Problems for a Platonic Idea of Logic

Derek Li, HMC ’23

What is logic? If Plato, in observing instances of virtuous actions, posited an encompassing Form of the Good, then shall we not, by discerning in every piece of reasoning the course of logic, acknowledge a Form of Logic on which is predicated every idea? No one has ever reconciled a contradiction or understood an illogical thought. A concept, to state that it has meaning, that it is conceivable, implies it is logical. Logic, by which I understand deductive logic,¹ is but another name for concept; it is a relation of concepts immanent in the concepts themselves.
Such is the Platonic conception of logic, as I’ll refer to it. I do not intend it as an interpretation of Plato’s view of logic but wish to employ the name to characterize the universal applicability of logic and an affirmation of concept as a primary epistemic entity. This view, of logic as the necessary relation of concepts, is itself a simple concept. That poor bachelor invoked by logic instructors expects us, by the nature his name indicates, to understand that he has no wife. Say you introduce to me your father and your daughter, I cannot be helped from the conclusion that this man is the grandfather of this girl. And, of course, mathematicians undertook great inquiries in logic: just as the ancients admired mathematics as the model of philosophical knowledge, the subject of modern logic was motivated by an examination of mathematical reasoning.² One preliminary question is regarding the ambit of logic. Does deduction pertain solely to mathematical concepts³ or more fundamentally to any idea which has sufficient clarity to be called a concept? I will treat this question more fully in the second section of the essay, though I expect the foregoing examples of the bachelor and the grandparent to express my position that the inferences drawn in these cases, accepting such premises as a person cannot simultaneously be present in multiple places, is no less deductive than the mathematical statement that an odd number cannot have four as a divisor. All that we require of a concept is that it be of sufficient clarity so that deduction may operate upon it. Let this be a necessary condition we assume for a concept—amenable to deductive reasoning.

I. Priority of Concept

Yet herein a problem reveals itself: what is a concept? Can a definition be provided which encapsulates its essence? What common content can we gather under the summary term of a concept? It appears reasonable to consider any notion with sufficient clarity as a concept. But, as a consequence, nothing substantive is capable of being ascribed to the word concept itself, rendering it as vacuous as the term being.⁴ Nonetheless, if we are capable of rendering a definition that subsumes the nature of all concepts, we may confront the foundation of logic and all intellectual inquiry. Well, what would such a definition look like? Likely, as all of our definitions, expressed as a sentence, which involves concepts. If this definitive sentence doesn’t invoke the definiendum itself, it must construct it via other relevant concepts. This then is a regress in that to articulate the essence of all concepts, we must presume other concepts to be understood. Let us refer to this as the problem of definition. We cannot define ‘concept’ without assuming some set of concepts. The quality of mathematics that constitutes it as a deductive subject is its specification of definitions, from which all its knowledge derives. Yet what are the foundations of definitions? A triangle can be defined as the figure closed by three mutually nonparallel lines in a Euclidean plane. What then is a Euclidean plane? Meno, in objecting to Socrates’s definition of shape as “that which alone follows color” by asking, “if someone were to say that he did not know what color is,”⁵ may be voicing a kindred sentiment. Let us presently waive the difficulties of communicating the concept implied by a definition to others, a problem long presented by the Greek sophist Gorgias. It appears that the concept of a Euclidean plane⁶ is so lucid that it requires no further clarification. Is this then a sure foundation on which we would rest our understanding? The apparent necessity of entertaining certain unspecified concepts is the problem of definition, and it invites us toward the question on the foundation of our conception.
An attempt to address this fundamental question considers logic as a natural phenomenon. When my mind moves through the lines of a proof, proceeding to deduce consequences from definitions, are the relevant ideas not necessarily conceived in my mind? When I think of a Euclidean plane, does not that present thought realize and thereby contain the concept? To negate this position is to concede that there is no validity in reasoning, and that our thinking bears no relation to the concepts supposedly grasped by thought. The philosophical identification of ourselves as rational beings, then, has no meaning. Consequently, the empiricist is disposed to observe that whatever meaning is to the idea of a triangle, it is what occurs in my mind when I do mentally conceive of a triangle and perform deductive operations thereon. So, whatever a triangle is, it must be contained in the thought of it when I conceive it in my mind.
This is a psychological⁷ position, reducing logic to the natural phenomenon of thinking, which I’ll refer to as the empirical reduction of logic. To be constructive, it of course needs to rid itself of that “whatever that occurs in the mind” and render the phrase meaningful by specifying what is supposedly captured in that phenomenon of thinking. Then it may proceed to explain with sophisticated scientific theory capable of constructing the conceptas a mental phenomenon. This reduction of a concept to physics⁸ broaches two domains of philosophy—ontology, of existence or reality in our world, and logic, of concepts, which are considered as independent of physical reality—commonly treated as independent. The question is whether such a psychological position which breaches this distinction is viable. Another consequent problem of this empiricist position is subjectivity. Is my triangle the same as your triangle? Though the mental image you have of a triangle—your subjective idea of a triangle—differs from mine, is it sensical to say that the conceptof triangle itself depends on the particularity of the mind in which it is conceived? In other words, though the ideais subjective, can we proceed to assert subjectivity of the concept.
Nevertheless, the empirical reduction engages a deeper problem in its attempted mode of exposition; to wit, can we articulate a concept without using one? Can we define logic without assuming it? Does not the empirical reduction, which is a physical position, imagine a concept as a phenomenon that occurs within some cognitive apparatus or constitution, namely the mind, and hope to capture it there? But whatever specific concepts we use here to construct the cognitive constitution, to be used in the definition, cannot be explained by the account itself, which assumes such a cognitive constitution. As great a psychologist as you are, when you expound to me your theory of the concept, it perforce involves concepts complex enough to describe the nature of thought. Probably, you will assume concepts such as the laws of nature. And I, a humble philosophical questioner, cannot accept this as an ex nihiloconstruction of logic or of concept, given that the very power and elegance of your theory derives from the clarity of your foundational concepts, which you assume. Put simply, as any ontological account, that is, an explanation of some phenomenon, is construed via concepts, we cannot therein seek an absolute definition of concept. From the problem of definition, there is no non-conceptual way of constructing a concept.
There is a more fundamental way of establishing the epistemic priority of concepts—that it is absolutely necessary to any meaningful reasoning to assume concepts objectively.⁹ By objective I claim not that my concept objectively represents some physical reality. But, in regarding concepts as objective, I only state that it is impossible to conduct reasoning at all without taking concepts by their own meanings. Let us demonstrate the objectivity of concepts in relation to the problem of subjectivity prompted by the empirical reduction: are concepts subjective to the thinker? When considered as thoughts, it seems that concepts exist, strictly speaking, only when they are thought of, and the nature of the concept is that instantiating thought. This suggests that my triangle can be different from yours and from that which I conceived moments ago. Very well. Consider my or your triangle, then. Is there anything in my concept of the triangle which renders it peculiar to my mind? Is it the line? the enclosing? Or is it the Euclidean plane? There is nothing additional to the concept of a triangle. What can be the meaning of ‘subjective’ in this context? To make it meaningful, it is necessary to introduce a concept of cognitive agents, with constitutions having the potential of conceiving different ideas of a triangle. But now subjectivity is in the concept of the variation of cognitive agents, which is merely an additional concept. Would you say that, indeed, this concept of subjectivity is subjective too? But it is impossible to give a meaning to subjectivity without introducing it as an additional concept, the subjectivity of which now cannot be established. This is a regress and the word subjectivityhas no independent meaning. There is nothing subjective in the concepts themselves, but, for the term subjectivity to have meaning, a concept must construct it.¹⁰
This is different from the physical possibility of subjective experience.¹¹ The matter here is the subjectivity of concepts. It is possible for me to conceive of a transcendent state of being which projects onto my subjective perception. This possibility is justified by the concept of such a reality and the limitations in my perception. Yet it is not possible for me to imagine a state of being not described by objective concepts. To illustrate, suppose, to the contrary, you say my concept of a triangle is subjective. How? Because it is conceived within my peculiar mental constitution. But if you intend to use this as a piece of reasoning—that it follows from different cognitive constitutions that the conceptions may be different—you are precisely taking ‘cognitive constitution’ as an objective concept in that you infer, in virtue of the concept itself, differential conception. I cannot imagine how otherwise you can obtain such an inference. To state that it is because anything else but the concept of different cognitive constitutions is nonsensical. The inference cannot be drawn from anything else but from the concept itself, which is the definition of deduction, operating objectively on the concept. There is no such thing as subjective deduction, so concepts cannot be regarded as subjective.
Indeed, when we say anything of a thing, what is the ‘thing’? We cannot say it is my subjective concept of that thing, where the term ‘subjective’ is undefined. Instead, if we go forth to recognize that thing at all, it has to be a concept to be taken with its peculiar meaning, and in this it is objective. Otherwise, we cannot reason at all: a triangle has three sides in virtue of its being a triangle—I cannot analyze this deduction more. Any other explanation which does not suppose an objective concept of a triangle with the property of three sides is capable only of incoherence. So concept, the unit of reasoning, is of necessity objective, so long as we reason. This situation can be compared with that of the sceptic, who, in order to convince us of the subjectivity of our perceptions, posits a transcendent reality not directly accessible to our senses. By this logical possibility, we are drawn to doubt the correspondence of our sensations with reality. Yet whereas a hidden reality is logically possible, to discard the objectivity of concept cannot have any logical warrant, for so it does away with logic and reasoning as well.
Succeeding our analysis above, we may summarize our tenets succinctly. Reasoning is in the strict sense, such as in mathematics, and in the loose sense, such as the opening examples of the bachelor and the boy, deductive. Deduction assumes and operates on concepts based solely on their own meanings. Thus, concepts are necessarily assumed for reasoning, so are irreducible, and they are taken solely by their own meanings, thus are objective. As there is no other mode of logical reasoning other than deduction, these conditions of deduction establish the absolute priority of concept to reasoning.¹² It was Frege who introduced the distinction between logic and psychology and exposed the problems and insufficiency of psychological descriptions employed as definitions of number. What I attempted above are more direct arguments against the subjectivity of the psychological position. Naturally disposed as we are to ascribe thinking to a cognitive phenomenon, which is nothing but an affirmation of fact, we are obliged to acknowledge that never can we hope of extricating ourselves from the supposition of objective concepts, which is the method of our thought. Beyond it I can see naught, and herein, I believe, is the limit of epistemology. Within it, we shall seek ever greater clarity of our concepts in philosophy and ever finer understanding of our cognitive constitution in psychology. Alas, Platonists we must be toward concepts.
Let me conclude this section by addressing the question regarding the existence of concepts, which might have troubled the reader for some time. The difficulty here is similar to philosophical questions concerning the term existence in general: what do we mean by existence? If we were to adopt the phrase, existence of concept, the existence here is clearly different from the existence of a physical object, which exists in space and time and is observable. Is it more akin to the existence of a law of nature, which is not directly observed but requires an epistemic leap to construct? But the purview of such a law is still physics—it purports to exert something of the reality which comes to be perceived. Yet a concept, say three-dimensional geometry, although its origin may be from our senses, once the concept is abstracted to stand on its own, such as the Euclidean space in analytical geometry, it has no relation whatsoever to anything physical. When I state that infinitely many different triangles can be formed within a Euclidean space, I am asserting nothing that is physically happening but only remarking on the meaning of the concept of Euclidean space itself. Indeed, Kant’s observation that the word existence contributes no additional meaning seems the most germane when we say, “a concept exists.” Saying that a triangle exists is merely to restate what a triangle is. We feel there is something profound behind the fact that some concepts are possible while others not, and so we think that, for this, concepts exist in a special sense. But then, we are reminded of the priority of concepts, that it is impossible to seek the origin of concepts beyond the concepts themselves—so the existence of some concepts need always be assumed—and the word existence, when it has meaning, is but another concept.

II. Problem of Form

Is logic a meta-conceptual discipline? Are such axioms of non-contradiction and the excluded middle prior to the mathematical theorems which apply them? It is true that in mathematical, scientific, and any theoretical reasoning such laws of logic obtain, but need this entail that there is a separate, more fundamental realm in which logical truths are established apart from the specific definitions and concepts which constitute the subject? How indeed is the validity of such laws established? These coupled questions let us denote the problem of form.
The problem of form can be aptly represented in the student’s disaffection with the definition of the conditional. The definition of the conditional in propositional logic is as a function of the truth values of the atomic propositions. We may accentuate this feature by writing p ⊃ q in the functional form ⊃ (p,q). To the mathematician, who uses ⊃ to symbolize valid deduction, this definition brings out none of the idea that the antecedent conceptually implies the consequent, such as A is a triangle implies that A is a planar figure. Of course, we have no reservation to the proposition p ⊃ p, or if we prefer a more meaningful statement, (p ⊃ q) ⊃ (¬q ⊃ ¬p). The difficulty, herein, is that this system of logic has no more fundamental concepts than the atomic propositions p,q. So, any logic that is contained in a mathematical proposition which we like to designate p, say A is a triangle, is not represented in the logical system and so cannot be reasoned. Yet this naturally prompts the questions: are such logical laws, e.g., the contraposition (p ⊃ q) ⊃ (¬q ⊃ ¬p) , not also results of deduction; and on what grounds do we regard them as prior and also applicable to mathematics and other disciplines? Propositional logic, constructed in the truth-functional approach as for our example of the conditional above, is simply a mathematical system of functions on the binary set {T, F}. It is due to the preciseness of the mathematical concepts of functions and sets the celebrated clarity of this system of logic derives. Hence it is strictly no more than any mathematical system with clear definitions.

A student familiar with the natural deductive approach to propositional logic may agree with us in disapprobation of the truth-functional approach as representing logic, and recommend that logical truth be grounded in the system of natural deduction. Let us examine natural deduction then and hope to find the prior realm of logic there. That system declares a set of primitive rules of inference, an example of which is simple disjunction elimination, which states that when we have some set of propositions Δ which necessarily leads to the truth of either proposition p or that of proposition q, formally Δ ⊢ p ∨ q, and some other set of propositions Φ which falsifies the proposition p, formally Φ ⊢ ¬p, then combining the set of premises Δ, Φ, it follows that q is true, or formally Δ, Φ ⊢ q. This rule is merely a formalization of our mode of arguing by cases: when one of two things are true and one is false, the other must be true. Here the symbol ⊢ means derives or necessarily follows, and it symbolizes valid deduction in the logical system. The meaning of the symbol is functionally defined by the set of inference rules, which dictates when we may obtain a statement of the form Θ ⊢ s. Indeed, a proof done in natural deduction aptly encapsulates our process of reasoning, so may appear more distinctly logical.
Nevertheless, our question is of the soundness of this system of logic, which leads us to inquire the soundness of the primitive inference rules. How do we know that they are true? The problem is that we cannot address this question without defining what are meant by propositions p,q and such connectives as ∨ and ⊃. Yet to these the system of natural deduction does not answer, since such rules are effectively definitions. There is no additional definition of an atomic sentence p, and we cannot revert to the simple definition given in the truth functional approach of p as having either a truth value of true or false. Also, there is no definition of the connectives besides how they may be applied, which is given by the rules of inference. We are disposed to declare such rules, like disjunction elimination, as self-evident: obviously if one of two statements is true and one of them is false, the other is true! Yet this leads us straight back to the problem of providing a definition for what concepts are—what are such p and q that we’re talking about, and what does it mean by taking their conjunction? Unless we can provide such an abstract definition and prove its applicability to all the different species of reasoning, we cannot definitively assert meaningfully that such rules of inference are sound. What indeed can soundness mean here? Put another way, the only method of reasoning that definitively establishes results is deduction. And to employ deduction, we require well-defined concepts as premises from which to deduce. To make a study of deduction itself, we must attempt to encapsulate the process of deduction in a set of concepts, which we call a logical system, and then proceed to deduce from these concepts. Thus, the purview of the results we establish is, just as in any mathematical system, within this set of concepts that constitute the logical system.
Several objections are in order for us to answer. The reader observes, truly, that we have only established that the system of propositional logic is a system that operates on specific mathematical definitions, but we have not demonstrated that prior logical truths cannot exist. Surely, the law of non-contradiction, ¬(p ∧ ¬p), applies universally! Let us take this logical law then. What mental justification do we actually think of when we consider this law to be self-evident? Usually, something like this object has a uniform color of white so it cannot simultaneously be black, or that a circle cannot be a square. Yet what does non-contradiction mean in these contexts but that two particular concepts are incompatible, incompatible as defined by those particular concepts! That the rule of non-contradiction applies universally does not entail that contradiction (or non-contradiction) is a fundamental concept that has a meaning apart from the particular cases of its application. I would be gratefully edified if anyone can justify non-contradiction in a higher level with an abstract notion of a concept. This requires, as belabored, an abstract concept for concept itself. Furthermore, it requires a definition of negation, which is a non-trivial matter, for it must consider the entire space of concepts and select only those which necessarily refutes the given concept, i.e., to exclude the middle. In light of our earlier demonstration of the priority of concepts to any reasoning whatsoever, the plenary word ‘concept’ subsuming all concepts is really vacuous, for if not, the set of concepts that constitute ‘concept’ must underlie each and every concept. I for one struggle to identify any common concept between a bachelor and a triangle that suffices in showing that the former cannot be married and that the latter is incapable of having four sides, without using the nature of these concepts themselves. Figuratively, there can be no separate Platonic form of logic from the other forms. Why do we think there is such a logical principle of non-contradiction presiding over all reasoning? We observe an instance of it in the former reasoning and another instance in the latter. By an instinct for generalization, we fashion to ourselves that the reason of the similarity is that this law operates at the level of ‘concepts’, so extends to all reasoning. But when obliged to clarify what we mean by concept in general, we make specific definitions as in propositional logic and reason about these and forgo thinking of concepts in general. This means that our systems of logic are truly models which approximate our common forms of reasoning, so are a formal subject in the sense that it does not warrant, by itself, validity in applying to these other systems of reasoning.
Another objection that leads directly to the heart of the subject is as follows: if logic pertains to deduction proper, it should not be studied as a deductive system, i.e., using deduction, but of necessity requires a more prior consideration. My question here is simply what this ‘prior consideration’ is. Deduction is the only certain mode of reasoning. In fact, if we analyze any argument that is only based on the meaning of relevant concepts, the only meaningful components which provide justification are precisely the deductive elements. All arguments are deductive, strictly or loosely. Thus, I know not of any other mode of ideation which can establish the soundness of the laws of logic, especially when we conceive of such laws to be prior to all theorems of mathematics and thus more fundamentally necessary. Rather, it occurs more direct and reasonable to my mind that what we denote as the laws of logic are patterns of deduction which appear universally in our theoretical reasoning in various subjects. Their validity is separately established in these respective subjects using the particular concepts with which they are concerned.
The remaining question is, of course, whence arise such patterns of universal applicability? A simple though not superficial response is that they arise from the very process of classification and synthesis of our modes of reasoning, where we seek patterns of argument. The modern study of logic originated with a close examination of mathematical reasoning. Aristotle invented his syllogistic logic by considering related types of arguments and abstracting away the respective contents by constructing a common form. To verify that an intuitively accepted logical pattern, say the principle of non-contradiction, on some concepts, we need only instantiate it in terms of the concepts involved. For the case of an even number, having a remainder of zero when dividing by two excludes the possibility of having a remainder of one, so non-contradiction applies.
I have referred to the view presented in this essay of logic as Platonic, by which I mean to emphasize, first, the universality of logic—that it applies to all concepts; and second, its priority to reasoning—that a concept cannot be reduced to anything else, e.g., physical phenomenon, but must always be assumed to reason. The misnomer may be that I do not think there is a separate realm for logic, i.e., a separate form of logic, which is independent of particular definitions and concepts, but it is to the nature of the concepts themselves logic ultimately pertains.

Endnotes

I will justify this identification after some illustrations of my concept of logic. The objection, I anticipate, is our common use of the term ‘inductive logic,’ which truly concerns matters of fact regarding existence rather than matters of concept, of which logic is properly concerned.
Gottlob Frege, a reputed founder of modern logic, writes in the opening section of his Foundations of Arithmetic, “After declaring for a time the old Euclidean standard of rigour, mathematics is now returning to them, and even making an effort to go beyond them.” Gottlob Frege, The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, trans. J. L. Austin (New York, NY: Harper & Brothers, 1950), 1.
One who holds such a strict position regarding deduction needs to explain what constitutes a mathematical concept. Mathematics has seen such expansion and development since modernity that the terms ‘number’ and ‘geometry’ are no longer adequate to provide a functional definition of the range of concepts studied in mathematics. Thus, it appears to me that by a mathematical concept, we mean a concept of such clarity that we are capable of distinctly grasping and reasoning about it, in the rigorous form of deduction.
The consideration of existence as an abstract entity originated with the Greek philosopher Parmenides, who declared that being cannot originate from unbeing (nothing), or at least this is inconceivable. Richard D. McKirahan, Philosophy Before Socrates: An Introduction with Texts and Commentary (Indianapolis, IN: Hackett Publishing Company, Inc., 2011), 145. It was Immanuel Kant who observed that being is not a property and has no conceptually distinct property but is always only assumed. Immanuel Kant, “Critique of Pure Reason” in Modern Classical Philosophers, ed. Benjamin Rand (Boston, MA: Houghton Mifflin, 1924), 450-451.
Plato, “Meno,” trans. G.M.A. Grube, in Plato: Complete Works, eds. John M. Cooper and Douglas S. Hutchinson (Indianapolis, IN: Hackett Publishing Company, Inc., 1997), 875.
There probably exists mathematical theory which constructs the concept of a Euclidean plane from more general concepts. There, the pertinent question is whether generality entails their being more elementary. Even so, the problem of definition still pertains to the definition of those more elementary concepts.
‘Cognitive’ is the more modern parlance. Yet ‘psychological’ has become an established term to signify views of logic as a natural phenomenon due to human physiology, subject to empirical investigations of science. For an example, see Frege, Foundations of Arithmetic, 33-36, §26.
I use the words physical, natural, and ontological interchangeably, with no special qualification attached to each, to refer to reality. They are contrasted with logical, conceptual, or analytical. For simplicity, I’ll mostly be using physicaland logical.
It sounds awkward to say “assume concepts.” What does it mean to assume a concept? By this awkward phrasing, my intent is to avoid the more problematic one, “assume concepts exist.” The problem there, which I’ll address later, is the meaning to the existence of a concept is utterly different from the usual meaning of the term that refers to a physical entity. To assume a concept is not to accept that the concept is true or false but to simply construe it and allow it to stand without further definition. When you say to me, “a Euclidean plane,” I grasp the concept in my thought clearly and do not require from you further clarification.
I do not at all deny that what occurs to mind when the word ‘triangle’ is uttered may be different from that which occurs to you. I do not even doubt that such essential thoughts which you rely on to deduce a triangle has three sides are different from mine. The brain activities may be different. What is, albeit, the point of argument is that if any person, in reasoning about a concept can sensically incorporate a meaning of subjectivity, which is not introduced as a separate concept, but is somehow interjected into all concepts concerned. ‘Objective’ here is not a claim on the uniformity of the thoughts of all people, which is a physical claim, but merely stating the irrelevance of the term ‘subjective’ in referring to the concepts themselves.
Again, using ‘physical’ as relating to existence, or ‘ontological.’ The subjectivity of experience is a metaphysical problem in that one can never have experience out of oneself, so experience is subjective by definition.
Our use of the phrase ‘inductive logic’ seems to present a contradiction. But if logic is the discipline concerned with the relation of the meaning of concepts and not a science that concerns physical reality, such a name as ‘inductive logic’ is truly a misnomer. There is nothing logical in the inference, the sun has always risen so it will rise tomorrow. Whether it would or not is solely a question about reality, and there is nothing in a simple concept of the sun as a hot radiating object that necessitates its future behavior. Thus, inductive reasoning, as it is essential to the generalization of observations to the formulation of scientific theories, is a topic for epistemology, which studies, in part, the validity of our claims about reality, and does not properly pertain to logic.