The Cardinal Differences Within the Field of Meaning>
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Up until now I have aimed at a systematic combination of simple and compound judgment-configurations, whose differences were purely determined by matter, namely, in such a way that any drawing in of concepts through contrasting matter and quality and the different modifications that the quality can undergo within each genus (such as judgment, presumption) was still to be avoided. Only once have I of necessity violated this principle, namely, in order not to have to separate existential judgments as judgments of being from their equivalents in the domain of particular and universal judgments. Likewise, I have also barred all complications and modifications that arise when concepts are drawn in that have arisen in reflecting upon the matter or upon the whole configurations and such, therefore, concepts such as “proposition”, “presentation”, and so all purely logical concepts. The first transformations, however, are especially of great importance and to some extent hard to clarify. And they are those that can be called modal distinctions in the concise sense. Historical logic presents a heading “modality”, and it is handled so unclearly that it includes the forms interesting us here. Kant arbitrarily grouped judgments into four groups from the standpoint of quality, quantity, relation, and modality, something which again gives rise to a false idea of equivalent treatment that is in no way present.
<§ 43. The Modal Distinctions>1
217Up until now I have aimed at a systematic combination of simple and compound judgment-configurations, whose differences were purely determined by matter, namely, in such a way that any drawing in of concepts through contrasting matter and quality and the different modifications that the quality can undergo within each genus (such as judgment, presumption) was still to be avoided. Only once have I of necessity violated this principle, namely, in order not to have to separate existential judgments as judgments of being from their equivalents in the domain of particular and universal judgments. Likewise, I have also barred all complications and modifications that arise when concepts are drawn in that have arisen in reflecting upon the matter or upon the whole configurations and such, therefore, concepts such as “proposition”, “presentation”, and so all purely logical concepts. The first transformations, however, are especially of great importance and to some extent hard to clarify. And they are those that can be called modal distinctions in the concise sense. Historical logic presents a heading “modality”, and it is handled so unclearly that it includes the forms interesting us here. Kant arbitrarily grouped judgments into four groups from the standpoint of quality, quantity, relation, and modality, something which again gives rise to a false idea of equivalent treatment that is in no way present.
In introducing modal differences, people commonly say that the judgment-mode, the mode of connection between S and p in the categorical judgment (which really counts as basic type) could be different depending on the mode of validity intended. Either “S is certainly p, it is actually the case that S is p”, or “S is possibly p”, or “S is necessarily p”. And, the three Kantian headings, assertoric, problematical, apodictic judgments, arise accordingly.
218First of all, with respect to these judgment- or statement-forms, any mixing up of psychological and meaning-logical distinctions is again now to be avoided. It may therefore not be said that it is a matter here of expressing our certainty, our considering-possible, necessary, as if that should be predicated about. Naturally, one can predicate thus, but that is of no concern to us here, and in discourse, reality or certainty, necessity, and such are normally not intended.
<a) The Possibility-Statement in the Sense of Seeming and in the Sense of Not-Being-Excluded>
If we remain within the pure theory of meaning, then for the time being we come to the actually completely justified forms I dealt with earlier: “It is so, that S is p, it is actually so”−or “That S is p, that is, that is certain, that is true”. “Certain” and “true” do not signify exactly the same thing here, but it indeed appears that the “certain” and the pure Is mean the same thing. It is the objective correlate of certainty brought to predication. Certainty is, however, nothing other than character of judgment, that is, naturally of judgment in the meaning-theoretical sense. When it comes to validity, this “certain” is equivalent to “It is true”.
The precisely analogous statements for the sphere of possibility-statements2 and probability-statements would be “It is possible that S is p”, “It is probable that S is p”, and for questions, “It is questionable−”, “It is doubtful−”. The same thing holds of all those judgment-forms, as was explained for the reality judgments with that-propositions in the subject position, which naturally do not confer any being-value upon the merely thought “states-of-affairs”.
Very important now is an equivocation in possibility-statements. (1) In one case, we have the meaning just now indicated, “Something is to be said for that”, which plays its role in the theory of probability. The “It is possible” is the correlate of a seeming, just as “It is probable” is the <correlate> of a presumption. (2) <In the other case, we have> the totally different meaning that in certain spheres makes the possibility-judgment totally equivalent to particular judgments. In mathematics, one says, “It is possible”. For example, “It is possible for a triangle to be isosceles”, as much as “There is such a triangle”. Here possibility and being do not coincide conceptually, but in equivalence. How does this equivalence come about?
219On its own, the meaning of “It is possible” is frequently “It is not excluded”. Here “not excluded” would naturally signify excluded by the axioms, which alone are what excludes in mathematics. Likewise, “It is impossible” <would signify> it is excluded (by the axioms). The difference between the two earlier meanings of “It is possible”, between the one associated with seeming and the present one that signifies “not being excluded”, consists precisely of the fact that in one case the meaning is positive, “Something speaks in favor of that”, in the other, negative, “Nothing speaks against that”. The two are not, however, equivalent. It plainly holds of mathematics that every proposition that <is> not ruled out by the axioms is true (while in natural science something analogous would hold if, with respect to the basic laws of nature, we knew that all being in nature strictly obeys laws, and if we had at our disposal a complete set of laws of which we could say that they were all basic laws).
<b) Necessity as Apodicticity and Relative Necessity. Its Relationship to Laws>
220Closely related to these judgments about possibility and impossibility are <judgments> of necessity and conformity to laws and their opposites. To be distinguished as well is the case of the statement, “That M is, is a law” and the statement, “That M is, is a necessity”, although the modes of expression become very mixed up. Every necessity points to conformity to a law. Necessity and conformity to laws are correlates, and this gives a concept of necessity a fixed sense. The “It must be so” expresses that what was stated may count as a conclusion, as a particular case of a law, for example, if we say of the quartic equation under consideration that it would have to have four roots, namely, bearing in mind the law that every algebraic equation of nth degree has n roots. The opposite of necessity would there be a more definite concept of possibility, namely, related to the same matter. In this sense, “It is possible that the points of a triangle indicated (point of intersection of the altitudes, the centroids) coincide” was tantamount to saying: It is not a law of triangles that these points always coincide. And so, conveying the meaning more loosely, “It is possible” and “It is necessary, that an A is B” are often tantamount to “It is universally so” and “It is not universally so” (which, since the negation is merely directed against the universality, does not imply that it is so in the particular case).
In relation to all that is then, however, a concise concept of necessity, that of apodicticity, which is a genuine modal concept, but equivalent to the concept of validity on the basis of a law. If, for example, we really and perspicuously carry out an inference, then the relationship of the conclusion to its terms is there before our eyes, but not only as existing in the universal nature of judgment, but in addition characterized in specific ways, precisely as the necessity-meaning. We are thereby not thinking of a law, but a law pertains to this. One cannot reasonably carry out an inference without recognizing that inferring in that way is universally valid and sanctioned by laws, that substituting terms by indefinites therefore yields an unconditionally valid law-judgment.
In meaning, corresponding to this consciousness of necessity is a specific coloration of the Is that expressed predicatively results in the reflexive judgment, “That M is, that is a necessity”, “That S is p−this being of a state-of-affairs−is a necessary being”. We also, however, say equivocally, it is a necessity that a + 1 = 1 + a is, that two straight lines intersect in one point, and so on. Those are, however, laws. The equivocation lies in the fact that we call the laws as source of necessity itself a necessity. Actually, we would only have to say in the particular case, “That these two straight lines must intersect at a point is a necessity”. We in fact see that without thinking about the law, in the consciousness of the apodicticity alone. At the same time, we observe that every individual case of an axiom yields an apodictic necessity and that necessities of arbitrary universal forms occur, as also laws either have a hypothetical content or do not.
All individuations of laws are necessities in themselves, absolute necessities. Along with this are relative necessities, namely, a proposition also figures for us in specific ways as apodictically characterized if it is proven to be a “necessary consequence”, therefore is a component part of an inferential necessity. Necessity-consciousness is related here, albeit nevertheless something different in comparison with the case in which a proposition is merely an individual case of a law. Relatively necessary judgments are only characterized as necessary in relationship to their premises.
221So, each necessity in certain ways indeed has a relationship to laws, but remaining is the clear-cut distinction according to which the one has a correlate in a law whose individual case is that indicated as necessary, while in the other case, the necessity that is relative to the conclusion, and consequently brings with it a relationship to premises of a certain determined kind whereby the law of necessity lies in the relevant law of inference according to which this conclusion from the premises could be characterized as being in conformity with laws. The validity of the predicate “necessity” is still equivalent to the validity of the pertinent law.
<§ 44. The Idea of Law. An Apodictic Necessity Corresponds Only to Pure Laws in the Individual Case>
Where do things then stand with the Idea of law that has played a specific role here? Can we not characterize it more precisely? It is clear that a law is a universal proposition. Conversely, not every universal proposition is a law. The word “law” is furthermore in many cases used in the sense of a universal requirement, that is, of a community-requirement, as proceeding from the bearer of the community-will, as proceeding from an authority, which has to place requirements on the community to which those living in <the> community feel bound to submit. Those would be laws in the political sense, church-religious laws, also mores, and so forth. That does not concern us here, since we are in the realm of judgments and not in the realm of practical requirements. If we remain in the realm of judgments, then a proposition’s universality can be empirical or pure (unconditional) universality. In both cases, a “law” requires that the universality be infinite, not confined to any thesis of a single individual. (“All roses in this garden”−“all things in the world” and so on.) Laid down then is the concept of pure law purely rooted in the concepts occurring in it.
222Someone will perhaps say: Law is a universal proposition that is stated in the consciousness of unconditional universality of validity. This subjective expression is, however, not completely clear. Can we state the same universal proposition−for example “All humans are mortal”, “All trees in the forest are lindens”−in a dual consciousness, on one occasion, in conditional, and then in unconditional universality, and naturally do so retaining the content, namely, of the identical function? That naturally does not work. On one occasion, we can speak loosely of universality, actually merely believing it holds for the most part in that way, and on another occasion of real universality, but then the first judgment is just only a particular judgment. Clarity with respect to the law-concept requires that we naturally already have definite, complete meanings before us and that we therefore do not operate with loose terms.
The distinction <between> unconditional and not-unconditional universality−which is often called “empirical universality”−must therefore refer to something else. We can thereby rather set aside talk of consciousness which, as we saw, so easily leads astray, since everything that interests us must be demonstrable in the realm of meaning. The universality, that precisely characterizes a judgment as general, universal, can be restricted or unrestricted. It is restricted when we speak of all Europeans, i.e. all the people of Europe, provided the qualifier “living in Europe” occurs, something which conceptual universality no longer allows to appear as pure. Such restricting does not admit the presentation that we associate with the word “law”. So if we were to have understood the talk of “all humans” to mean actual humans living in historical time or those on Earth, then a pertinent universal proposition with “all humans” could no longer count as a law. Full universality in the sense of law requires that it absolutely read, “All humans”, disregarding any limitation to any individual existence whatsoever. No nominal presentation with an individual-nucleus may therefore occur in a law-judgment, no proper name, and likewise no “this” that refers to individual existence, and in turn no particular term of the kind that would refer to individual being may occur in the law.
In short, pure laws = universal judgments−but the universality must be pure, unconditional, not limited by any overt or hidden individual-positing and particular positing.
223We therefore see that consideration of the difference of nuclei has a role in determining the law-concept. Only pure general-nuclei are admissible. All individual-nuclei are excluded. In this purest sense, all mathematical-universal, as well as arithmetic, geometrical, kinematical, theorems are laws, and so forth, as also are the universal propositions of the theory of meaning, in addition also the laws belonging to the essence of nature in general, not though, the particular laws of nature, which are therefore not laws in the pure sense. They still implicitly contain reference to the actual world, which is only one and the same in all judgments of the natural sciences, so that here too we speak of laws.
An (absolutely) pure apodictic necessity corresponds to every pure law of this kind in the individual case. A law does not correspond to every statement containing a reference to individual existence, and consequently not everyone can <be> made with the consciousness of necessity, or have the validity-character of necessity either, thus, for example <the statement> that the weather is beautiful today, likewise <the statement> that all bodies have gravity, and so on. Being able to be characterized as necessary and valid is therefore a definite property of a proposition. For example, “These two apples and those two apples together make four apples”, or “This thing there is extended, has causal properties”−are necessary, obviously hold purely by virtue of laws. If a judgment is made in the consciousness of purely apodictic necessity, the restricting terms, as opposed to the bearers of necessity, must (as <is> easy to see) in certain ways be used as the restricting terms in a particular consciousness, in the consciousness of contingency or actuality, namely, of the contingent restriction of necessity.
It is further to be observed that even purely mathematical propositions that do not include anything about individual existence can also be made in the consciousness of necessity, precisely by virtue of the relationship to a higher consciousness of laws, for example, the <relationship> of number 2 + number 3 = <the relationship> of number 3 + number 3. Naturally, that holds for finite numbers generally and in conformity with laws. And so, we are likewise in general able to have woven the universal into the particular, to see or to suppose it to be necessary, not merely something valid, but something valid as an individual case of the law, but without our predicatively stating or thinking that this proposition is anything particular, etc.
224It is moreover to be said in addition that singular mathematical judgments have the property of not being judgments about facts, and yet they are not laws. But they are nevertheless equivalent to laws. More universally: There are pure concept-judgments that exclusively contain pure concepts, pure general-nuclei and that are of course not in themselves of the nature of laws, but yet can always be rephrased in law-sanctioned universality, in full equivalence. If the sound c, taken as idea, is lower than sound d, then it is naturally to be said with unconditional universality that every real or possible individual sound that corresponds to the Idea c, is lower than the very same individual sound of the quality-Idea d, and so in general.
Therefore, the distinction between pure laws and pure concept-judgments (law-sanctioned states-of-affairs and states-of-essences) is not as essential as the difference between the two kinds of judgments and factual judgments. We need the law-concept in order to mark out a kind of universalities−<the> purely conceptual universalities, especially in contrast to the necessary particulars. But, it is otherwise important to bear in mind that every essence-judgment, every purely conceptual one, has the value of a law, though not always the form of a law.
<§ 45. a) The Distinction Between Judgments in Pure Concept-Judgments and Factual Judgments (A Priori and Empirical Judgments)>
By an empirical judgment, a factual judgment, I understand a judgment in which individual existence is posited, in whatever way, whether definitely or indefinitely. Belonging there are, therefore, ordinary existential judgments, but not, for instance, those of pure mathematics, in which pure numbers are posited. Every categorical judgment of the empirical sphere, such as “Today the sky is blue”, and so forth, also belongs here.
225On the opposite side are the pure concept-judgments that simply do not posit existence in any overt or implicit form, therefore, all the purely general judgments and their equivalents. Bear well in mind that we are not identifying the concepts “general” and “universal judgment” here. Every functional-judgment is universal that has the form of what is universal in general, whereas a judgment about ideal objects is general. Corresponding to every purely conceptual nucleus, as we know, is a nominal proper-presentation, for example, corresponding to the nucleus of the adjective “red” is the noun “red”, where the object named there, the “essence red”, is the Idea. Every judgment, then, that judges about such objects, general objects, is a general judgment. And such a judgment can be universal and particular, singular and plural. It can have each of our logical forms, something of which we can certainly assure ourselves in sciences such as the mathematical sciences.
Every general judgment can be changed in such a way that it no longer judges about general objects, but rather about individual ones, but in unconditional universality (therefore, not as existential judgment). Since this change is grounded in the universal essence of such judgments, it is not then very important if we also call these judgments changed in this way general, therefore, do not rigorously keep general and pure concept-judgments apart, although nevertheless differences must again be seen.
I called the difference between existential judgments and pure concept-judgments a cardinal difference. One immediately sees that accordingly all sciences separate in a clear-cut way into ones in which only pure concept-judgments occur, in the others existential judgments as well. The former would investigate what holds “a priori”, what is grounded in the essence of pure concepts and holds for particulars in general in unconditional universality, precisely exclusively insofar as they are particulars of the pure concepts concerned. The others would investigate matters of fact, quite simply, what holds for matters of fact. If it has already been ascertained that they are particulars of certain pure concepts, then pure concept-judgments or pure laws apply to them. However, investigation into matters of fact does not begin with such things, but first ascertains what holds for what is given in experience, which alone can give matters of fact, i.e. under which concepts it is to be grasped and how it is to be further determined in accord with the essence-insights of pure concept-judgments.
226I described the difference between pure concept-judgments and factual judgments also as a difference between a priori and a posteriori <judgments>, wherewith a fundamental, really the most fundamental, meaning of this ambiguous distinction is specified. The difference is actually indicated in this way with respect to substantiation, etc. Kant also talks of a priori and a posteriori concepts, just as the tradition prior to him had. Corresponding to them in the realm of my meaning analyses is naturally the conceptual distinction between essence-concepts, pure concepts and individual concepts with which we have become acquainted as the difference between pure nuclei. (Every nominal presentation has its stuff and its form, and eventually, we precisely come to differences between presentations in terms of their ultimate stuff, therefore, <to> differences such as those between the presentation-content “Socrates”−or the concept, in the sense of the nucleus−and the content “green”, “similar”, and so on. Both can be linked in complex presentations. However, as soon as an individual-content occurs, the concept is impure.) Therefore, it is the difference between concepts that is in turn the basis for the distinction between a priori and a posteriori judgments.
A very important–indeed fundamental for all of philosophy–special case of a priori judgments is denoted by the heading “analytic judgments”. And the very controversial distinction between analytic and synthetic judgments, which Kant related to the distinction between a priori and a posteriori judgments, plays a major role. Before examining this more closely, it is to be borne in mind that the universal distinctions pertaining to the sphere of pure concept- and law-judgments recur in the sphere of analytic judgments.
Therefore, we have to distinguish analytic laws, or analytic pure concept-propositions from analytic necessities. And again, in the sphere of existential propositions, we have to distinguish those having the character of analytic necessity and those not having it. Thus, any denial of an explicit contradiction is an analytic necessity and any explicit contradiction itself an analytic impossibility. For example: That the theater is full and not full is an analytic impossibility, and if we state that that is not true, then that is a necessity. The corresponding universal law “It is not true that any object a is and at the same time is not” is then an analytic law.
<b) Concept-Truths−Factual Truths. Analytic−Synthetic Truths>
227The outcome of the reflections of my last lecture3 was as follows: Truths break down A) into pure concept-truths, truths that do not include any kind of positing of individual existence; B) <into> ones that do include some such thing, namely, factual truths, existential truths. Or epistemologically expressed: Truths break down into a priori truths and a posteriori truths (or <into> non-empirical truths and empirical truths).
The pure concept-truths are either pure laws or to be converted equivalently into pure laws. There are various gradations here. Pure laws can relate to singular particulars that are themselves of a purely conceptual nature (namely, ideal objectivities), or they can relate in unconditional universality to individual objectivities. The latter holds indirectly for all pure laws, insofar as ideal particulars, such as, for instance, the Idea “two”, the Idea “sound c”, and so on have a pure extension of empirical particulars.
Pure categorial concept-truths as a whole. <These> essentially belong together and form a single system of scientific disciplines, all of which I deal with under the broadest heading of “formal logic” or “analytics” (= mathesis universalis in Leibniz’ sense).
228If we now look at the pure concept-truths that do not exclusively contain that remarkable group of pure concepts that I called formal categories, then they again break down into two groups: first of all, those that remain truths when we replace their material concepts by pure categories; second, those for which that is not the case. For example, “Two colors the same as a third are the same as one another” is only a special case of “Two objects the same as one and the same third one are the same as one another”. Or, “If one spatial distance is greater than a second one, then the latter is smaller than the former”. That is naturally a special case of a universal principle regarding magnitudes that holds for any kind of magnitude and not merely for spatial distances. “Magnitude” is however a categorial concept. All a priori individuations of pure laws of meaning belong here, all a priori individuations of arithmetic laws, and so forth.
Every special case of a law is a necessity. Therefore, these pure concept-statements are analytic necessities. Therefore, that characterizes the one group of a priori truths in completely definite ways. The others are the a priori truths that are not analytic necessities−such as, for example, geometrical and kinematic axioms. We call these synthetic or synthetic a priori concept-truths.
<Regarding B> As concerns factual truths, they too can be categorized into two groups: mere individuations of a priori truths and those that are not. a) Those which are empirical individuations of analytic concept-truths, i.e. arise through substitution of a term positing existence for a pure concept; b) those which are not, but are empirical individuations of synthetic (synthetic a priori) concept-truths; and c) those which are neither of the two, are pure factual truths.
By “analytic truths” in the broadest sense, I at one point understand, analytic concept-truths, therefore all pure categorial truths, therefore, the entire pure mathesis, pure logic, then however, also their a priori and empirical individuations, therefore, the analytic necessities. In the case of empirical individuation, we have a mixture of what is a priori and what is empirical, insofar as the carrying over to the case of an individual existence–for example, <the carrying over> of a logical law to empirical propositions–precisely brings in empirical existence, but thereby at the same time necessity, necessity of the being-so, in this case hic et nunc. We therefore then have analytic-necessary existential truths.
229In like manner, by synthetic truths, on one occasion, I understand synthetic concept-truths, such as, for example, the axioms of the pure theory of time, of pure geometry, and so forth, then their empirical individuations, therefore, the synthetic existential truths, which are empirical owing to the introduction of existence, but otherwise express the fact that what is existing is necessarily this way or that, namely, according to a synthetic necessity (therefore corresponding to a pure non-categorial law). Therefore, the carrying over of any geometrical law to a given instance of nature also belongs here.
If one understands the heading “analytic” and “synthetic” so broadly, then one must nevertheless constantly keep in mind the categorizations falling under it, for they are the most important. To bear in mind in addition is that if someone says that all analytic truths are eo ipso a priori, but not all synthetic truths a posteriori, then that is to be taken cum granis salis. An analytic existential proposition holds by a priori necessity. It states something that holds hic et nunc, because in general it holds by virtue of analytic laws. Provided that it is not then to be stated about the particular function of such a proposition that what is existing actually is, but that it is necessarily like this and that, talk of the a priori validity of the proposition is justified. For example, “This thing is extended” − obviously because thingness is an extended existence, and the fact that an extended existence is extended is a particular case of the logical law of the tautology that what is an a being b is also an a. Otherwise, the entire proposition as it is there is, however, a posteriori, provided it is speaking of this thing. And, if in stating it we had succumbed to a hallucination, it would be false. The fact that the subject-positing is valid can naturally not guarantee the analytic law. Only experience and empirical substantiation can show that.
230Now, “mere” factual truths that are altogether “contingent”, where precisely no component of necessity is <given>, separate themselves <off> from such analytic existential truths. These separations affect the objective nature of truths. They turn into separations of judgments when we substitute the supposed truths for the actual truths, and then therefore distinguish judgments that are supposed as concept-truths of an analytic or synthetic kind, are supposed as existential truths and as analytic necessities with respect to their being-so-composition. Then, also judgments in the meaning-theoretical sense acquire difference and classified separation, with a minor alteration, which I am not going to go into more closely.
<§ 46. The Laws of Apophantic Logic and Those of Formal Ontology>
A new distinction that I make within the analytic sphere, whereby we can limit ourselves to analytic concept-truths, is now of the greatest importance from the standpoint of formal logic, but then in addition also from the standpoint of higher philosophical interests.
The separation is determined by that of the formal categories themselves into the two groups correlatively corresponding to each other: meaning categories and object categories. On the one side are concepts such as “meaning”, “judgment”, “proposition”, “nominal presentation”, in short, all the concepts we acquired in the theory of forms of meanings. On the other side are concepts such as “object”, “property”, “relation”, “unit”, “multiplicity”, “cardinal number”, “whole and part”, “magnitude”, and so forth. From the perspective of the theory of meaning, we start with the Idea of judgment and pursue the different judgment-forms. The laws we obtain in so doing are laws for meanings, as the concepts that originarily and immediately arising here are concepts of meanings. Two kinds of laws are then possible.
<a) Purely-Grammatical Laws and Laws of the Apophantic Theory of Validity>
231To begin with, the laws with which we have been concerned in extenso up until now: the laws of the theory of forms of meanings. Knowledge of them is fundamental for both logicians and grammarians, and in the latter respect, I also call them purely grammatical laws. In accordance with our terminology originating in the nature of the matter, these laws are also to be called analytic laws. If neither Kant, nor one of his successors, thought of such a priori truths under the heading “analytic laws”, that is because they allowed themselves to be led by very narrow considerations and did not arrive at the cases of natural unity of the Idea of the “analytic”. Above all, however, it is because they did not in general see, and for this reason never dealt with, the distinctive nature of the meaning-laws of the indicated groups.
The theory of validity directly attaching to the theory of forms of meanings–which I also discriminatingly call the apophantic theory of validity and combine with the theory of forms under the heading of apophantics–forms a second area of analytics in the broader sense. If at one point we examine the different judgment-forms in the theory of forms, we can then ask whether−viewed as forms for universal judgments–they unconditionally permit the formulation of universal truths or falsehoods. In fact, there are several kinds of such law-truths. If, for example, we take the pure judgment-configuration “S is p and is not p”, then for this purpose, we can state the law that, for any S and p that would be substituted in it, every judgment of this form is false. Every contradictory categorical judgment is false. Likewise, every tautological judgment, every judgment of the form “Sp is p” is correct, if the judgment “Sp exists” is correct. Again, we can state that if two judgments of the form “All A are b” and “All b are C” are valid, the corresponding judgment “All A are C” is implied as a necessary consequence. The entire analytic theory of inference belongs in the domain of apophantic logic. Every analytic law of inference states that in two judgments of pure forms, for instance U1, U2, with regard to definite terms, the conclusion of pure form U3 is implied.
It is, however, to be noted here that in all laws of this sphere, the defining concepts are indeed formal categories, but categories that definitely belong to the class of meaning categories. For example, in the law of inference “If all A are B and all B are C, then All A are C”, the concept defining the extension of the variables is the concept of the concept itself. That means that the hypothetical law is valid for any three concepts A, B, C. If we infer in accordance with the formal law, “If from proposition M, proposition N follows, and from proposition N, proposition P follows, then from proposition M, proposition P follows”, then we are inferring in accordance with a law where the terms are propositions, i.e. for any propositions M, N, P, this mode of inference is valid. Once again, “proposition” is a category of meaning.
<b) The Formal-Ontological Laws as Equivalent Rephrasing of the Apophantic Laws of Validity and as Laws for Objects That Arise Through Nominalization of Dependent Proposition-Forms>
232In comparison, there is, however, a wealth of laws that do not make pronouncements for meanings in general as regards their categories, but make pronouncements about objects in general, properties and relations in general, about sets in general, about what holds for sets in general with regard to their containing one another or being mutually exclusive, or what holds for numbers in general with regard to their different relationships grounding in the essence of numbers. Likewise, <there are> propositions about the relations between whole and part, about sequences and ordinals, and so forth. That is the field of formal ontology. A great percentage of such propositions simply arise from the fact that we equivalently rephrase apophantic laws of validity into laws for objects, for properties, states-of-affairs, and so forth, because one can do that a priori, as is immediately obvious−instead of speaking about arbitrary concepts as meanings and about what holds for propositions of one form or another with respect to their truth and falsehood. If they are satisfied by such and such concepts, we can speak of arbitrary properties and states-of-affairs in relation to obtaining and not obtaining. Instead of saying, for example, that every proposition of the form “S is p and not p” is false, one can say, “For no object does it hold that it may have some property and not have the same property”. Likewise, the ontological proposition that every object has some predicate (or that an object without a predicate is a Widersinn) can be looked at as the equivalent rephrasing of the meaning-theoretical proposition: If the proposition-form “X is something” is satisfied for some X, then there is a concept-meaning a that satisfies the proposition-form “the same X is, is a”.
233In addition to that comes, however, a multiplicity of new laws of the formal ontological sphere that pertain to all the concepts that originate in judgment-forms through the nominalization of moments. The form of the indefinite plural can be nominalized in such a manner that the concept of multiplicity results, whereby multiplicity turns into the object (<and> multiplicities fall under this concept as objects). Likewise, the form of “two” that, for instance, occurs in the definite plural form “Two A are B” can be nominalized into the number “two”, where “two” turns into the object. The objects that we call numbers of the number series in general arise in this way. And then belonging to these new concepts (which, as one sees, have their essential relationship to meaning-forms) are pure concept-propositions that judge about sets in general, about numbers in general, and so forth. Likewise, nominalization of the form of the identity and relations judgments “identical”, “equal”, “unequal” yields the hypostatizations equality, identity, difference, inequality. The relation-form of the “in something” yields inexistence (being contained), the concepts “whole” and “part”, and so forth. The concepts “sequence” and “ordinal”, the concepts “quantity”, “magnitude” arise in like manner. And, the formal mathematical concepts in general arise in this way. And, in formal mathematics, judgments are then made about cardinal numbers, ordinal numbers, and other pure numbers in general in formal universality, likewise about sets, orders, and order types in general, about combinations and permutations, about quantities in general, and so forth.
The theory of forms is now no longer directly at the basis of the judgments, but one is operating within a realm of concepts that ultimately lead us back to certain meaning categories, but arise out of these through a certain transformation, through a certain nominalization. And, in addition to that come indirect concept-formations having a relationship to meaning-forms, such as, for instance, the concept of quantity in the original sense, as that of a whole of equal parts.
234That may suffice to show that with formal mathematics we do not actually enter into an essentially new domain, but are here dealing with a field of pure concept-truths whose conceptual matter is inseparably linked to the original matter of the logic of meaning. The philosophical import of this realization can of course not be assessed here. The great interest attaching to discovering and soundly determining the fundamental demarcation in the overall realm of knowledge cannot in general be discussed here. Such a fundamental demarcation, which ranks as the lowermost and perhaps the most important, is that which separates the domain of analytica priori knowledge from synthetic-a posteriori knowledge and isolates the entire domain of a formal a priori ontology in a clear-cut way under the former heading.
<c) Disagreement with Kant>
235Kant, who first saw the difference between analytic and synthetic a priori and yet did not understand its full scope and essential demarcation, not in vain called it a classic distinction for transcendental philosophy. What he lacked was the genuine concept of the analytical, which is determined by the conceptual sphere that I called that of the formal categories. What he further lacked was the understanding of the essence of formal logic, of traditional logic to begin with. If we look at what it offered to purely logical truths and, therefore, just as Kant wished, separate the logical a priori from everything empirical-methodological, especially from everything that is a matter of the theory of the art of logic, then we soon realize that what is purely logical according to the tradition exclusively belongs in the apophantic sphere, and that therefore formal logic to a certain extent sought to be apophantics, without being able to define its goals and its natural limits scientifically. Apart from Leibniz and a few of those influenced by him, no one had suspected−and especially not Kant–that pure arithmetic and all the disciplines essentially related to it, such as pure combinatorics, pure theory of magnitudes, and so on, intrinsically belong together with the old formal logic. Unmistakable for Kant’s readers is furthermore the fact that for him the concept of the analytical extends as far as his concept of pure logic, i.e. to be precise, of apophantic logic. One does not at first notice that his definition of analytic judgments oddly limits itself to categorical judgments (as though Kant himself would not have coordinated categorical, hypothetical, disjunctive judgments under the heading “relation”) and implies that every categorical judgment is analytic whose predicate concept is overtly or covertly contained in the subject concept, therefore, for example, “A body is extended”. The denial of such a statement produces a “contradiction”, and Kant also said in the Prolegomena that the principle of all analytic judgments is the law of contradiction. Exactly this, however, leads us to an understanding of the implicit identification of analytic judgments and all formal logical judgments, more precisely, of analytic laws with formal logical laws and of analytic necessities with formal logical necessities. It is in particular a tenet of the apophantic theory of validity that by virtue of its basic principles–to which the law of contradiction itself belongs–, the denial of the law of contradiction is deducible from the denial of every purely analytic concept-proposition (analytic law) and that, consequently, the denial of any analytic truth, i.e. the assertion of an analytic falsehood, is traceable back to an analytic contradiction. One can accordingly also say that what is characteristic of analytic truths is that a contradiction may be derived from them, or that they evidently “imply” a contradiction, whereby of course the validity itself is and must be analytic. Connected with that therefore is the fact that for Kant analytic judgments and formal logical judgments (in the dual sense of law and necessity) coincide. And, it is consequently understandable why he did not call arithmetic truths analytic, but synthetic. Actually, no arithmetic proposition can be understood apophantically-logically. At least that has never been demonstrated up until now. Only the fact that every theorem follows from the arithmetic axioms can be deduced formal-logically. The synthetic (in the Kantian sense) is then, however, implied in the axioms, and from there is carried over to the content of the theorems themselves. However right Kant’s observation is then in itself (when one takes into consideration the separation between what is merely apophantic and what is ontological in the broader sense), by means of it, Kant blocked the way to an incomparably more important realization, namely, to the fundamental separation that dissevers everything belonging to the realm of formal category from the sphere of the non-formal a priori. The consequence of this for Kant’s critique of reason is a completely inadmissible equation and equal treatment of the arithmetic disciplines with the remaining purely mathematical disciplines, with geometry and chronology, and the combined severing of both from what he called pure natural science. Nevertheless, I must not dwell longer on this point.
Subsequent note by Husserl, “From the start, the basis for classifying modality is to be discussed before beginning the theory of forms: Forms that do not draw in ‘qualities’, that are not reflexive and <ones> that are”. Editor’s note.
Subsequent note by Husserl, “‘Impossibility’ must be precisely discussed: ‘compatibility’, ‘incompatibility’; <a> law that <when> an a and <a> b <are> united, a whole of the form (αβγ) is possible; a law that such a thing is not possible, that there is no such whole in unconditional generality. There is no entity that this entity unites, etc.” Editor’s note.
The text is published as Appendix XV. (Editor’s note).