Introducing the Scandal of Deduction
In response to some feedback on my article about resolving the Scandal of Deduction, I have decided to put together a quick, simple introduction to the problem.
The Scandal of Deduction
Philosophers have often prized deductive reasoning above inductive reasoning. This is because it is generally accepted that, if we are using the correct tools of deduction, any conclusion that follows from true premises must also be true. If we know that all men are mortal (premise A) and that Socrates is a man (premise B), then it follows that Socrates is mortal (conclusion, C). That is, A and B imply C.
However, this implies that, in any deductively valid argument, the conclusions must already be implicit in the premises. From this, it follows that no new information is generated by deduction. In our example, the information that “Socrates is mortal,” is contained in our premises. Given our premises, our conclusion follows with a probability of 100%. Likewise, deterministic computation of the sort carried out by our digital computers also does not create new information. Thus, it seems we learn nothing new from deduction!
On the face of it, this widely accepted proposition, the “Scandal of Deduction,” seems ridiculous. Deductive arguments appear to carry information; they reduce our uncertainty. Data analysis software seems to tell us new things about our data, etc. We do not memorize Euclid’s axioms and thereby know the answers to all geometry problems, even though all the solutions follow from the axioms.
This problem is closely related to the assertion that all of mathematics is tautological, i.e., the examination of truths that are true by definition. However, while it is obvious that 2 + 2 is 4, the value of 1,9173 ÷ √529 is much less obvious. The question posed by the “Scandal” is: how can we vindicate our intuition that computation and deduction are informative? In what sense is mathematics informative?
Below we will survey common solutions to this problem:
Deduction is Actually Induction in Disguise
J.S. Mill, the key proponent of this response, says that we only know that “all men are mortal,” through experience. Thus, our “deduction” is just extending this empirical knowledge to a new case. Our inference was actually complete when we decided that all men were mortal, an empirical claim.
Likewise, our knowledge of mathematics comes from experience. Axioms are experimental truths; generalizations from observation. But then why should some deductions be so obvious and others so challenging? Mill claims that our problem is actually just deciding whether the particular case we are analyzing falls within the domain of prior inductive findings. A deduction can be informative if requires such a complex set of inferences that it is difficult to tell which established findings the case falls under.
This view has not been particularly popular because many philosophers do want to assert that deduction differs fundamentally from induction, in part because of the Problem of Induction.
Psychological Solutions
In this view, all the information in any valid deductive conclusion is in the premises. However, we may not be psychologically aware of the implications. Logical reasoning simply helps us “bring out” the full content of our premises, organizing it so that it is easy for us to inspect (C.G. Hempel was a proponent of this view).
This position is easier to understand if we take a quick detour into the philosophy of language. A big question in that field is: “how do sentences get their meaning?” We do not have time to delve into the many hypotheses that have been proffered. It is enough to note that the shapes of letters do not seem to obviously imply their meanings. After all, if we do not know Greek, a Greek sentence does not convey its meaning to us when we see it. In the same way, when we read premises, it is not clear that this implies that we extract all the information contained in them. It is possible to misunderstand language, or fail to grasp a premise’s meaning, either in part or in whole. Deduction then, helps us extract information from our premises.
In his early work, Ludwig Wittgenstein advances a similar solution. “When the truth of one proposition follows from the truth of others, we can see this from the structure of the propositions,”¹ and “in a suitable notation we can in fact recognize the formal properties of propositions by mere inspection of the propositions themselves.”²Thus, given an “ideal notation,” every deductive truth will be readily apparent.
The main problem philosophers have with these solutions to the Scandal is that they are too vague. They do not explain why some deductions are obvious and some are not, nor how any sort of “ideal notation,” could be constructed.
Formal Solutions
Formal solutions try to rigorously define why it is that some deductions are less obvious than others. These solutions try to analyze logic itself to determine if there are ways in which deduction can indeed create new information, or determine how deduction makes “hidden” information readily apparent.
Jaakko Hintikka’s solution has been the most influential of these. This solution distinguishes between depth information, the maximal amount of information we can extract from a set of premises, and surface information, the amount of information that is trivially available given the premises. For example, Aristotelian syllogisms only present surface information.
These sorts of solutions, such as those invoking the concept of virtual information, are quite complex, and cannot be covered in depth here. However, we can note that, to date, no one formal solution has been widely accepted as resolving the Scandal. One difficulty for these solutions is that even very simple deductions can sometimes be surprising to us, just as the results of fairly simple arithmetic can also surprise us. Thus, it is not readily apparent that the Scandal only applies to certain types of deduction, e.g. those invoking certain inference rules. Appeals to computational complexity, as defined in computer science, have a similar problem. An extremely simple computation can still take even a supercomputer an arbitrarily long amount of time to compute if the input is long enough.
The Solution from Physics
Another solution, the one I propose here, involves looking at the physical processes that underly computation in digital computers and reasoning in the human brain. Both thought and computation only occur over time. That is, entailments are not known as eternal Platonic entities, but rather logical consequence is instantiated in an essentially stepwise process through the means of some physical medium.
While most contemporary work on logic focuses on formal systems, it’s important to recall that logic was once largely considered to be “the study of the most general features of thoughts or judgments, or the form of thoughts or judgments.” For various reasons, philosophy has revised the way it tends to think about logic, but this does not diminish the fact that deduction is something that occurs in the mind. Or in the case of machines, in a physical system the mind has crafted so as to have it carry out deduction.
Computation itself involves communication. A Turing Machine head must read symbols off a tape, and instructions for its states must somehow be coded into it. Likewise, neurons working together compute, but in doing so they also must communicate. In communications, a message is not received before it is sent, and so a conclusion cannot be known until computations on the premises are finished. This is because communication is a causal process. This fact is widely recognized because we do not tend to think of communications in the totally abstract way that we often think of computation. But it is very easy to slip into thinking of the mathematical relationships involved in computation as existing eternally, outside spacetime.
This may be the case, I do not mean to argue the merits of mathematical Platonism here. But it is also the case that we, and our computers, and anything doing deduction in our world, do exist in spacetime. Thus, we should never expect that any conclusion becomes apparent in “no time at all,” simply from having a set of premises “before us.” Afterall, a system can only come to know of such premises through some form of causal interaction, one which must, like all causal processes, occur over time. That it seems like some deductions are trivial is simply due to the way in which most of the vast amount of computation involved in creating our conscious perceptions is hidden from us, as detailed in this post.
- Tractatus Logico-Philosophicus , 5.13
- Tractatus Logico-Philosophicus, 6.122
