enigmas, (Duckworth, 1978)
Dummett, M., ‘The Justification of Deduction’, in Truth and Other Enigmas, (Duckworth, 1978)
Goodman, N., ‘The New Riddle of Induction’, in Fact Fiction and Forecast, (Harvard University Press, 1983)
Haack, S., ‘The Justification of Deduction’, in Mind, Vol.85, No.337, (Oxford University Press, 1976)
Quine, W.V., ‘Carnap and Logical Truth’, in Synthese, Vol.12, No.4, (Springer, 1960)
Thomson, J. F., ‘What Achilles should have said to the Tortoise’, in Ratio, (1963)
Supervised by Mr J Barfield, Whitgift School.
Haack, S., ‘The Justification of Deduction’, in Mind, Vol.85, No.337, (Oxford University Press, 1976), p.114
Carroll, L., ‘What the Tortoise said to Achilles’, in Mind, Vol.4, No.14, (Oxford University Press, 1895)
* The original paper uses an argument about triangles drawn from Euclid, and labels the propositions (including the conclusion) with letters rather than numbers. However, my version is structurally identical.
Thomson, J. F., ‘What Achilles should have said to the Tortoise’, in Ratio, (1963)
Haack, S., op. cit., p.116
Brown, H. I., ‘Circular Justifications’, in Philosophy of Science Association, Vol.1 (University of Chicago Press, 1994), p.407
Dummett, M., ‘The Justification of Deduction’, in Truth and Other Enigmas, (Duckworth, 1978), p.296
Dummett, M., ‘Is Logic Empirical’, in Truth and Other Enigmas, (Duckworth, 1978), p.271
Goodman, N., ‘The New Riddle of Induction’, in Fact Fiction and Forecast, (Harvard University Press, 1983)
