Logical Inference and Derivation
Axel Arturo Barceló Aspeitia
Indiana University, Bloomington
December 10, 1998
The following text is an excerpt from the Ph.D. dissertation I am currently working on. The main subject of such dissertation is the role of the notion of set of rules in Wittgenstein's philosophy of mathematics during the early thirties. As a philosopher, Wittgenstein has the fortunate virtue of basing his theories in commonsensical views about our linguistic uses. his whole philosophical view can be seen as a complex elaboration of these views. As such, we can enjoy of some of the fruits of his analysis without having to submerge ourselves into the depths of his philosophical thought. Such is the case of his analysis of logical inference. There is no need to know anything about Wittgenstein's philosophy to understand it (Hence, there is no need to know anything about it in order to understand the following text either). As a matter of fact, there is no thesis in Wittgenstein's analysis which departs from common sense ideas about logic and its practice. At most, what he does is to organize those commonsensical views into a complex but very clear picture. That is what I am offering here: Wittgenstein's clear picture of what everybody who has done at least some very little of the most simple logic already knows. I am not offering any complicated philosophical theory of logic inference, but only presenting a clear picture of logical inference which organizes our commonsensical views on the topic in such a way that, I hope, will help us dissolve some common confusions about it. In particular, I am interested in the confusion between logical inference and derivation. This confusion is certainly not rare and it is just a case of a more general confusion between mathematical processes and their results. In our everyday life it may not be common to mistake the process of cooking with the resulting meal, or the process of opening a door with the open door itself. However, in the case of logic or mathematics, this point may not be that easy to see. This point may be hard to grasp because, in many cases, we are not very used to making such distinction in our language. Sometime we use the same word for both the names of processes and the name of their results. We use for example the same word 'addition' to refer both to the mathematical operation between numbers and the process of calculating its result. Still, they are radically different things. Addition in the later sense is, for example, what accountants do. Additions in the former sense are mathematical propositions. Mathematics tell us the correct way to add only in so far as it tells us what would be its correct result, that is, a correct addition in the first sense.
1. Inference.
Something similar happens in the case of logical inference, where we have two different notions sharing the same name. Compare the following two sentences:
1. It took me a long time to come up with the inference we had for homework.
2. The inference we had to do for homework took me a long time.
The word 'inference' is used in a different sense in each sentence. In the first sense, a logical inference consists in the transition(1) from one assertion (utterance or sentence) to another(2). But this transition must not be mistaken for some sort of process or event that happens in time(3). What took me a long time, according to sentence (1) was not the inference itself, but to come up with it. To come up with it took me a long time. This is because an inference in this first sense is not a process but the result of one. This process, in turn, is the meaning of 'inference' in the second sense. This is the sense of 'inference' in sentence (2). What I call 'to come up with the inference' in sentence (1) is what I call 'inference' in sentence (2). An inference in the second sense is the process of coming up with an inference in the first sense. This inference does happen in time and is performed by someone (person or, in some cases, calculating machine).
Once we have recognized between inferences-as-processes and inferences-as-results, we must also realize that there is no one-to-one correspondence between them. It is clear that different processes can have the same result. Therefore, the same inference-as-result can be brought up through different inferences-as-process. In formal logic, for example, we come up with our inference through symbolization and the application of rules of derivation. But there are many other processes of drawing a conclusion besides formal derivation. Contrast a formal inference with what we call inferences in our everyday life. Here perhaps inferences like the following come to mind: "the stove is smoking, so the chimney is out of order again". In this case, just as in a formal inference, the inference consists in the same thing: the transition of one expression from another(4).
This may go on paper, orally or 'in the head'. --The conclusion may however also be drawn in such a way that the one proposition is uttered after the other, without any such process; or the process may consist merely in our saying "Therefore" or "It follows from this", or something of the kind(5).
2. Derivation
A related problem to that of distinguishing processes and their results is that, in many cases, it is not very easy to say what is the result of a process. Derivation is a good example of this. What is the result of a derivation? We may be tempted to say that the result is the conclusion. But we could also say that the result is the inference. Also, we may say that the derivation itself is the result. However, when we speak this way, we are actually speaking about three different processes. Consider the following example:
A Þ (B Þ A)
A Þ ((B Þ A) Þ A)
A Þ ((B Þ A) Þ A) Þ ((A Þ (B Þ A)) Þ (A Þ A))
(A Þ (B Þ A)) Þ (A Þ A)
A Þ A
Under different viewpoints, we can say that the result is the concluding proposition (AÞA), or that it is the inference of (A Þ A) from A Þ (B Þ A), A Þ ((B Þ A) Þ A) and A Þ ((B Þ A) Þ A) Þ ((A Þ (B Þ A)) Þ (A Þ A)), or that the result is the whole derivation itself, including its intermediate steps. In fact, in each case we are talking about different processes, each one with its own result. In order to answer the question 'what is the result of this?', we must first ask ourselves what do we mean by 'this'. We have already talked about derivation as a process for the formation of logical inferences. As such, if by 'this' we mean an inference-as-process, its result is the inference of the conclusion from the premises Sometimes, when we say that the result of a derivation is its conclusion, we are speaking improperly, because we do not see that the conclusion is correctly derived only from certain propositions, and hence, that the actual result of the derivation is the inference of the conclusion from the premises. What the derivation is telling us is that it is valid to infer the conclusion from the premises. On the other hand, if by 'this' we mean the proof of a theorem, then the result is the conclusion. In this case, what the proof is telling us is that (A Þ A) is a theorem of our theory, in this case, the axiomatic system L4 of Kleene's Introduction of Mathematics (Van Nostrand, 1952). Finally, if by 'this' we mean the construction of the derivation, then of course that the derivation itself is the result. However, it is very important to distinguish between all these processes. Each one has a different set of rules. In each case, there are different reasons why the result is correct, and 'correct' means different things. In the first case, the derivation is correct because each step follows the rules of derivation. In turn, the inference is valid because we can form a correct derivation of it. Finally, the proposition is a theorem because it can be correctly inferred from the axioms of the theory. One easy way to see that each set of rules is different is to look at the rules which it contains. The set of rules of the derivation, of course, includes what we call the rules of derivation (in this case, only Modus Ponens). The set of rules of inference, on the other hand, includes all valid inferences. Finally, in the formation of the proof, the rules include the axioms and the rule that everything that can be correctly inferred from them is a theorem. In this case, the set of rules is the axiomatic theory itself.
3. Inference and Derivation.
As you can see from the previous paragraphs, the set of rules of inference and the set of rules of derivation form a very tight link. Logical derivation holds a privileged place among the processes for forming inferences. It is told apart from other processes in so far as it not only forms an inference, but also provides a proof for its validity. To say that an inference is valid in a formal system is nothing but to say that it can be derived inside a formal system of derivation. This does not mean that valid inferences (not even those valid in a formal system) are only those which in fact are formed through formal derivation. Derivation is not the only correct way to form inferences. As I said before, any method which gives the same results would do. Logic does not tell us that we must always form our inferences through the rules of derivation. Think of the way we teach formal logic, specially at the basic levels. What we learn in our formal logic courses is not how to make inferences. In a certain sense, we already know how to draw conclusions. We do it all the time in our everyday life. As a matter of fact, that is why it is important to study formal logic. What we learn in formal logic is neither how to infer (as if we had never done it before) nor how we to infer correctly (as if we never did it correctly before). It is not that, in order to make correct inferences, we must go through our everyday lives doing Modus Ponens and the like.
Logic tells us how to recognize a valid inference. It tells us which inferences to call 'valid'. In this sense, we can say that it provides criteria for validity. If we are learning anything new, it is not so much how to infer, but how to recognize a valid inference. That is why we learn to derive. That is why we learn to provide formal proofs for the validity of our inferences. We learn to derive, because derivation gives us some strict criteria for validity. In formal calculus, an inference is valid if we can produce a formal derivation of it in our calculus. Different logical analysis give us different tools for recognizing the validity of inferences. The tool that formal logic gives us is derivation. But even within formal logical, different calculi give us different criteria for validity, so the word 'valid' has a different meaning with regards to different calculi. "Valid" in propositional calculus does not mean the same thing that "valid" in quantificational or modal logic. Furthermore, each use of the word 'valid' has a very specific and -we may say- technical sense and, as such, each is just a part of our intuitive notion of what is a good argument. To have a correct formal derivation of an inference in a logic calculus is to have a proof that the inference is valid. However, the inverse is not true. A valid inference does not necessarily have to be valid in some formal calculus (at least not of the calculus we have today. It is still an open philosophical question whether it will ever be otherwise). That is why it is important to tell apart the validity of an inference and the correctness of a derivation. Derivations are very precise and constrained tools for the recognition of validity, but, as such, cannot give us the fool picture of what is a good argument. Just like not being able to hammer a nail inside a concrete wall does not mean that the wall is impenetrable, we cannot say that an inference is invalid because we cannot construct a formal derivation of it in, say, propositional calculus. Even though the task is one -to recognize the validity of an inference-, the tools are many. Each one has its own power and its own use. Derivation is just one of them: a very powerful one indeed, but still just one.
In summary, the rules of inference do not tell us how to infer, but how to recognize a valid inference. These rules have nothing to do with the how of the formation of the inference, but with its validity. Different processes for the formation of inferences may be appropriate as long as they produce valid inferences. One of these processes is derivation. Derivation obeys different rules than inference. The rules of derivation are different from the rules of inference. Among the many processes for the formation of inferences, derivation holds a special place because it not only forms the inference, but provides a formal proof for its validity. The proofs that formal derivations give us are conclusive only in one sense. They tell us what counts as a valid inference, but are inconclusive regarding invalidity. Derivation is just one of the many tools we have for the recognition of validity. However, the picture of validity it supplies is incomplete. Logical validity is more rich and complex. The work of Logic is much more complicated than mere formal calculation.
Mexico City, Thursday, December 10, 1998
1. 0 Wittgenstein, Ludwig: Remarks on the Foundations of Mathematics [RFM] §9
2. 0 RFM §6