FORMAL REASONING
AN ESSAY ON LOGIC
0.
There is a philosophical debate about the status of logic as a subject. On the one hand logic is connected with reasoning in general, and in particular the kind of reasoning connected with information processing in human beings. On the other hand logic is a normative subject. It teaches how people should think, and not only how they actually do it. In this essay we shall give an outline of how logic finds it place in the information processing in humans. First a sketch of this outline. We consider five levels of information processing. From the top down to the bottom we have
We imagine this as a pyramid with the bottom level by far the most pervasive. The upper levels are resting on the lower levels.
The information processing goes through four states
There are the following transitions
We now go on to describe what happens on each level in more detail. We start with the bottom level and move upwards.
1.
The vegetative level is the basic level. One could imagine that it has just two states - sensation in and motions out. Below we shall argue that there is also the internediate state which we have called kind. First about the input - the sensations. With sensations we mean raw, unprocessed sensations coming from the outside world or from your own body. Note the following about sensations
We emphasize the fullness of sensations. Sensations are just what gets at you when you sit down and let the world seep in. No description can fully describe everything in a sensation. Some sensations come from wavepatterns. From a physical point of view it is obvious that a sensation is not something which belongs to an instant of time. It must be a time interval. This is backed up by psychological experiments. Each sense has a sensory register - a short term registration of the last second or so of sensation. In vision this is like an afterimage. In hearing the beginning of a sound is still part of a sensation even when we have come to the middle of the sound. The registration is echoic - there is no processing involved. The uniqueness of each sensation follows from the fullness. No sensation can be repeated. It occur only once.
As the output we have motions. This may be muscular motions or other kind of movements giving raw material to new sensations. We do not have much more to say about motions. For our story it is more important to concentrate on the intermediate state between sensation and motion. There are three arguments for such an intermediate state
On the vegetative level we learn by stimulus response. But this cannot work if each sensation is unique. There must be processing of sensations where similar sensations are lumped together. In this way we get from a sensation to a kind. So learning is impossible if we do not have a kind level. Similarly with memory. There is no way we can remember just sensations. Jorge Luis Borges has a short story about Funes who did just that - and with miserable result. Newer results about the physiology of vision has backed this up. There is much information processing going on in our sensory apparatus. We do not see pixels but lines, edges, circles, colours, movements etc.
The transition from sensation to kind is hidden deep in our biology. We could use the Kantian term "Verstand" for this transition. The transition is there and we cannot do much about it. The processing done on the vegetative level gives a background in our consciousness. It is just there and we do not need to attend to it. Most emotions seems to be of this kind.
2.
The difference between the vegetative level and the mimetic level lies in the relation to the outside world. At the vegetative level the outside world is very weakly represented. There are just some traces of it in the particular transitions from sensations to kind that is used. But of course there is much more to the outside world than that. The kind gives just a snapshot - a schematic view seen from a particular angle. In the outside world we do not only see from one angle, but there are also expectations about how things would look from other angles. Try to imagine yourself first looking at a house and then at a theater decoration representing a house. The sensation may be the same in both cases. Perhaps also the kinds. But there is certainly a difference in how the things are seen. This difference becomes particularly clear when we consider visual jokes like the wittgensteinian picture of a duck/rabbit.
We use the term object for such an internal representation of an external thing. An object consists of
Talking about views gives a visual flavour to the objects. As an antidote to that it may be useful to think about the musical objects in a piece of music. In the expectation structure there are among other things
Note the following difference between kinds and objects. There is a decidable equality between kinds. We know when two kinds are equal. This is not so among objects. We may think that two objects are the same, but by digging into the expectation structure, getting new wievs we suddenly discover that the two objects are not the same. Conversely we may have two seemingly different objects which turn out to be the same. With objects we get the possibility of deception.
The transition from sensation to kind - der Verstand - is automatic. It just happens. This is not so with the transition from kind to object. Here we must jump. This becomes clear considering the visual jokes where one and the same kind may give one object or another. In each instant there is only one object given. We could follow up our kantian terminology (with "der Verstand") and call this transition for "der Vernunft". To use "der Vernunft" we must single out the sensation with our attention. There is a directedness in "der Vernunft" which is not in our "Verstand".
At the mimetic level we have started to divide the world into objects and we also get actions. An action is a composite transition from an object to a motion to a sensation to a kind and then to a new object. We start to develop "ein praktische Vernunft" - a competence in how to act in a world of objects. We believe that this is done at this level and not only at the higher levels where we have language. This is certainly my experience watching my children as babies.
3.
A symbol is something representing something else. It is given by
In principle the connection is arbitrary. There is no way which gives you automatically the signified from the signifier. A different problem is how such a connection can be learned and how it can be remembered. At the outset we usually rely on some kind of resemblance and hints.
Information processing at the symbolic level gives new power. To me it is remarkable that there seems to be no pictures (abstract or concrete) made by animals or by early humans. (I have seen mentioned 40 000 years as the age of the earliest known instances.) Before pictures were made it seems unlikely to me that there were any use of symbols.
At the symbolic level we cannot use the symbols as much more than names for objects. But it is of course a magical thing that by using simple names one can evoke complicated objects which may not even be present.
4.
Above the symbolic level we have the compositional one. Here we compose symbols making new symbols and we get language. Composition is not just made by putting symbols beside each other. The crucial thing is - as noted by Frege - functional composition. Consider how a child learns to speak. At the vegetative level the baby receives sensations, filters them through "der Verstand" and gives out motions. Then comes the mimetic level where the baby starts to constitute a world consisting of objects and actions. After a year it starts to make auditory symbols signifying either objects or actions. The crucial step now is to observe that
One of the favourite games betwen children 1-2 years old and their parents is to point at things and say: red chair, red book, blue chair, red cube, blue ball, red ball ..... In this way the incomplete symbols " red --" and "blue --" is learned. Following Frege we have indicated with "--" that there is an open position where a complete symbol should be inserted. In this way we get a new kind of symbols. They are not representing actions or objects but can be made to represent such if symbols are inserted in the appropriate positions. Let us call such a symbol for a compositional symbol and the symbols at the symbolic level for complete symbols. To guide the correct insertion of symbols, the compositional symbols have a type assigned.
We think that natural languages are recent inventions in the history of mankind. The type assignment is given by the grammar. The parts are individual words, prefixes, suffixes, declinations etc. We must remember that the individual languages have a history. There is no reason to assume that each individual language can be presented in such a way that the parts and the type assignment is exhibited in an explicit way. Our claim is a more modest one. We claim that the cognitive processes at the compositional level were necessary and sufficient to get natural languages started. How the structure of a particular natural language actually is, is buried in its history.
We have talked about objects as being undecidable. Objects have a backside which we can start to unravel but will never finish with. Two objects may turn out to be the same after a long unravelling. Our signifiers are also objects. But they are objects of a simple kind. We express our symbols in such media that it is decidable whether two signifiers are the same or not. If in doubt the problem is with the medium and not the name.
There are a number of attempts of using concepts of linguistics in other fields (anthropology, litterary theory, sociology). This is really only useful as long as the other fields concerns phenomena on the compositional level and not only the symbolic level. They rarely do.
Using composition we get an unlimited number of new symbols by making symbolic expressions. Such symbolic expressions is said to belong to the syntax while the signified belong to the semantics. Following Frege we claim that one cannot get from syntax to semantics by syntactical means alone. There is no way to do this automatically. We had bring in ourselves as human beings already knowing some symbols.This means that we both have a gap between syntax and semantics and much of what goes under the name of semantics should rather be seen as an extended syntax.
Philosophers have used the gap between syntax and semantics to give some limitations to what computers can do. The argument is simply that computers are syntax machines and we cannot use them to get semantical results. Later we shall show that this argument does not at all adress the important question about what computers can and cannot do.
5.
The formal level is our last and topmost level. At the symbolic level we introduced the naming relation between signifier (the name) and the signified. We have a picture theory of meaning. This theory is usually attempted to bring up to the compositional level. Consider for example our baby hearing uttered "red ball". What should be the picture of "red"? The usual answer is that it is an impression of redness. Already here it is clear that something is wrong. There is no reasonable way to compose such different categories as "impression of redness" with "ball" to get "red ball". But the real breakdown of the picture theory of meaning came with formal reasoning.
Consider proof by contradiction as for example given in the usual proof that there is an infinite number of primes. Assume there is a finite number of primes. Following the picture theory of meaning we make a picture where there are only a finite number of primes. Then comes some intermediate steps where this picture is given more and more details. Then at last we get a contradiction. What we thought was a picture, was not a picture at all.
This proof baffled the Greeks. I think that in spelling out what was involved they discovered logic and formal reasoning. There are certainly remnants of formal reasoning before. It is for example so in some elementary mathematical arguments. The crucial step here is to have a theory of how to think from absurd assumptions. Other cultures have theories of how to think from reasonable assumptions. The point about proof by contradictions is that the assumptions are just plainly wrong.
At the formal level we think from assumptions. This has been explained in an exemplary way by Gentzens natural deduction which of course fits nicely in with Freges theory of functional composition. Let us spell this out. The main requirement is that we had to have type assignments where we know whether something belongs to a type or not. This is a requirement for the compositional level. Then we must be able to use variables of the different types. An assumption could then be like
Then we can use this X further down in our argument. This is for example how we solve equations. Let X be a number such that 2X+3=3X+1.
Then X=2. Other equations may result in two or more candidates for X, or perhaps no solution at all (like a proof from contradiction).
What are the intermediate steps after having made an assumption? This is reasonably clear when we solve an equation. Frege came up with the idea of a syntactical rule. A syntactical rule changes a symbolic expression (the premiss) into another symbolic expression (the conclusion). Frege is very careful that his syntactical rules should do more. They should also transform an interpretation of the premiss into an interpretation of the conclusion. To be a little more precise. Assume the premiss has variables X Y Z. Then given any interpretation of the premiss where we have used particular interpretations of X Y Z, we get an interpretation of the conclusion using again the interpretations of X Y Z. The interpretations of the assumptions propagate through the intermediate steps and leads to an interpretation of the conclusion.
Let us see what goes wrong with the philosophers attempt to use the gap between syntax and semantics to give limitations about what computers cannot do. Computers work on syntax alone. So how can we make computers do semantic work. The answer is they can do semantic work in exactly the same way Freges semantic rules can do. Consider a computer calculating 5+7. It is obvious that the computer cannot do it alone. There must be a human interpreting the input and the output. The human gives the semantic of the expression to the computer when he types in the expression. Then he starts the execution and gets the answer 12 on the screen which he gives the correct semantics. The reason why this works is because the program is correct. The main ingredient in a proof of correctness is of course to show that the semantics of the input is propagated using syntactical rules to a semantics of the output. The problem with Searles Chinese room is not the gap between syntax and semantics, but whether we can get syntactical rules doing the required job (which there are no reason to believe that we can).
Putting the formal level at the top may give the impression that this is the most important form of reasoning. That is not my intention. One way to counteract that is to remind the reader of the pyramid of levels where the vegetative level is by far the largest. Another way is to talk about open and closed expressions. This is a difference in how the semantics for the expressions behave. In a closed expression the semantics is given in advance. In an open expression the semantics can be given while you try to understand more of it. Consider "love" as a word in a poem. The semantics is certainly not clear when you first hear it, but most of us are willing to go along and listen to more of the poem as we take with us our own experiences as humans. Then in a number of cases the word did not create any problem. In other cases you may even get an extended semantics for "love".We claim that the open semantics is the typical - and most useful case. The rethorical tradition deals with open semantics.
What is closed semantics? If everything is given in advance, is there more to be said? An important starting point for Frege was closed semantics. He considered the case where we have two arithmetical expressions and asked whether they were equal or not. He would like to distinguish between the equality 4=4 and an equality like (1+3)(5-2)-8=4. In the first case the two sides have the same "Sinn". In the second case they have different "Sinn" but the same "Bedeutung". For Frege it was important to distinguish between the symbolic expressions such as they were ready for computation and the result of the computations. The same distinction is also important for the semantics of computer programs.
6.
So far we have mostly taken the development of information processing as an internal affair. As something that happens inside each of us. This is obviously not so. The environment enters at the very start. "Der Verstand" may be good for some environments and bad for others. With development of "peaktische Vernunft" with objects and actions we influence our environment. And so on for the other levels. Here I shall discuss some connections between our organizing of environments and the usefullness of closed expressions.
Consider what we often call the civilization process, This is the process going through stages like
Now I want to make the following important point. All these changes and the other changes connected with the civilization process makes it possible to describe new parts in our environment using closed expressions. The civiliztion process is connected with getting better contol over the environment making it more predictable. Here expressions where the semantics can be given in advance enters, and hence closed expressions. We organize the environment such that we can do book keeping, can do our own calculations.
Now another important point. One may get the impression that the civilization process is a straightforward process. If one looks at the details it is certainly not so. Take for example the introduction of money economy. A number of things got a price tag. But at the same time there was a discussion about what could not be measured in money. Similarly for workers in a factory. At the same time as strictly organized work hours were introduced there also appeared unstructured leisure time. There is a dynamic here that it is important to understand. Throughout history there are examples of attempts of structuring our environment which did not work out. The civilization process led to many traumas. Some were given cultural expression through litterature and other art forms.
Above we discussed some philosophers handling of the question about what the computers can and what they cannot do. We argued that considerations of the gap between syntax and semantics were not important. In our view it is important to look at the history of the civilization process. To put it crudely: in the old stone age there were not much a computer could do, bot with the organization of society there were more room for closed expression and also for possible uses of computers. Considering the historical details we know a lot about which areas could be more strictly organized and which areas resisted such organization.
7.
We started this essay by asking about the status of logic as a subject - whether it was normative or part of psychology. Our answer is that it is normative. One does not need much additional justification than the justification needed for composing words with words in our natural language. There is then room for formal reasoning.
Our argument and our position is quite close to Freges position. Frege wanted to show that true arithmetical statements were analytic and not only synthetic a priori as Kant maintained. Going through Freges argument he seems to have almost succeeded. There are two flaws in Freges system
The truth values do not fit well into the functional composition which is the backbone of his system. It can easily be given up and his arguments translated - reasonably faithfully - into a natural deduction system. To get numbers as objects of ground type Frege had to translate the natural higher order definition into a ground type using a comprehension axiom. He knew that this was problematic and admitted that this was a point that he was not able to justify. If we give up this, then Freges argument can be followed to show that 5+7=12 is an analytic statement. It follows by formal reasoning alone.
By Goedels incompleteness theorem it follows that for any consistent formal system S (of arithmetic) there are true universal statements F which are true but cannot be proved in S. One can reasonably call such a statement for synthetic a priori. It must be justified by other means than just inserting the definitions. We may need to analyze well foundedness properties of certain orderings. This is in line with Freges argument that true geometrical statements were synthetic a priori.
Herman Ruge Jervell
Philadelphia
December 1993