|Logic and Philosophy of
Logic: An Empirical Study of A Priori Truths
1. A Priori Knowledge and A Priori Truths
In reflecting on our knowledge of logic, I was puzzled because logical knowledge seems to have incompatible characteristics. This knowledge has some claim to a priori status, but logical systems are also developed and "tried out" to capture linguistic practice. Can an a priori body of knowledge have an empirical character? To answer this, we must consider what it is to be a priori.
A priori knowledge has traditionally been conceived to be the product of insight and reasoning. Some truths are simply evident to someone who understands them and reflects on them. These truths are known to be such without being checked in experience. Other a priori knowledge is inferred by evidently correct reasoning (this is deductively correct reasoning) which begins from a priori knowledge. That a priori knowledge which is not evident must be obtained by chains of reasoning which ultimately begin with evident premisses. A priori knowledge is the knowledge which Hume claimed to be "either intuitively or demonstratively certain." (An Inquiry Concerning Human Understanding)
It isnt clear what there is about the objects of a priori knowledge that makes a priori knowledge possible. If we have a faculty of rational insight, on what does it "operate"? Descartes thought a priori knowledge resulted from the study of our innate ideas; Hume thought ideas were the source of the knowledge, but he didnt think these were innate. Someone else might think that a priori knowledge comes about because we have direct cognitive access to Reality, or to its fundamental features.
Mathematics has traditionally been regarded as the clearest case of a body of knowledge which is a priori. Fundamental mathematical propositions/principles are evident. The non-evident truths are deduced from the evident ones. If this is actually the status of mathematical knowledge, then some episodes in the history of mathematics are difficult to understand. In the nineteenth century, there were controversies in mathematics about both negative and complex (imaginary) numbers. Many mathematicians thought there are no such things as negative numbers, and similarly for complex numbers. Eventually, both negative numbers and complex numbers carried the day; the mathematical community now unhesitatingly accepts both kinds of numbers. If mathematical knowledge is simply a collection of evident truths and their deductive consequences, how could competent mathematicians have disagreed about negative numbers? Or about the axiom of choice, for that matter?
What makes a piece of knowledge to be a priori is the way it is obtained or justified. If Brown has a priori knowledge that P, Smith may not know that P. It is also conceivable for Brown to have a priori knowledge that P, while Jones has non-a priori knowledge that P. Perhaps Jones has her knowledge on the basis of observations and measurements.
I think there really is a priori knowledge as traditionally conceived. But I dont think this knowledge is obtained by reflecting on ideas, whatever they are, whether innate or otherwise. Nor do I think we have special cognitive access to Reality or its fundamental features. To understand what is going on with a priori knowledge, I think it is helpful to recognize a class of a priori truths, or a priori statements. An a priori statement is one which "reflects" fundamental features of the conceptual framework within which it is framed. The following sentences state a priori truths:
It is a feature of our numerical framework that seven comes "after" and is larger than three. Understanding how to count, and to use numerals for saying how many things of a kind there are, is all that it takes for a person to recognize that seven is greater than three. The other statements are also a priori, with respect to their conceptual frameworks.
An a priori statement either expresses/describes semantic relations between concepts of a framework, or it expresses/characterizes the application of the framework to experience and the world. That a person has mastered a conceptual framework does not mean that she has articulated for herself the statements that are a priori with respect to that framework. If she has not, then such statements are not part of her explicit knowledge, and so do not constitute a priori knowledge for her. But she might reflect, and then acquire a priori knowledge as traditionally conceived of some of the a priori truths/statements. However, there is no reason to think it possible to obtain a priori knowledge of all a priori truths. It is quite conceivable that some a priori truths are neither evident nor reachable by cogent deductive inferences beginning from statements that are evident.
2. Some Adjustments
A priori knowledge is called that because it is knowledge obtained without "consulting" experience. If Hume were right, we would need experience to provide us with our ideas, but what we learn by inspecting the ideas wouldnt need to be supported by experiential evidence. The expression a priori seems appropriate for statements which reflect features of their conceptual frameworks, for one adopts these statements in adopting the conceptual framework. Once the framework is in place, and accepted, there is no point in checking these statements. The conceptual framework "stacks the deck" in favor of the a priori statements, so that no evidence against them can turn up (so long as one continues to operate "within" the framework).
Since a statement is a priori with respect to a conceptual framework, it seems possible that a single statement might be a priori with respect to one framework but not a priori with respect to another. But a statement is made by employing concepts belonging to a conceptual framework. The concepts in a conceptual framework are characterized and constituted by their relations to other concepts in the framework, and by the relation of the conceptual framework to experience. So it may appear that one statement cannot belong to different conceptual frameworks unless one framework is a subframework of the other. However, two conceptual frameworks can overlap even though neither contains the other. A concept in one framework can be essentially the same as a concept in a different framework, in spite of the fact that they have somewhat different relations to other concepts in the respective frameworks. So, for example, the sentence:
might once have been used to make a true statement that was not a priori, while the sentence is now used to make a statement that is a priori with respect to the conceptual frameworks of physical/chemical theory. Although the statement would certainly have been understood somewhat differently when it was not a priori than it is today, the different statements are essentially similar, and are instances of a single rather specific kind of statement. The criteria for determining that two actual statements are occurrences of a single (kind of) statement rather than of distinct statements are to some extent arbitrary, but it seems reasonable to allow slight changes in content in different occurrences of one statement.
I have labeled statements that are a priori with respect to their conceptual frameworks a priori truths. But there is a difficulty with this. A statement might be a priori with respect to its conceptual framework without being true. A conceptual framework is developed and applied to experience in order to make sense of experience. But it is possible to come up with a conceptual framework that has a poor "fit" with experience. The framework might be accepted and used for a time, but its poor fit lead people to develop an alternative framework with a better fit. The poorly fitting framework, at least in retrospect, is inappropriate and incorrect. Some or all of its a priori statements will be incorrect/untrue. Something like this has happened with the conceptual frameworks of phlogiston theories of combustion. With respect to such a framework, it will be a priori that a burning object emits phlogiston. So long as the theory was accepted, its a priori statements would be regarded as true. But we no longer think they are true, for we have a different, and better, understanding of combustion.
When a theory and its conceptual framework have a bad fit with experience, it is tempting to hold that the statements which are a priori with respect to the framework are actually true, though they have no application to the world. But this is a temptation which ought to be resisted. A theory and its conceptual framework are designed to make sense of experience and the world of which we have experience. Their application to experience and the world are constitutive of the theory/framework. If the fit of the theory is bad, then various concepts will also have bad fits, but they remain concepts of the things to which they are applied.
Statements are a priori with respect to conceptual frameworks, but conceptual frameworks are artifacts, human creations, not part of our innate endowment. Conceptual frameworks are invented, adopted, abandoned, and modified by human beings. This is also true of the theories/conceptual frameworks developed by mathematicians. The relation of a mathematical theory/conceptual framework to experience and the world we experience is more remote, and indirect, than the relations of scientific or (even) philosophical theories, but this relation still matters. What else can explain the difference between theories/frameworks that become part of official mathematics and those that do not? Still, mathematical knowledge does come close to a priori knowledge as traditionally conceived, once the theories/frameworks become entrenched so that their fundamental principles are evident to those working within the frameworks.
3. Empirical Knowledge
The original and official "opposite" to what is a priori is a posteriori. But that is a mouthful, and it is common to substitute other expressions such as factual or empirical. Let us use empirical. In contrast to a priori knowledge, empirical knowledge must be checked in, and rest upon, experience. In contrast to a priori truths, empirical truths do not have their status settled by the conceptual frameworks to which they belong. If an empirical truth had turned out otherwise, the conceptual framework would not have been affected.
There are problems with this terminology. In considering the character of empirical statements, we can equally well attribute this empirical character to whole theories/conceptual frameworks. These are frameworks devised in order to make sense of experience and the world we encounter in experience, and to "organize" this experience. Theories/conceptual frameworks are developed and tested in experience; when we do this, we must be prepared for the theory to be either satisfactory or not. Its truth is not determined by our other conceptual frameworks; nor can we tell by reflecting on the theory/framework that it will do the job we want done.
As for empirical knowledge, if this is the precarious knowledge that could in principle be defeated by experience, then much (perhaps all) a priori knowledge is so only relative to our conceptual frameworks, which can always be reformulated or rejected. This means that a priori knowledge is also precarious and, hence, empirical.
With respect to statements, I think it best not to use empirical as the contrast term for a priori. I will use empirical for theories/frameworks and for statements that capture, or try to, features of experience and the world. So an a priori statement might be empirical, as part of an empirical theory. Even mathematical statements might be empirical, to the extent that mathematical theories/frameworks are connected to the world. I will not go back to a posteriori as the contrast term for a priori. Instead, within a conceptual framework I will distinguish conceptual from non-conceptual truths, and statements. I will not use the terms conceptual and non-conceptual for whole frameworks, but only for statements, with respect to a conceptual framework.
When it comes to knowledge, I also favor using empirical so broadly that some or all a priori knowledge can be considered empirical. Empirical knowledge must fit, and "rest upon," experience. Empirical knowledge must be checked to see that it fits. Typically it is a whole body of knowledge that is empirical. Such a body will usually have an a priori component. But this body of knowledge, including its a priori component must be derived from, and supported by, experience and the world we experience. Perhaps there is some body of knowledge that is wholly a priori, and has no empirical character. I doubt this, but dont now have the time to explore the exact character of mathematical knowledge.
A logical system contains three elements: an artificial language, a semantic account giving truth conditions of sentences in this language, and a deductive system for establishing that some sentences or formulas or arguments or etc. of the language are theorems. The artificial-language sentences, unlike natural-language sentences, are not used by people to think with or to communicate with each other. But our actual linguistic practice is constituted by speech acts, or linguistic acts, performed when we use expressions for thinking, speaking, and writing (as well as for listening and reading). Artificial logical languages are not used in our ordinary linguistic practice. Instead of treating the artificial logical language as a genuine language, it is most appropriate to regard sentences and other expressions of this language as representations of speech acts.
A sentence [A v B] is not a sentence someone will speak or think (even if we replace the letters A and B by actual sentences), nor is it a symbolic "rewrite" of an actual sentence, as 2 + 3 might be considered a rewrite of two plus three. Such a sentence is used to represent a disjunctive statement that is true if one or both component statements are true. We use the artificial sentence to explore the semantic and inferential connections of the disjunctive statement.
A logical system can be a theory of (a part of) our actual linguistic practice. The artificial-language sentences represent natural-language statements, the semantic account gives truth conditions of these statements, and the deductive system codifies representations of statements/arguments/etc. that have a logically distinguished character. For example, if theorems of the deductive system are artificial-language sentences, these theorems represent analytically true statements that can be picked out (identified) on the basis of artificial-language forms (which makes the statements logically true).
A logical system might also be used to represent a linguistic practice which we dont actually practice. This is in fact the case with conventional first-order languages. For material conditional sentences do not represent actual conditional statements. And quantified first-order sentences are not accurate representations of the kinds of statements we customarily make. (The statement made with Every student in the class is failing would commonly be "translated" with something like this first-order sentence:
But the translation is not a good representation of the sentence it translates; the first-order sentence comes much closer to being a good representation of the statement made with For everything, if it is a student in the class, then it is failing, although the representation isnt perfect because of the bad fit of horseshoe sentences to real conditionals.) First-order sentences commonly represent statements which merely approximate the natural-language statements they are used to analyze.
However, unless there are deficiencies in our actual linguistic practice, and logical systems are devised to remedy these deficiencies, it seems most appropriate that logical systems "capture" our ordinary linguistic practice. I know of no such deficiencies, and there are few philosophers/logicians who seriously claim that logical systems are a cure for such. (When he wrote "The Concept of Truth in Formalized Languages," Tarski might have been willing to make the claim.) Since standard systems of logic dont represent our actual statements, and no one seems to care, it might seem a stretch to insist that logic should try to capture ordinary practice. But logical systems are used to evaluate this practice (both mathematical and non-mathematical), as well as to analyze and develop mathematical theories/models. (First-order logical systems come closer to representing ordinary mathematical statements than they do our non-mathematical ones.) And many logicians and textbook writers think either that customary logical systems bring to light hidden features of our ordinary language, or that they provide symbolic rewrites of our ordinary statements. These views are mistaken, but they reveal a belief that logical systems either do or should have application to ordinary linguistic practice.
And the many attempts to develop a satisfactory logical system for conditional statements must be considered so many (unsatisfactory) empirical theories of an actual linguistic practice. Developing such systems and presenting them for evaluation is a trial and error procedure. Except for internal considerations like consistency or coherence, we check these systems by reflecting on what people do or would say in certain circumstances. But we also consider what they ought to say. Our ordinary linguistic practice is a normative practice. There are norms for correct talking and for correct inferring/arguing. These norms are not always adhered to, but we discover these norms in linguistic practice and use them to evaluate the linguistic acts of ourselves and others.
A logical system is an attempt to represent, to capture, our ordinary linguistic, normative practice. It is an attempt to represent and explore features of our ordinary conceptual frameworks. But these features are a priori statements and principles of the frameworks. Even though we develop the systems to deal with the a priori, we dont come up with these systems by intuiting evident truths, and then unraveling their deductive consequences. Our knowledge of logical systems does not constitute a priori knowledge of our linguistic practice. Nor can we tell by a simple inspection that a logical system is adequate to its task. We make up the system and then check its fit, just as we would do with a physical or chemical theory. However, the logical theory is intended to fit our conceptual frameworks, not the world of independent physical and chemical processes. This shows the logical system to be an empirical theory of our linguistic practice. Although logical knowledge certainly has some a priori components, this knowledge is not, as a whole, a priori. It is, however, wholly empirical. Logical knowledge is empirical knowledge of a priori statements and principles, and logical systems are empirical theories of the statements and principles.