Degradation of Logical Form
Axiomathes, Vol 8, nos. 1-3, 1997, pp. 31-52
Wahrheiten zu entdecken, ist Aufgabe aller Wissenschaften: der Logik kommt es zu, die Gesetze des Wahrseins zu erkennen.”
“Dierein-logische Gesetze....aber sind rein theoretische Wahrheiten idealer Art, rein in ihrem Bedeutungsgehalt wurzelnd und nie über ihn hinausgehend. Sie können also durch keine wirkliche oder fictive Aenderund in der Welt des matter of fact berührt werden.”
The title is meant to emphasize the immense loss of status I take logic to have undergone in recent decades, and to suggest something about its causes. The loss is most obvious in the context of higher education, where almost no post-secondary institutions now have effectual general requirements in standard formal logic, as that was easily understood thirty or more years ago. Courses in so-called ‘critical thinking’ are, with rare and noble exceptions, only a further illustration of the point, for many of them, if not most, say nothing at all about logical form and formal logic, and proceed as if thought and discourse could be critically understood and appraised in total ignorance of their formal aspects.
But the decline is also seen within the field of philosophy itself, where logic--thought not long ago to be the very heart of philosophy--has become an area of arcane specialization, and most students and faculty have as little to do with that as possible. “Logic” as a special field of study is regarded as something which they can omit without harm to their work in philosophy because it has no essential connection with anything they must be responsible to in their practice.
The now-arcane specialization is perhaps experiencing the effects of decades of scorn poured upon the despised ‘traditional’ or Aristotelian logic--mistakenly and misleadingly branded as “non-symbolic” in contrast to “symbolic” logic--and of the idea fostered within “symbolic” logic, as it came to be practiced, that there are many logics, perhaps an infinitely large number of them, which are only language systems one may pick and choose from, depending upon what kind of result one is interested in.
This idea contains an important element of truth. But, once loose in the environment--even in the philosophical environment, not to mention general intellectual culture--it easily fell prey (like Einsteinian ‘relativity’) to powerful motifs already there, such as inclinations to regard reasoning and its laws as culture bound: gender or race specific, or at any rate as only a historically conditioned phenomena.
Logic the specialization can no doubt find solace in its associations with mathematics, computer science and engineering. But on the other hand life and thought are left without a recognized body of logical principles to which everyone is responsible, and philosophy to the passing array of momentarily enticing or provoking issues and personalities.
I do not wish to blame quite all of the troubles of the end of the 20th Century upon the degradation of logic, but it might improve many of our prospects if at least our leaders in significant positions, from parents to presidents, had some cultivated sense of logical relevance as a general feature of thought and discourse, and possibly of specific logical relations such as implication and contradiction. For they need to know what is and is not the case.
Plato thought that leaders should be trained in geometry because in its demonstrations“it forces the soul to contemplate being,” and therefore “would serve as a winch to hoist the soul to truth.“3 He may have exaggerated a bit. But how many leaders today, in any field, could recognize a demonstration, in Plato’s sense, or are even aware that there are such things--much less that they are in some important way responsible for them? How many current leaders even in the field of education understand the centrality of the formal aspects of thought and the critical appreciation thereof to pedagogy?
Information (and misinformation) comes in units, e.g. that 8 is greater than 5 or that Sue’s dress is red. We shall call these units of information propositions.4 Frege called them simply “thoughts.” Each proposition relates to some other propositions in such a way that its truth values (true or false) necessitate one or the other truth value in those other propositions. We shall here speak of such relations between propositions as logical relations.
Chief among logical relations are entailment or implication, where if P entails Q and is true, then Q must be true, and if P and Q are contradictions they must have opposite truth values.
Whatever logical relations may hold between propositions do so in virtue of certain properties or characteristics of the propositions concerned. (This is true of any relations whatever, between any entities whatever.) Whether or not P and Q have logical relation R depends, accordingly, on the kinds of propositions they are. And we might then decide to say that those characteristics--whatever they may turn our to be--in virtue of which a proposition has the logical relations that it does in fact have make up the “LOGICAL FORM” of that proposition. This would give some initial clarity to the common assertion that some propositions (and therefore the associated sentences) are true or false in virtue of their form, and that arguments when valid (or invalid) are valid (or invalid) in virtue of their form.
But what, exactly, is to be made of this “logical form”? Which properties, precisely, constitute logical form, and, especially, what are those properties properties of. In short, what is logical form? This has long been and remains a highly contested matter. But the fate of logic as a field of knowledge rests upon whether and how we respond to it. It is divergence in this area that makes it possible for logic to be degraded from the dignity proper to it in the domain of human knowledge and life.
In approaching the issue of the nature of logical form we note that there is something which we might call “pre-logical insight” into simple cases of logical relations. That is, it is possible for straightforward thinking to discover the presence or absence of such relations--to know, for example, that if a certain proposition is true a certain other one must be true (or false)--without knowing what logical relations are in general, and without knowing anything about logical form as such.
This type of insight seems to have been what was functioning in the Platonic dialectic, for example, where there was manifested a clear awareness that if some bits of “information” are true, others must be false--but where that awareness was not based upon any general understanding or theory of the types of propositions involved.5 The insight into the necessitation of truth (or into contradiction) was due only to reflecting on the particular matters at issue. The appreciation of the presence of logical relations seems, in some cases, not to depend upon an appreciation of logical form in general.
If, for example, we know that all swans are living beings, a bit of focussed reflection upon what it is that we know in that case will suffice to assure us that no thing that is not living is a swan. Similarly, if we know that no swans are geese, it will not be hard to discover that no geese can then be swans, just by thinking about what is at issue.
In actual fact the reflecting that goes on in these cases may turn in one or more directions, not always clearly distinguished. One might focus upon relationships between qualities of “swanness” and “being alive,” for example, or upon the classes of swans, living things, geese, etc. and their relationships, or even upon circle diagrams representing classes. But one can also focus upon one’s own thinking--upon what is thought, upon the information content of one’s thought--in thinking that all swans are living beings etc. It is possible to realize by reflective analysis that the truth of the thought that all swans are living things requires the truth of the thought that no thing that is not a living being is a swan. That is, this can be recognized as so merely by analysis of what ones thinking is about and of how the subject matter is represented in the thinking.
Now for the understanding of logic it is never too early to point out that swanness (the property), classes of swans and units of information about swans--not to mention circle diagrams, inferences, statements about swans and utterances of statements about swans--are all quite different from each other, all having properties that the others do not have. Failure to keep them all distinct and then properly to interrelate them has had devastating consequences for logic as a field of knowledge, and indirectly upon all fields of knowledge dependent upon it.5a
Logical Form and Generalization
Pre-logical apprehension of logical relations is valuable in its own right, and plays an indispensable role in the development of logic as a field of knowledge. But it is extremely limited in scope. It presents us with nothing remotely resembling a discipline, a field of systematic research, or a system of logical knowledge. Development toward a systematic discipline awaits a certain generalization that eventually expresses itself in talk of logical form. The crucial step is generally thought to have been made by Aristotle. Thus, as Bochenski says, “Aristotle was the first formal logician.“6 Kneale and Kneale in substance agree.7 Whether or not he was actually the first, certainly he is the first on record, and he did take the crucial step, developing it very powerfully. This is marked above all by his introduction of variables, which seems to have occurred somewhere between his Topics and the two Analytics.
What the variables do is to enable us to conceptualize and express generalizations about kinds of propositions and arguments: specifically, generalizations which involve no reference to determinate subject matters that the propositions may or may not be about. The general contrast of matter with form then makes it easy, if somewhat misleading, to speak of the logical form of propositions or units of information--as opposed, one might think, to their “logical matters.” On this interpretation there is really little mystery about logical form. It consists simply of certain features of units of information, namely, those that determine how truth values can or must be distributed over those units and certain other units without regard to which specific objects they deal with.
To illustrate what this means, we return to our thought of (or proposition about) all swans being living things, and the corresponding thought that no thing not living is a swan. Further reflection will show us that the first of these propositions is of a certain type: the type, namely, where a specific property is ascribed to all things of a certain kind. Aristotle’s formulation of this is: B belongs to all A, and the standard form long used in English is All A is B. Now it is possible to see by reflective analysis--i.e. by thinking about the matter--that if any proposition of that type is true, the corresponding proposition, which indicates that no thing which lacks that property is of that kind, is also true. I. e., No non-B is A. It is possible also to realize that it makes no difference what particular propositions of these types are about, so far as the logical relation in question is concerned. So long as a proposition is of the general type indicated, the corresponding proposition must be true (or false, as the case may be), given only that it is of the general type of proposition it is and is about the same things--geese, swans, etc.--whatever they may be.
“Formal” logic would, thus, provide a general analysis and understanding of the logical relations between propositions solely on the basis of their general types as specified without reference to subject matter. Every proposition has some logical relations to other propositions, though there may be reasonable disagreements about what they are in given cases. Indeed, there are ways of classifying propositions that have no bearing upon their logical relations in the sense indicated above.8 But it remains true that some ways of classifying propositions have a direct and conclusive bearing upon logical relations they may have.
It is well known that generalized descriptions of propositions which reveal their logical interrelationships are not restricted to non-compound propositions and their components--whether terms, quantifiers, modal operators, etc. And the distinction between pre-logical and logical reflection and insight applies to “propositional” as well as quantificational logic. The story of Stoic logic makes this clear.8a
Pre-logical insight into the presence of logical relations no doubt occurs with specific thought or information complexes corresponding to familiar rules of propositional logic such as Modus Ponens, Modus Tollens, Disjunctive Syllogism and various forms of the Dilemma. Such insight may even extend to strategies such as the Reductio, and to some of the simpler replacement rules such as commutation. How far, precisely, it extends may vary from individual to individual. But it surely does reliably cover a rather extensive area of relationships between unanalyzed atomic propositions and their combinations.
And the move to abstractive reflection upon the specific propositions involved also yields here, as it did with ‘term’ logic, generalizations about logical relationship between propositions of certain types, regardless of what the propositions are about. As Tarski indicated, all of the familiar formulae of propositional logic are to be read as generalizations about propositions in certain standardized relations or contexts.9
In the case of Modus Ponens, for example: for any propositions p and q, if the conditional proposition, (pq) is true, and the corresponding p proposition is true, the corresponding q proposition must be true. Or consider one of the De Morgan formulae: for any propositions p and q, if the denial of a conjunction of p and q is true, then the disjunction of the denial of p with the denial of q is true, and conversely.
The general descriptions of propositions basic to quantificational and propositional logic arguably do not exhaust the ones relevant to the determination of logical relations between propositions. Those that single out modal and epistemic operators are widely accepted as fundamental to the development of a more adequate logical theory, and others with even less claim to deal with the strictly “formal” aspects of propositions have been advocated, e.g. the temporal. And the issue of how, in general, to draw the line between the features of propositions that are “formal” and those that are not remains unsettled.
The General Concept of Logical Form
These observations seem to me to capture the sense of “logical form” that has been fundamental to the development of logic as a science, a field of knowledge, and the sense in which logical relationships between propositions, and formal validity in arguments, are matters of the logical form of propositions involved. Many questions and further developments remain, some of which attach to the very idea of logical form and call into question what I have just now said about it. However it seems to me reasonable to think that the outstanding questions and developments should not cast serious doubt upon the general account of logical relations and logical form stated, nor upon the illustrative cases mentioned. Moreover, I take it that the existence and knowledge of logical relations and logical form in the sense explained is something of immense significance for knowledge and human life. Human beings are capable of extensive and intricate practices without logical understanding. But is it too much to say that mastery of a field of knowledge is impossible without some explicit appreciation of logical form and logical relations? And is it possible to have a rational outlook on reality as a whole without such an appreciation?
On the other hand, an accurate understanding of logical form and logical relations is dependent upon how one resolves some of the deepest problems in ontology and the philosophy of mind and language. This is why the attempt to develop logic as a field of knowledge independently of an explicit ontology--specifically, one that provides an ontological placement of propositions (“units of information”) and of what the propositions are ‘about’--will ultimately fail. I think that the history of logic in the 20th century confirms this.
Aristotle makes a statement at the very opening of his book On Interpretation that has great importance for the project of understanding logical form. He comments that spoken and written language is “not the same for all men,” but that what spoken and written words are “in the first place signs of--affections of the soul--are the same for all; and what these affections are likenesses of--actual things--are also the same.“10 What follows in the text makes it clear that the “affections” in the soul in question are to be understood as thoughts, and that it is these which, given appropriate formation, are true or false. They are what we have called “units of information” or “propositions” above; and they, like the objective states of affairs which they represent, are according to Aristotle “the same for all.”
Now this need not mean that the experiences in which the identical units of information are grasped and the corresponding states of affairs apprehended must be identical--which of course they cannot be--or even totally similar. They may and often do, in fact, differ in very significant ways. All that is required, if Aristotle is to be right, is that the information content be capable of being present--as the very same unit of information, with all its essential properties--in many minds, and that presence in different minds is not by itself grounds for saying that it is not the same proposition. The information content is a genuine universal.
There may, then, be specific grounds for doubting or denying the identity of the ‘thoughts’ or propositions involved in specific cases of thinking, but those grounds cannot consist merely in the fact that two different minds are involved, or that two different races, cultures, genders, etc. are involved. That is the point of Aristotle’s “logical universalism,” as we might call it. The essential features of these thoughts, including their logical structure, will also be universals, as will the logical relations founded upon them. They will be the same for all--always.
But all of this is by no means agreed upon. Referring to Book I, chapter 1 of the Prior Analytics, Bochenski comments that “What emerges from that text is the complete neutrality of the technical expressions ‘terms’, ‘premiss’, ‘syllogism’, relative to any philosophical interpretation. For the premiss consists of terms, the syllogism of premisses, and premisses are logio, which can equally well mean utterances or thoughts or objective contents, so that the way is open to a formalist, psychological or objectivist interpretation. All these interpretations are permissible in regard to Aristotelian logic; the purely logical system excludes none of them. Guided by his original intuition the founder of formal logic so chose his terminology as to rise above the clash of interpretations to the level of pure logic.“11 Bochenski was mainly concerned to emphasize that Aristotle was not unaware of the importance of language, which is surely right. And that was an important point to make in the period when Bochenski wrote his History. It is also true that the term “logoi” can be understood in various ways. But I think he goes too far in saying that “the practice of Aristotelian logic was undoubtedly to regard meaningful words as its subject matter.” If faced with the issues as they stand in the 20th Century, I cannot imagine Aristotle would make such a choice. And as Bochenski himself acknowledges, for Aristotle demonstration “is addressed, not to outward speech, but to that within the soul.“12 If meaningful words are the subject matter of logic, the logical universalism of Aristotle at least dictates that the words or language will, in general, have no essential involvement with that subject matter. That the language would make a logical difference was surely undreamed of by Aristotle. He never gives the least hint that Egyptians, Persians and Greeks were not capable of the same logoi or that a syllogism for one might fail to be a syllogism for the other. (What we call an ‘invalid’ syllogism was simply no syllogism at all for him.)
And a similar point holds in the renewal of logic in Frege and Russell many centuries later. They too were logical universalists. Their concerns for language was not based upon the idea that it was the subject matter of logic, but upon the awareness of the power of language to obscure or conceal the true character of propositions--or alternatively to assist and guide thought into conformity with the true formal characteristics of propositions and their genuine logical interrelations. The concern which they and others of the late 19th Century had for laying bare the logical forms of propositions and finding a language adequate to express them led to the renewal of a discipline that, one hundred years earlier, Immanuel Kant had, famously, judged to be “completed and perfect.“13
We return to Bochenski’s suggestion that logic as a cognitive discipline is neutral with reference to whether its subject matter is language, thoughts or objective contents. Surely this cannot be true, and here is why. The factors listed simply are not the same things, however closely intertwined they may be, and no discipline can be neutral as to what, basically, it is about, its subject matter. Such neutrality is finally impossible. What one can say, perhaps, is that a significant degree of logical insight and systematization can be attained without settling the issue of what logical relationships between propositions precisely are and essentially involve. But this very clarification and understanding of that very insight will require dispelling the initial ‘neutrality’. The laws of logic cannot be equally derived from and applied to thought, discourse and objective entities, for these are not only non-identical but vastly different in character from one another.
In his efforts to arrive at a correct understanding of logic as a field of knowledge, Edmund Husserl found himself forced to distinguish several strata within our overall epistemic engagement with entities. To each stratum there is a unique pattern or order of connection between factors of that stratum, and corresponding laws that govern the order:
1. The stratum of concrete psychical events in the minds of particular investigators, involving “a psychological pattern of connection among the presentations, judgements, insights, surmises, questions etc., in which research is carried out.“14
2. The stratum of objective reality, involving “a pattern of connection among the matters investigated and theoretically known in the science, which constitute its sphere a domain. The pattern of connection within investigation and knowing is plainly quite different from that within what is investigated and known.“15
3. The stratum of concepts, propositions and theory, involving “the logical pattern of connection, i.e. the specific pattern of connection of the theoretical Ideas in which the unity of the truths of a scientific discipline, and those, in particular, of a scientific theory or proof or inference are constituted.... The logical pattern of connection is the ideal form for the sake of which we speak in specie of the same truth, the same syllogism or proof, the same theory and rational discipline, by whomsoever these ‘same things’ may be thought.“16
4. The stratum (at least in some cases) of specialized symbolisms or algorithms, involving standardized patterns of spatial shapes and orders that can be operated by attending to the standardized patterns and the conventional rules governing them, hence without attention to a corresponding subject matter or even to the logical relationships between propositions bearing upon it.17 Here arises the contrast between the uninterpreted or (in quite a new sense) “formalized” elements of the algorithm and their meanings when interpreted. This contrast between the sign as a mere physical entity and its meaning when interpreted came to replace in many quarters the older contrast between form and matter in logical analysis.
5. The stratum of “natural language,” involving the patterns of “meaningful words” of languages such as German, French and English. Husserl very early on saw clearly that language is not an algorithm and an algorithm is not a language.18
With reference to #4 and #5 one has to distinguish, once more, between the concrete entities and events of their actual employment and the general structures or essences which are present in such an employment and make it possible. However, while symbolic systems or algorithms were thought by Husserl to have great and essential significance for the development of theoretical science as well as general culture, neither they nor ‘natural’ languages were regarded by him as essential to acts of consciousness or to knowledge as such.
To give a general description of Husserl’s line of argument, developed over a long period of years and in many texts, he held that as long as 1-5 are kept clearly distinct--which they must be--analyses of what the laws of “pure logic” are about, and the kind of evidence upon which they are based--i.e. of their sense or “content”--show that they deal with pattern #3. It is concepts and propositions that come together through logical relations to constitute proofs and theories.19 The factors in the other patterns do not have the characteristics required to permit relationships between them to be logical relations and hence to make up proofs and theories. The ‘forms’ of those factors are, whatever else, not logical forms, though logical forms may be associated with them in various ways. To see this one need only attend to them, consider them in detail. Then it will be clear that logical relations have characteristics that cannot be derived from those other strata, and that logical laws are not about them.20
By contrast the intentional bearings or meanings which concepts and propositions simply are (stratum #3), and which they therefore impart to concrete acts of thought and discourse by being instanced in them, are the foundations of truth values and of the laws governing them. Propositions are of and about corresponding objectivities, and about them as being arranged in specific manners. (E.g. Swans and living beings once more.) If those objectivities are as presented in the propositions, the propositions in question are true, and otherwise they are false. And if they are as presented by the propositions then they must also be (or perhaps cannot be) as presented in certain other propositions--those that are logically related to the original propositions. Thus logical relations emerge from the structure of intentionality and truth.21
So the outcome of Husserl’s investigations is that logic is a science of objective and totally invariant structures of thought, and, indirectly, of objective and totally invariant structures of the world which thought is of and which knowledge actually grasps. This high view of logic is the one which has now been generally lost. Almost no one now defends it. But the basis for it lies unchanged in the sense of the familiar principles of traditional formal logic: in what they affirm or say.
From this result Husserl proceeds to deal with a number of very important issues in the theory of logic.
First, he explains how logical relations serve as guides and norms of the concrete thought events that make up stratum #1. The logical relations between concepts and propositions are imported into the flow of concrete acts of thinking because concepts and propositions are properties of those acts. Those relations do not govern which of such acts can occur. But insight into those relations and the “laws” which express them enable us to know, within limits, which among various actual acts of thought (“judgments”) must be, can be, or cannot be true (or false), given the truth (or falsity) of others. We can thus reject a particular inference or argument as invalid because it is of a type whose premisses very well could be true and its conclusion false, or see that a particular judgment or assertion (at a particular time and place) must be true because the negation of the proposition it involves is false.22
Second, he places the stratum of logical connections or “theory” into a rigorous correlation with the “categorical” or “formal” structures of the world, of the domains of possible research, knowledge and being. Thus a formal ontology goes hand in hand with a formal logic. His “breakthrough” with reference to the problems about the objectivity of knowledge that haunted his earliest researches had to do with the viewing of the essential correlation between the formal concepts of logic and corresponding structures of objects and objectivities.23 The “breakthrough” did not constitute a full phenomenology of knowledge, but it did lay out the possible forms an object of consciousness merely as such could take, within which all actual being must fall. When conjoined with his analysis of “fulfillment” or verification, a general foundation was laid for answering “in principle” all possible questions about what there is.
There is, Husserl found, an objective interconnection that pervades scientific theorizing. But it can be understood in two ways: “as an interconnection of the things to which our thought experiences (actual or possible) are intentionally directed, or...as an interconnection of truths, in which this unity of things comes to count objectively as being what it is. These two interconnections are given together priori, and are mutually inseparable. Nothing can be without being thus or thus determined, and that it is, and that it is thus and thus determined, is the self-subsistent truth which is the necessary correlate of the self-subsistent being. What holds of single truths, or single states of affairs, plainly also holds of networks of truths or of states of affairs. This self-evident inseparability is not, however, identity.... Truths which hold of truths do not coincide with truths that hold of the things posited in such truths.“24
This approach to ontology is not blind, in the manner of much recent philosophy, which selects a “theory” on the basis of various considerations--empirical, pragmatic and aesthetic--and then looks to see what kinds of objects fall within the range of the bound variables that occur in it. Rather, Husserl’s view is that, just as the essential logical forms can be “themselves” fully present to insight, so can the essential formations of objects, along with the correlation between them and the logical forms. We can see, for example, what it is to be such and such, to be identical, to be part of a whole, to stand in a relation, etc. all of which are structures common to every possible entity.
Third, Husserl explains how it is that algorithms (“formal” languages) enable us to know and deal with domains of entities, e.g. numbers and physical objects, and to do so far beyond the scope of any possible intuition or experience, or even explicit conceptualization, by human beings. The analysis of knowledge that is based on the blind use of symbols--as in mathematics for millennia, and then in “formalized” theories of inference generally--was a concern that drove his earliest researches in the philosophy of mathematics, and somewhat later his entire program of a phenomenology of knowledge.25
The categorical forms of entities run parallel to the forms of propositions, proofs and theories, allowing us to determine by logical inference what is the case with entities that we have not examined, and possibly cannot actually examine. Similarly, algorithms can be set up to parallel the logical relationships of concepts and propositions. This parallelism permits us to disregard the interpretations or meanings of symbols, i.e. the corresponding concepts, propositions and logical relations, and, instead of inferring, simply to calculate. Without this ability to calculate, most of what is accepted as knowledge in the developed sciences simply would be impossible for us. The general superiority of scientific methods of calculation over those of inference was something Husserl understood from his earliest investigations.26 How they work, precisely, and what justifies them, was another matter, and one which he paid much subsequent attention to. He found its fundamental clarification in the fact that the operations on symbols were set up to run parallel to the logical relations among propositions.
It should be emphasized that Husserl’s investigations concerning the “formalized languages” were epistemological. The project that culminated in the famous incompleteness results of Gödel and Church was quite different from his, though not wholly unrelated. Given a formalized theory that is “complete,” one would still have to answer the question of how the algorithm, the blind manipulation of symbols, enables us to know what is the case with entities we are not even thinking about as we work the algorithm. That was Husserl’s question, and he thought he had answered it decisively and completely by reference to the two-fold parallelism of objective categories to logical forms and relations and of the latter to rules of symbolic (‘formal’) procedure.
The Disappearance of Stratum #3: Propositions
Around the beginning of the 20th Century there was a fairly strong consensus within logical theory that the view of logic as formulating objective laws of the logical relations of propositions was the correct view. Husserl, however, was not the central figure in the consensus. His treatment of Psychologism in the “Prolegomena” was known and respected, though generally misunderstood.27 And his painstaking refutation of “formalism” in logical theory was hardly known at all.27a
The center of the consensus about logic was held by Bertrand Russell, Frege, and the early Wittgenstein. For all their brilliance and creativity, however, they were unable to distinguish and successfully interrelate the five distinct strata of our cognitive involvement with entities in knowledge. Russell sometimes just runs them together. In the chapter on “Propositional Functions” in his Introduction to Mathematical Philosophy, for example, a propositional function is treated as a linguistic expression, a concept, and a quality.28 Frege does, I think, conflate concept and quality, but in general he simply ignores the problem of relating the mind (person) to the mental (or linguistic) act or content, of relating the act to the proposition or concept, and of relating the proposition or concept to the object (referent).29 The early Wittgenstein certainly worries a great deal about how to relate all of these. But he comes up with no positive result, to say the least, and is finally defeated by the problem of how propositions and languages relate to “the world.” One upshot of this is a radical reinterpretation, by him, of what logic, the laws of logic and logical relations are.
All of this is not terribly surprising. The question of the precise subject matter of logical knowledge touches, as we have noted, upon the deepest of epistemological and ontological assumptions. These involve distinctions and relations that are inherently hard to sort out and keep straight. Frege and Russell never were able to develop a thorough and intelligible account of how algorithms or ideal ‘languages’ served the interests of logic as a science of the logical relations of propositions in general. They stand among the few logicians of great genius. But for all their insights and accomplishments they left the philosophy of logic in a highly unstable position. This was precisely because they took propositions and their truth value (or logical) relationships to be the basic subject matter of the science of logic, but were unable to give any satisfactory account of: (1) propositions, (2) their relations to the mind or person thinking them, (3) to the world which the proposition are about, (4) to natural language, or (5) to the logical calculi, algorithms or ‘ideal languages’ which are absolutely necessary for the full development of a science of logic.
The Two Historical Options
(depth grammar), or social (Wittgenstein), etc. On the former alternative the physical properties of appropriately “formalized” written sentences determine the logical relationships in which the sentences stand.31 Concerns about the bearings of logic on mind and world were sacrificed to the objective of getting rid of ‘strange’ entities, “Platonistic” ones, and accompanying strange ways of knowing--“strange,” at least, to the overwhelmingly empirical and naturalistic inclinations of the 20th Century. Importantly, the formalists did, by and large, retain the priority of logical form over logical relationships and, further, over inference--at least for a while. But for the linguisticists it was the possibilities and impossibilities of behavior (including inference) that are primary, and any notion of logical form must be derived from them. However, the functions, name relations, variables, “uses” and “rules” invoked by them remained, in spite of all protestations, about as strange as any Fregian proposition. What they are and how they do what they allegedly do--indeed, what language is and how it does what it does--is not a subject of agreement or possibly of significant understanding. The great ‘advantage’ was only that “Platonism” was ruled out. On either of the two options mentioned, the status of logic as a science of objective and totally invariant structures of thought--and, indirectly, of the world which thought grasps--was, simply, lost.
Rudolf Carnap and others attempted to center the basic logical concepts (logical form, logical relations, logical laws) on the physical (spatial) structure and order of written ‘sentences’ in a ‘formalized’ language. The important contrast now was not between the general character of a proposition (unit of information) and what the proposition was specifically about. It was between the sentence as a physical entity and its meaning. Indeed, the meaning is just the unit of information, and this was now totally set aside by the radically different sense of “form” that was introduced.32“Formal rules” were now explicitly understood as rules which do not refer to meaning. They refer only to the spatial shapes and orders of the symbols and rules based thereon.33
It was not (yet) denied that sentences have meanings, only that they are irrelevant to logic. One might think such an interpretation of logic was bound to fail, and it did. In fact, the ‘formal’ languages were never set up without regard to meaning. The “uninterpreted” formalisms were never truly uninterpreted. They were set up with an eye to logical forms and relations understood in the traditional sense. The distinction between the elements of the language, the formation rules and the transformation rules clearly mirrored the distinction between the logic of concept, judgment and inference that had long structured treatments of logic. Moreover, areas such as modality could not be put into syntactical terms understood ‘formalistically’, and the role of logic in actual thinking and acting was totally untouched, as the second Wittgenstein, Ryle and others relentlessly pointed out.33a
Over time, the ‘formalistic’ tendency was transformed in such as way that the ‘formalism’ was understood to deal with sets and operations on sets. Arnold Koslow’s brilliant treatment in his A Structuralist Theory of Logic34 is perhaps the best current representation of how the Post-Russellian ‘formalist’ tendency has developed. A ‘formal’ language is still developed, but it is used to define and express operations on sets and their elements, among which can be but need not be linguistic items. An “implication structure” is defined, and implication (treated as inference) is interpreted in terms of this structure, independently of particular logical operations and, specifically, of logical syntax as previously understood. Logical operations and modality are then reinterpreted or defined in terms of the “topic neutral” notion of an “implication structure.”
No one can fault the impressive elegance of such treatments, and no doubt this particular treatment by Koslow responds well to certain standing problems in attempts to underestand logical inference relying on syntax. That is all quite valuable. But it is hard to shake the impression that what we have here is simply another shift in subject matter, similar to the shift from “meaning” to spatial shape and order in the older formalists. What, exactly, does an “implication structure” as an abstract operation on a set have to do with logical relations and logical form as traditionally understood. One could reply that that is precisely the point. Logical relations and logical form were not understood, traditionally, and that the development of and “implication structure” in terms of the relations of sets, subsets and elements of sets shows us what they really are. But this seems a most unlikely story, and a philosophical account--not just an elegant formal development--is needed to show the relationship of the “implication structure” to validity of argument and logical relationship as something we have always dealt with, and continue to deal with, without a formal syntax or treatment in terms of sets and their abstract structures. Absence of such an account is what makes it seem to many--perhaps wrongly--that logic, the contemporary arcane discipline, is both arbitrary and irrelevant with reference to the serious work of thinking and living faced by both philosophers and ordinary citizens.
The second option that opens up after the collapse of the Frege/Husserl/Russell interpretation of logical forms and logical relations is to regard them as a function of behavior, generously understood. They are, specifically, to be taken as features of discourse, but discourse in the midst of life. For some the discourse is “ordinary” discourse and the life “ordinary.” For others the discourse and life that matters is that of the scientist. The debate over which discourse and life is most significant for philosophical understanding has been and continues to be one of the enduring currents of philosophy in the 20th century.
The problem with logical principles interpreted behaviorally is not so much the impression of arbitrariness and irrelevance, as it is the sense that they are only socially and historically determined, and hence could be--and, some hold, actually are--grounded on nothing but force or accident. It follows on this view that the laws of logic will change when social and historical conditions which support them change. Modus ponens or the law of non-contradiction, for example, might well prove to be false. And will that mean they are false now, and just not known to be? Or that they are true now and will be false then? Or--what? It is now commonly thought that massive shifts of conceptualization and linguistic practice are possible which would transform all structures of life and thinking, not excepting the laws of logic. We are also used to the idea that there are, even now, many incommensurable “language games,” with different and incommensurable “logics” or acceptable patterns of reasoning, and no one master logic to which they all must conform. In logic too, it seems, there is no “God’s eye view.”
Hilary Putnam has produced a series of papers over the years in which he has examined the question of whether or not the laws of logic can be “revised.35.” What is continuous throughout his discussions of this question is the assumption that the real issues is whether laws of logic can or should be “accepted,”“given up,” or revised. It is this assumption that places him with the, broadly, behavioral interpretations of logical laws--along with Quine and Wittgenstein, for example. He very explicitly and repeatedly rejects the “metaphysical baggage” that comes with “transcendental” interpretations of logic, as in Kant and Frege (certainly in Husserl and Russell as well).
So he considers the typical logician’s statement: “For all statements p, ‘-(p.-p)’ is true.“36 He comments on his reluctance to “give up” this claim, which he then rephrases as “I cannot imagine finding out that it is false.” (p. 250) But this leaves it possible that it might, some day, be found false. As at one time we did not know how to disconfirm Euclidian geometry, or even knew if anything could disconfirm it--though we do now--“similarly...we do not today know how to falsify or disconfirm” (p. 251) the logician’s sentence above. We do not know now how to show that some statement of the form p.-p is true. But someday we might, as with the parallel axiom in Euclidian geometry.
In short, the “unrevisability” of the laws of logic is, for Putnam, relative to the body of knowledge we now have. Given the body of knowledge we now have we cannot describe circumstances under which the belief that, for all p, ‘-(p .-p)’ is true would be falsified. “In such a case we cannot, I grant, say that B is ‘unrevisable’, but neither can we intelligibly say B can be ‘revised’.” (pp. 253-254)
Now taking the limits of imagination as the indication of the peculiar status of the laws of logic is not “gross psychologism” (p. 250) according to Putnam, because “it is not a mere question of what some people can imagine or not imagine; it is a question of what, given a conceptual scheme, one knows how to falsify or at least disconfirm. Prior to Lobachevski, Riemann, and others, no one knew how to disconfirm Euclidean geometry, or even knew if anything could disconfirm it. Similarly, I would argue, we do not today know how to falsify or disconfirm..., and we do not know if anything could (or would) disconfirm .” (p. 251)
Further, “logical truths do not have negations that we presently understand. It is not...that we can say that the theories of classical logic are ‘unrevisable’; it is that the question ‘Are they revisable?’ is one which we have not yet succeeded in giving a sense.... Saying that logic or arithmetic may be ‘revised’ does not have a sense, and will never have a sense, unless some concrete piece of theory building and applying gives it sense.” (p. 256)
The laws of logic are, then, necessary only in relation to a conceptual scheme. Given our conceptual scheme, the question “Can the laws of logic be revised?” or “Could there be other laws of logic than the now familar ones?” is said to be unintelligible.
But the obvious implication of all that Putnam says, the parallel with Euclidian geometry and so on, makes it clear that we have no reason to suppose that a shift that allows sentences of the form -(p .-p) to be true (‘acceptable’) could not occur. Similar shifts, apparently, have occurred. The laws of logic, and the corresponding logical forms and relations, are clearly, on Putnam’s view relative to a historical context which has changed dramatically and can still do so. This is, actually, a part of the “God’s eye view” which, though repeatedly disowned as such (p. 258), implicitly governs his entire philosophical outlook. Psychologism that is not “gross” remains psychologism, though with a difference. In “non-gross” psychologism the constraints and forces with the individual mind are simply replaced by constraints and forces that are socio-historical. That is the genius of the later Wittgenstein and all who go along with him--Putnam prominent among them.
It remains true, however, that not even Quine or Putnam would ever cite a behavioral fact (broadly interpreted) in a proof of a theorem of logic. Nor, when they are doing logic and not philosophizing about it, would they take a logical theorem to be about behavior. Psychologism, gross or otherwise, does not enter the sense and evidence of logician’s truths when logic is being seriously treated as a field of knowledge. It seems to me that, just as with the allegedly “uninterpreted formal languages” of the formalists, the presence of logical form and logical relations as traditionally conceived continually reasserts itself when one does logic rather than engages in general reflections about it, when epistemological and ontological assumptions take control.37
I have tried in the foregoing remarks to tell a story of how logic as a field of knowledge has come to be regarded as it currently is: generally, as irrelevant, arbitrary in one or several ways, and possibly even politically pernicious. But perhaps someone will say that, after all, some version of what I have all too crudely called “Formalism” or “Linguisticism” is just true. I must acknowledge, in response, that I have not dealt with the many serious considerations that lie back of these tendencies. Nevertheless, I cannot help but think that we do have, and have long had, a large body of knowledge (though not a complete one) about the logical relations between various well-known types of propositions, and that this knowledge is neither of nor dependent in its sense or evidence upon algorithms or behavior.
To show Formalism or Linguisticism to be wrong would be an incredibly difficult task. But I hope I have made it clear why, as practiced, they have had the effect of diminishing the perceived significance, for life and philosophy, of logical form and logical relations. I also hope to have made it clear that, in my opinion, the true significance of logical form and relations is very great.