The Analytic and Synthetic in Kant and Frege

In this brief and hasty post, I will present both Kant’s and Frege’s notions of the analytic/synthetic distinction.  Frege claims that he does not mean to assign new meanings to the analytic/synthetic, but rather state more precisely what earlier writers meant by them.  I will argue that if Kant had Fregean logic, rather than Aristotelian logic, in his toolkit, then he would have characterized the analytic/synthetic distinction differently.  In particular, he would have recognized the judgments of arithmetic are analytic and that analytic judgments can be ampliative.  So Frege’s version will, in fact, amount to a more precise rendition of what earlier writer meant by the distinction.

  1. Kant’s Analytic/Synthetic Distinction.

First we will characterize Kant’s distinction.  An analytic judgment is one in which the concept of the predicate is contained within the concept of the subject. For instance, consider the judgment “all bachelors are unmarried”.  Unmarried is contained within the concept of bachelor, for to deny that all bachelors are unmarried would be a contradiction; we would say that she just didn’t understand the meanings of the terms.  A synthetic judgment, in contrast, is one in which the concept of the predicate is not contained within the concept of the subject. For instance, consider the judgment, “all bodies are heavy.”  The concept of heavy is not contained within the concept of body, so there is no contradiction in denying the judgment. In this way, two concepts are synthesized in making this judgment.

This leads Kant to the conclusion that all mathematical judgments are synthetic a priori.  They are a priori because they are necessary.  For it cannot be the case that the internal angles of a triangle do not sum to 180 degrees, nor could it be the case that 7+5 does not equal 12.  Kant admits that it may seem as though 7+5=12 is analytic (and follows from the principles of non-contradiction).  But this is not so.  An analytic judgment is a judgment in which the concept of the predicate is contained within the concept of the subject, but no where in the concept of the combination of 7 and 5 do we find the concept of 12.  This is more apparent with a very large numbers: you could be thinking of 230987 and a moment later be considering the combination of 222634 and 8353, and not realize that you had been thinking of the name number.  So these judgments synthesize two concepts.  To form the judgment, then, we must appeal to something more than just the two concepts – we appeal to a priori intuition.  Mathematics, then, is characterized by the construction and synthesis of concepts in a priori intuition.  A priori intuitions are the pure forms of space and time (the formal requirements of any sensation or experience) – they are the canvas upon which we can have any experience at all.  Arithmetic is based on the pure intuition of time, and geometry on that of space.  So in a priori intuition we construct arithmetic propositions by manipulating and arranging units of time.

  1. Frege’s Analytic/Synthetic Distinction.

Now we will characterize Frege’s distinction.  For Frege, whether a judgment is analytic or synthetic is not dependent on the content of the judgment, but rather on the justification of the judgment.  For a judgment to be analytic, its truth and justification must rest on definitions and purely general logical laws.  For example, that 5+7=12 is true just in virtue of the definitions of 5, 7, and 12, and logical laws [1].  Any analytic judgment is traceable back to only definitions and purely logical laws.  If one of these definitions or laws is denied then anything could follow – an unfortunate situation.

A definition is a stipulation, not an assertion, so it cannot be denied.  But once a definition is stipulated, it is immediately transformed into an analytic judgment.  Consider the following. I stipulate that a “sachelor is an unmarried man”.  Immediately, I can judge sachelor to be an unmarried man, or even say that “a sachelor is a bachelor” (provided I am familiar with the concept of bachelor). These would be analytically true judgments, for I have defined the terms in the right way, and the rest relies on the law of identity.

A judgment is synthetic when the truth of the judgment appeals to, in addition to logical laws and definitions, a “special science” – that is, when there is appeal to some fact that is neither a law nor a definition.  So the truth of the proposition “most swans are white” is synthetic because it is not in the definition of a swan that it is white (to be a swan is not to be white).  Rather, the truth and justification of the proposition relies on at least partly on experience, which is not in the domain of definition or logical law.

The justification for an arithmetic judgment rests on its proof.  So for Frege, arithmetic judgments will be analytic.  This is because their proof rests only on definitions and general logical laws.

[1] Frege assumed that all mathematics was reducible to logic.  So mathematical operators like “+” will have their logical counterparts.

  1. Frege: why arithmetic judgments are analytic

Because I briefly mentioned Kant’s reasons for supposing that all analytic judgments are synthetic, I will quickly describe why Frege thinks that all arithmetic judgments are analytic.  I will not offer all of the arguments that Frege provides, because this post is more concerned with comparing the notions of analytic/synthetic and speculating how their respective logics influenced their characterizations.

If arithmetic judgments are to rely on a priori intuition, we must be able to conceive of Number in a priori intuition. In what way could we do this? We might think of Number as magnitudes of some kind, either spatial or temporal. Therefore, we must examine what it is to have an a priori intuition of magnitude. It is unclear what an a priori intuition of magnitude could possibly be. What things have magnitudes? Lines, numbers, areas, forces, e.g.. We could conceive of a line of a particular magnitude in a priori intuition, but we ought not confuse this with conceiving of magnitude. A line of some length can be brought under the concept of magnitude, as can numbers, areas, and so on. But that these all belong to the concept of magnitude is not to say that any one of them is an intuition of magnitude in general. In intuition we can consider things which belong to the concept of magnitude, but we cannot intuit magnitude itself. So what would an a priori intuition of magnitude or number be? What is an a priori intuition of 100,000? It cannot be “100,000”, for that just denotes 100,000; it is the name of the number, not the number. Again it cannot be a line or any other thing. We have no a priori intuition of 100,000. Worse yet, consider what your intuition of 0 could be – that is, what is your sensation of 0, what would it be to experience 0? Of 1? When we ask these questions no intuition of the number itself strikes us. Yet Kant claims that intuition must be sensible. Thus we have shown that there can be no a priori intuition of number. Because no intuition, empirical or a priori, can give us Number, arithmetic judgments cannot be synthetic.

Now for Frege’s positive argument for the analyticity of arithmetic.  While Frege does not disagree with Kant that geometric judgments are synthetic a priori, he does disagree that there is a substantive analogy between geometric and arithmetic judgments. Geometry is a special science whose domain is all that is spatially intuitable. Arithmetic, however, has a much wider domain – so much so such that it is intimately connected with the laws of logic in a way that geometry is not. The domain of geometry is all that can be represented in three-dimensions. Now consider a four-dimensional object – such a thing cannot be represented in spatial intuition. We can think – that is, conceptualize – a four-dimensional object, but should we represent it in intuition as an aid, then we are still working within three-dimensional space. Our representation of a four-dimensional object must still be three-dimensional. Our intuition merely represents or symbolizes the four-dimensional thing, but does not directly display it. It is this same notion that allows us to intuit a curved line as straight. Calling the line straight does not involve any contradiction. This is to say that we can still go on to prove things within that system despite our taking the curved thing as straight. This means that we can reject a geometric axiom without ending in contradiction, for the axioms are independent of each other. This makes geometry a special science.

In this way, geometry is unlike logic, for if you were to deny a logical law, you would wind up in contradiction. E.g. deny the law of non-contradiction, and you truly arrive in a contradiction, namely that a is not a. And it seems nothing intelligible can be thought after this. Recall that analytic truths are truths that can be traced back to just definitions and logical laws, and that if one of these were to be denied then nothing can properly follow. Logic is an analytic enterprise, then. But geometry relies on logical laws, definitions, and axioms (which can be denied and are what make it a special science). This is why geometry is synthetic.

But we cannot say the same for arithmetic as we can for geometry. One cannot assume the contrary of a fundamental proposition of arithmetic [2] and proceed to reason successfully – we cannot perform the same conceptual magic tricks as we do in geometry. This shows that arithmetic must be intimately connected with the laws of logic. For Frege, logic is the most general domain – the laws of logic are the laws of thought, and so apply to every domain of thought, even the special sciences. Language has a truth-functional, logical structure in which the truth conditions of an utterance are built up out of the truth-conditions of its component parts.  The reason that language has this structure is that the thought requires this structure. And, moreover, the domain of arithmetic is much wider than that of geometry, governing all that is numerable – and all that is numerable is thinkable, vice versa (for you can map one number to every unique [item of] thought). Because arithmetic is connected to logic in this way, arithmetic must be analytic, for to deny its fundamental propositions would lead to contradiction in the same way as for logic. And so arithmetic must proceed only from general logical laws and definitions. And so arithmetic must be analytic.

[2] It is unclear exactly what Frege means by this phrase.  My guess would be the Peano axioms.

  1. Comparing Kant and Frege’s distinctions.

Having shown both Kant’s and Frege’s conceptions of the analytic/synthetic distinction, we are now in a position to ask whether their conceptions of the distinction are equivalent.  Here’s what’s at stake.  If their conceptions differ, then Frege’s arguments against Kant that arithmetic judgments are analytic become empty, as Frege is using “analytic” in a special sense and so is merely “talking past” Kant, as it were.  This is because he takes himself to be arguing against the synthetic-ness of arithmetic judgments in Kant’s sense; so his arguments must pertain to Kant’s distinction.  If their conceptions are the same, however, then arithmetic judgments are analytic a priori (if Frege’s arguments are successful), contrary to Kant’s claim that they are synthetic a priori.

The most apparent discrepancy between Frege’s and Kant’s conceptions is that one conception concerns the justification of the judgment (Frege), whereas the other conception concerns the conceptual content of the judgment (Kant).  So it may appear that their senses of analyticity are distinct.  But this is not so.

It is important to consider the fact that Kant was working with Aristotelian logic which is based on subject and predicate.  Frege was, unsurprisingly, working with Fregean logic, which is based on function and argument and is thus much more powerful.  When Kant asserts that a judgment is synthetic, he is saying that the judgment must appeal to something outside the meanings of the terms.  When Frege asserts that a judgment is synthetic, he is saying that the judgment must appeal to some special science (or particular fact).

That a judgment appeals to something outside the meanings of the terms, however, is just to say that a judgment is not based solely on definitions and logical laws, relying on some particular fact or other.  When we are talking about concepts that are not contained within their subjects, we are appealing to some particular fact or special science.  For example, to justify “most swans are white” is to observe the matter of fact that most swans are white, with some exception – it is not to reason from the terms.  This is wholly consistent with Frege’s account.

However, Kant also maintains that analytic judgments are explicative, whereas synthetic judgments are ampliative.  To be explicative is just to explicate knowledge that we already have – to make explicit things known; to be ampliative is just to amplify knowledge – that is, extend our knowledge about the concept.  But if Frege is in fact just stating more precisely Kant’s distinction, then analytic judgments can be ampliative.  Complex formal logical judgments can lead to conclusions that are not apparent, but surprising, and thus amplify our knowledge of concepts.  For instance,


If you look at (1) and (2), then at the formulae in (13), your knowledge has been amplified, for prima facie you cannot see the relation between them.  One cannot see just from looking at at certain numerical propositions all the derivations that proceed from them, even though those derivations follow just from general logical laws and definitions, so those conclusions must be ampliative.  We learn important new relations regarding relevant concepts.  Kant cannot maintain that analytic judgments all are explicative and that synthetic judgments are all ampliative, if his analytic/synthetic distinction is the same as Frege’s.  So we must show why Kant would and should give up his claim that analytic judgments are ampliative.

Consider the fact that Kant does not present his distinction in the context of logical examples, like Frege, but rather in the context of concepts in natural language.  E.g. he provides the example that “all bodies are extended”.  Clearly this is analytic and does not amplify our concept of body.  In contrast, “all bodies have weight” is synthetic and does amplify our concept of body by attaching a new, distinct concept of weight to it.  These examples are reminiscent of Aristotelian logic or syllogism.  All men are mortal, Socrates is a man; therefore, Socrates is mortal.  An Aristotelian syllogism has a concept/subject structure – concept B holds of subject A.  This kind of structure is amenable to the analysis of concepts, subjects, and arguments in and of natural language, items like, “body”, “weight”, “extension”, “mortal”, “man”, “green”, etc…  And these are the kinds of items that Kant has in mind when he first presents the distinction – items familiar to natural language.  And these are the kinds of items that will turn out to be explicative in analytic judgments (A holds of A) and ampliative in synthetic judgments (A holds of B).  

But we can see that Frege’s notion of analyticity still captures all of the natural language examples that Kant has in mind.  To see how, consider a Fregean treatment of the judgment that “all bodies are extended”.  Is this a matter of just laws and definitions?  Let’s see.  Define a body as a thing which occupies space.  Define to-be-extended as to occupy space.  Now substitute in our definitions and we get the judgment, “all things which occupy space, occupy space”.  And this seems trivial – so trivial that the only way to deny it is to deny the law of identity.  This is to say that one could only reasonably hold this judgment as false if things that occupy space didn’t occupy space!  A flat-out contradiction.  So in virtue of only (1) the definition of body, (2) the definition of extension, (3) the law of identity.  So this judgment will be analytic in Frege’s sense.  We can see that Frege’s sense of the distinction captures the natural language examples that Kant had in mind in first presenting his distinction.  It is also explicative as opposed to ampliative, for no new knowledge was about any of the relevant concepts was obtained.  What about Kant’s synthetic proposition, “all bodies have weight”?  Define weight as the downward force an object exerts.  Substitute, and: “all things that occupy space, exert a downward force”.  This is synthetic.  Logical laws, like the law of identity, and definitions do not suffice to justify this judgment (for downward force is not identical to occupation of space); empirical facts about the world must come into the justification.  Moreover, this will be ampliative, for we extend our knowledge of bodies when we learn they have weight.  So Frege’s conception of the analytic/synthetic distinction capture the natural language cases Kant primarily considered.

  1. Were Kant to have Fregean logic

While Kant may have presented his distinction with natural language in mind, Frege presents his distinction of the analytic/synthetic with logic and arithmetic in mind.  Analytic natural language judgments will continue to be explicative for both Kant and Frege.  But it’s in the realm of logic and arithmetic that analytic judgments are sometimes going to be ampliative – that is, that they are going to extend our knowledge of the relevant concepts (recall the complexity of formal logical arguments).  Because Kant was more concerned with judgments in natural language, it seems plausible that he would abandon his ampliative/explicative distinction in light of the ampliative nature of analytic mathematical judgments.  For if Kant had been working the more powerful Fregean logic, rather than the Aristotelian, he would have recognized some arithmetic and logical judgments as analytic and ampliative.  Recall that Aristotelian logic was based on subject/concept arguments.  These were basic in structure, so its conclusions seemed rather explicative.  Recall our earlier syllogism, and notice that each premise is a stipulation and conclusion seems rather explicative.  We have two definitions, (1) all men are mortal and (2) Socrates is a man.  The law of identity will allow to instantiate Socrates in (1) to get: Socrates is mortal.  On Frege’s picture, this is analytic and ampliative.  And it is just a containment (logical) relation between concepts, and so conforms to Kant’s conception of the analytic/synthetic distinction.




Frege, Gottlob. The Foundations of Arithmetic; a Logico-Mathematical Enquiry into the Concept of Number. Evanston, IL: Northwestern UP, 1968. Print.

Kant, Immanuel. Critique of Pure Reason. Trans. Norman Kemp Smith. New York: St. Martin’s, 1965. Print.


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