Florilegium

Ante Christum

300s BCE

Socrates: Have you heard what they say nowadays that knowing is?

Theatetus: Perhaps; however, I do not remember just at this moment.

Socrates: They say it is having knowledge.

Theatetus: True.

Socrates: Let us make a slight change and say possessing knowledge.

Theatetus: Why, how will you claim that the one differs from the other?

Socrates: Well, then, having does not seem to me the same as possessing. For instance, if a man bought a cloak and had it under his control, but did not wear it, we should certainly not say that he had it, but that possessed it.

Theatetus: And rightly.

Socrates: Now see whether it is possible in the same way for one who possesses knowledge not to have it, as, for instance, if a man should catch wild birds–pigeons or the like–and should arrange an aviary at home and keep them in it, we might in a way assert that he always has them because he possesses them, might we not?

Theatetus: Yes.

Socrates: And yet in another way that he has none of them, but that he has acquired power over them, since he has brought them under his control in his own enclosure, to take them and hold them whenever he likes, by catching whichever bird he pleases, and to let them go again; and he can do this as often as he pleases.

—Theatetus, Plato, 300s BC

Of things that reciprocate as to implication of being, that which is in some way the cause of the other’s being might perfectly sensibly be called prior in nature. And that there are some such cases is clear. For there being a human reciprocates as to implication of being with the true statement about it: if there is a human, the statement whereby we say that there is a human is true, and reciprocally–since if the statement whereby we say there is a human is true, there is a human. And whereas the true statement is in no way the cause of the thing’s being, the thing does seem in some way to be the cause of the statement’s being true. For it is because of the thing’s being or not being that the statement is called true or false.

—Categories , Aristotle, 300s BC

“Being” is said in many ways.

—Metaphysics, Aristotle, 300s BC

The positive premise necessarily converts, thought not universally but in part. For instance, if every pleasure is a good, then some good will be a pleasure.

—Prior Analytics, Aristotle, 300s BC

When both P and R belong to every S, it results of necessity that P will belong to some R. For since the positive premise converts, S will belong to some R; consequently, since P belongs to every S and S to some R, it is necessary for P to belong to some R (for a deduction through the first figure comes about).

—Prior Analytics, Aristotle, 300s BC

Anno Domini

1100s

If ‘every stone-man is a stone’ is true, also its conversion per accidens is true (’some stones are stone-men’). But no stone is a stone-man, because neither this man nor that man etc. is a stone. But also this ‘a certain stone-man is not a stone’ is false by necessity, since it is impossible to suppose it is true

—Dialectica, Peter Abelard, 1115 AD

There seem to be two necessities of consequences, one in a larger sense, if namely that what is maintained in the antecedent cannot be the case without that what is maintained in the consequent; the other in a narrower sense, if namely not only the antecedent cannot be true without the consequent, but if also the antecedent by itself requires the consequent.

—Dialectica, Peter Abelard, 1115 AD

1600s

All that up to the present time I have accepted as most true and certain I have learned either from the senses or through the senses; but it is sometimes proved to me that these senses are deceptive, and it is wiser not to trust entirely to anything by which we have once been deceived.

But it may be that although the senses sometimes deceive us concerning things which are hardly perceptible, or very far away, there are yet many others to be met with as to which we cannot reasonably have any doubt, although we recognize them by their means. For example, there is the fact that I am here, seated by the fire, attired in a dressing gown, having this paper in my hands and other similar matters. And how could I deny that these hands and this body are mine, were it not perhaps that I compare myself to certain persons, devoid of sense, whose cerebella are so troubled and clouded by violent vapours of black bile, that they constantly assure us that they think they are kings when they are really quite poor, or that they are clothed in purple when they are really without covering, or who imagine that they have an earthernware head or are nothing but pumpkins or are made of glass. But they are mad, and I should be any the less insane were I to follow examples so extravagant.

At the same time I must remember that I am a man, and that consequently I am in the habit of sleeping, and in my dreams representing to myself those same things or sometimes even less probable things, than do those who are insane in their waking moments. How often has it happened to me that in the night I dreamt that I found myself in this particular place, that I was dressed and seated near the fire, whilst in reality I was lying undressed in bed! At this moment it does indeed seem to me that it is with eyes awake that I am looking at this paper; that this head which I move is not asleep, that it is deliberately and of set purpose that I extend my hand and perceive it; what happens in sleep does not appear so clear nor so distinct as does all this. But in thinking over this I remind myself that on many occasions I have in sleep been deceived by similar illusions, and in dwelling carefully on this reflection I see so manifestly that there are no certain indications by which we may clearly distinguish wakefulness from sleep that I am lost in astonishment. And my astonishment is such that it is almost capable of persuading me that I now dream.

I suppose, then, that all the things I see are false; I persuade myself that nothing has ever existed of all that my fallacious memory represents to me. I consider that I possess no senses; I imagine that body, figure, extension, movement and place are but fictions of my mind. What, then, can be esteemed as true? Perhaps nothing at all, unless that there is nothing in this world that is certain.

—Meditations on First Philosophy, Rene Descartes, 1641 AD

To be brief, I hold as axiomatic the identical proposition which varies only in emphasis: that what is not truly one Seiendes is not truly one Seiendes either.

—Leibniz Letters, Gottfried Wilhelm Leibniz, 1687 AD

1700s

That all our knowledge begins with experience there can be no doubt. For how should the faculty of knowledge be called into activity, if not by objects which affect our senses and which, on the one hand, produce representations by themselves or on the other, rouse the activity of our understanding to compare, connect, or to separate them and thus to convert the raw material of our sensible impressions into knowledge of objects, which we call experience? With respect to time, therefore, no knowledge within us is antecedent to experience, but all knowledge begins with it.

But though all our knowledge begins with experience, it does not follow that it all arises from experience. For it is quite possible that even our empirical knowledge is a compound of that which we perceive through impression, and of that which our own faculty of knowledge (incited by sense impressions) supplies from itself, a supplement which we do not distinguish from that raw material until long practice has rendered us capable of separating one from the other. It is therefore a question which deserves at least closer investigation and cannot be disposed of at first sight: Whether there is any knowledge independent of all experience and even of all impressions of the senses? Such knowledge is called “a priori” and is distinguished from empirical knowledge, which has its source “a posteriori”, that is, in experience…

—Critique of Pure Reason, Immanuel Kant, 1781 AD

1800s

“The World is my representation,” this is a truth valid with reference to every living and knowing being, although man alone can bring it into reflective, abstract consciousness. If he really does so, philosophical discernment has dawned on him. It then becomes clear and certain to him that he does not know a sun and an earth, but only an eye that sees a sun, a hand that feels an earth; that the World around him is there only as representation, in other words, only in reference to another thing, namely that which represents, and this is himself. If any truth can be expressed “a priori”, it is this; for it is the statement of that Form of all possible and conceivable experience, a form that is more general than all others, than time, space and causality, for all these presuppose it.

—World as Will and Representation, Arthur Schopenhaur, 1818 AD

I must here combat the view that, e.g. \(2 + 5\) and \(3 + 4\) are equal but not the same. This view is grounded in the same confusion of form and content, sign and thing signified. It is a though one wanted to regard the sweet-smelling violet as differing from Viola odorata because the names sound different. Difference of sign cannot by itself be a sufficient ground for difference of the thing signified. The only reason why in our case the matter is less obvious is that the Bedeutung of the numeral 17 is not anything perceptible to the senses. There is at present a very widespread tendency not to recognize as an object anything that cannot be perceived by means of the senses; this leads here to numerals’ being taken to be numbers, the proper objects of our discussion; and then, I admit, 7 and 2 + 5 would indeed be different. But such a conception is untenable, for we cannot speak of any arithmetical properties of numbers whatsoever without going back to the Bedeutung of the signs. For example, the property belonging to 1, of being the result of multiplying itself by itself, would be a mere myth; for no microscopical or chemical investigation, however far it was carried, could ever detect this property in the possession of the innocent character that we call a figure one. Perhaps there is talk of a definition; but no definition is creative in the sense of being able to endow a thing with properties that it has not already got – apart from the one property of expressing and signifying something in virtue of the definition. The characters we call numerals have, on the other hand, physical and chemical properties depending on the writing material. One could imagine the introduction some day of quite new numerals, just as, e.g., the Arabic numerals superseded the Roman. Nobody is seriously going to suppose that in this way we should get quite new numbers, quite new arithmetical objects, with properties still to be investigated. Thus we must distinguish between numerals and their Bedeutungen; and if so, we shall have to recognize that the expression \(2\), \(1 + 1\), \(3 - 1\), \(\frac{6}{3}\) all have the same :ref`bedeutung`, for it is quite inconceivable where the difference between them could lie. Perhaps you say, \(1 + 1\) is a sum, but \(\frac{6}{3}\) is a quotient. But what is \(\frac{6}{3}\)? The number that when multiplied by \(3\) gives the result \(6\). We say “the number”, not “a number”; by using the definite article, we indicate that there is only a single number.

—Function and Concept, Gottlob Frege, 1891 AD

Equality gives rise to challenging questions which are not altogether easy to answer. Is it a relation? A relation between objects, or between names or signs of objects? In my Begriffsschrift I assumed the latter. The reasons which seem to favour this are the following: \(a = a\) and \(a = b\) are obviously statements of differing cognitive value; \(a = a\) holds a priori and, according to Kant, is to be labeled analytic, while statements of the form \(a = b\) often contain very valuable extensions of our knowledge and cannot always be established a priori. The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even today the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names ‘a’ and ‘b’ designate (bedeuten), it would seem that \(a = b\) could not differ from a = a (i.e. provided \(a = b\) is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing. What is intended to be said by \(a = b\) seems to be that the signs or names ‘a’ and ‘b’ designate (bedeuten) the same thing, so that those signs themselves would be under discussion; a relation between them would be asserted. But this relation would hold between the names or signs only in so far as they named or designated something. It would be mediated by the connexion of each of the two signs with the same designated thing. But this is arbitrary. Nobody can be forbidden to use any arbitrarily producible event or object as a sign for something.

In that case the sentence \(a = b\) would no longer refer to the subject matter, but only to its mode of designation; we would express no proper knowledge by its means. But in many cases this is just what we want to do. If the sign ‘a’ is distinguished from the sign ‘b’ only as object (here, by means of its shape), not as sign (i.e. not by the manner in which it designates something), the cognitive value of \(a = a\) becomes essentially equal to that of \(a = b\), provided \(a = b\) is true. A difference can arise only if the difference between the signs corresponds to a difference in the mode of presentation of that which is designated. Let a, b, c be the lines connecting the vertices of a triangle with the midpoints of the opposite sides. The point of intersection of a and b is then the same as the point of intersection of b and c. So we have different designations for the same point, and these names (’point of intersection of a and b’, ‘point of intersection of b and c’) likewise indicate the mode of presentation; and hence the statement contains actual knowledge.

It is natural, now, to think of there being connected with a sign (name, combination of words, letter), besides that to which the sign refers, which may be called the Bedeutung of the sign, also what I should like to call the sense of the sign, wherein the mode of presentation is contained. In our example, accordingly, the Bedeutung of the expressions ‘point of intersection of a and b’ and ‘point of intersection of b and c’ would be the same, but not their senses. The Bedeutung of ‘evening star’ would be the same as that of ‘morning star’, but not the sense.

It is clear from the context that by ‘sign’ and ‘name’ I have here understood any designation representing a proper name, which thus has as its Bedeutung a definite object (this word taken in the widest range), but not a concept or a relation, which shall be discussed further in another article. The designation of a single object can also consist of several words or other signs. For brevity, let every such designation be called a proper name.

The sense of a proper name is grasped by everybody who is sufficiently familiar with the language or totality of designations to which it belongs; but this serves to illuminate only a single aspect of the Bedeutung, supposing it to have one. Comprehensive knowledge of the Bedeutung would require us to say immediately whether any given sense belongs to it. To such knowledge we never attain.

The regular connexion between a sign, its sense, and its Bedeutung is of such a kind that to the sign there corresponds a definite sense and to that in turn a definite reference, while to a given Bedeutung (an object) there does not belong only a single sign. The same sense has different expression in different languages or even in the same language. To be sure, exceptions to this regular behaviour occur. To every expression belonging to a complete totality of signs, there should certainly correspond a definite sense; but natural languages often do not satisfy this condition, and one must be content if the same word has the same sense in the same context. It may perhaps be granted that every grammatically well-formed expression representing a proper name always has a sense. But this is not to say that to the sense there also corresponds a Bedeutung. The words ‘the celestial body most distant from the Earth’ have a sense, but it is very doubtful if they also have a reference. The expression ‘the least rapidly convergent series’ has a sense; but it is known to have no reference, since for every given convergent series, another convergent, but less rapidly convergent, series can be found. In grasping a sense, one is not certainly assured of a reference.

– On Sense and Reference (On Sinn and Bedeutung), Gottlob Frege (Max Black translation), 1891 AD

A concept - as I understand the word - is predicative. On the other hand, a name of an object, a proper name, it quite incapable of being used as a grammatical predicate. This admittedly needs elucidation, otherwise it might appear false. Surely one can just as well assert of a thing that it is Alexander the Great, or is the number four, or is the planet Venus, as that it is green or is a mammal? If anybody thinks this, he is not distinguishing the uses of the word ‘is’. In the last two examples it serves as a copula, as a mere verbal sign of predication. As such it can sometimes be replaced by the mere personal suffix. Compare, e.g., ‘Dieses Blatt ist griin’ and ‘Dieses Blatt grunt’.We are here saying that something falls under a concept, and the grammatical predicate stands for (bedeutet) this concept. In the first three examples, on the other hand, ‘is’ is used like the ‘equals’ sign in arithmetic, to express an equation. In the sentence ‘The Morning Star is Venus’, we have two proper names, ‘Morning Star’ and ‘Venus’, for the same object. In the sentence ‘The Morning Star is a planet’ we have a proper name, ‘the Morning Star’, and a concept word, ‘planet’. So far as language goes, no more has happened than that ‘Venus’ has been replaced by ‘a planet’; but really the relation has become wholly different. An equation is reversible; an object’s falling under a concept is an irreversible relation. In the sentence ‘The Morning Star is Venus’, ‘is’ is obviously not the mere copula; its content is an essential part of the predicate, so that the word ‘Venus’ does not constitute the whole of the predicate. One might say instead: ‘The Morning Star is no other than Venus’; what was previously implicit in the single word ‘is’ is here set in four separate words, and in ‘is no other than’ the word ‘is’ now really is the mere copula. What is predicated here is thus not Venus but no other than Venus. These words stand for (bedeuteri) a concept; admitedly only one object falls under this, but such a concept must still always be distinguished from the object. We have here a word ‘Venus’ that can never be a proper predicate, although it can form part of a predicate. The Bedeutung of this word is thus something that can never incur as a concept, but only as an object.

—On Concept and Object, Gottlob Frege (Peter Gleach translation), 1892 AD

Dear colleague,

For a year and a half, I have been acquainted with your The Foundations of Arithmetic, but it is only now that I have been able to find the time for the thorough study I intended to make of your work. I find myself in complete agreement with you in all essentials, particularly when you reject any psychological element in logic and when you place a high value upon an ideography for the foundations of mathematics and of formal logic, which, incidentally, I find in your work discussions, distinctions, and definitions that one seeks in vain in the works of other logicians. Especially so far as function is concerned, I have been led on my own to views that are the same even in the details. There is just one point where I have encountered a difficulty. You state that a function, too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer, its opposite follows. Therefore, we must conclude that w is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection does not form a totality.

—Correspond with Gottlob Frege, Bertrand Russell

1900s

The universe consists of objects having various qualities and standing in various relations. Some of the objects which occur in the universe are complex. When an object is complex, it consists of interrelated parts. Let us consider a complex object composed of two parts a and b standing to each other in the relation R. The complex object “a-in-the-relation-R-to-b” may be capable of being perceived; when perceived, it is perceived as one object. Attention may show that it is complex; we then judge that a and b stand in the relation R. Such a judgement, being derived from perception by mere attention, may be called a “judgement of perception.” This judgement of perception, considered as an actual occurence, is a relation of four terms, namely a and b and R and the percipient. The percetpion, on the contrary, is a relation of two terms, namely “a-in-the-relation-R-to-b” and the percipient. Since an object of perception cannot be nothing, we cannot perceive “a-in-the-relation-R-to-b” unless a is in the relation R to b. Hence a judgement of perception, according to the above definition, must be true. This does not mean that, in a judgement which appears to us to be one of perception, we are sure of not being in error, since we may err in thinking that our judgement has really been derived merely by analysis of what was perceived. But if our judgement has been so derived, it must be true. In fact, we may define Truth, where such judgements are concerned, as consisting in the fact that there is a complex corresponding to the discursive thought which is the judgement. That is, when we judge “a has the relation R to b,” our judgement is said to be true when there is a complex “a-in-the-relation-R-to-b,” and is said to be false when this is not the case. This is a definition of Truth and falsehood in relation to judgements of this kind.

—Principia Mathematica, Bertrand Russell and Alfred Whitehead, 1910 AD

Vicious circles arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole. Thus, for example, the collection of propositions will be supposed to contain a proposition stating that “all propositions are either true or false.” It would seem, however, that such a statement could not be legitimate unless “all propositions” referred to some already definite collection, which it cannot do if new propositions are created by statements about “all propositions”. We shall, therefore, have to say that statements about “all propositions” are meaningless. More generally, given any set of objects such that, if we suppose the set to have a total, it will contain members which presuppose this total, then such a set cannot have a total. By saying that a set has “no total,” we mean, primarily, that no significant statement can be made about “all its members.”

• Principia Mathematica, Bertrand Russell and Alfred Whitehead, 1910 AD

Form is the possibility of structure.

—Tractatus Logico-Philosophicus, Ludwig Wittgenstein, 1921 AD

The Greek expression φαινόμενον, to which the term “phenomenon” goes back, is derived from the verb φαίνεσθαι which signifies “to show itself”. Thus φαινόμενον means that which shows itself, the manifest. φαίνεσθαι itself is a middle-voiced form which comes from φαίνω — to bring to the light of day, to put in the light. φαίνω comes from the stem φα — , like the light, that which is bright — in other words, that wherein something can become manifest, visible in itself. Thus we must keep in mind that the expression “phenomenon” signifies that which shows itself in itself the manifest. Accordingly the φαινόμενα or “phenomena” are the totality of what lies in the light of day or can be brought to the light — what the Greeks sometimes identified simply with τὰ ὄντα (entities). Now an entity can show itself from itself in many ways, depending in each case on the kind of access we have to it. Indeed it is even possible for an entity to show itself as something which in itself it is not. When it shows itself in this way, it “looks like something or other”. This kind of showing-itself is what we call “seeming”. Thus in Greek too the expression φαινόμενον (”phenomenon”) signifies that which looks like something, that which is “semblant”, “semblance”. φαινόμενον ὰγαθόν means something good which looks like, but “in actuality” is not, what it gives itself out to be. If we are to have any further understanding of the concept of phenomenon, everything depends on our seeing how what is designated in the first signification of φαινόμενον (”phenomenon” as that which shows itself) and what is designated in the second (”phenomenon” as semblance) are structurally interconnected. Only when the meaning of something is such that it makes a pretension of showing itself — that is, of being a phenomenon — can it show itself as something which it is not; only then can it “merely look like so-and-so”. When φαινόμενον signifies “semblance”, the primordial signification (the phenomenon as the manifest) is already included as that upon which the second signification is founded. We shall allot the term ‘phenomenon’ to this positive and primordial signification of φαινόμενον and distinguish “phenomenon” from “semblance”, which is the privative modification of “phenomenon” as thus defined. But what both these terms express has proximally nothing at all to do with what is called an ‘appearance’, or still less a ‘mere appearance’.

—Being and Time , Martin Heidegger, 1927

Dasein is an Seiendes which does not just occur among other Seienden. Rather it is ontically distinguished by the fact that, in its very Sein, that Sein is an issue for it. But in that case, this is a constitutive state of Dasein’s Sein, and this implies that Dasein, in its Sein, has a relationship towards that Sein— a relationship which itself is one of Sein. And this means further that there is some way in which Dasein understands itself in its Sein,, and that to some degree it does so explicitly. It is peculiar to this entity that with and through its Sein, this Sein, is disclosed to it. Understanding of Sein is itself a definite characteristic of Dasein’s Sein. Dasein is ontically distinctive in that it is ontological.

—Being and Time , Martin Heidegger, 1927

The Wesen of this entity lies in its Zu-sein , Its Was-sein (essentia) must, so far as we can speak of it at all, be conceived in terms of its Sein (existentia) . But here our ontological task is to show that when we choose to designate the Sein of this entity as “existence” (Existenz), this term does not and cannot have the ontological signification of the traditional term “existentia” ; ontologically, existentia is tantamount to Vorhandenheit, a kind of Sein which is essentially inappropriate to entities of Dasein ‘s character. To avoid getting bewildered, we shall always use the Interpretative expression “ Vorhandenheit “ for the term “existentia”, while the term “existence”, as a designation of Sein, will be allotted solely to Dasein.

The essence of Dasein lies in its existence. Accordingly those characteristics which can be exhibited in this entity are not ‘properties’ vorhanden of some entity which ‘looks’ so and so and is itself vorhanden; they are in each case possible ways for it to be, and no more than that. All the So-sein which this entity possesses is primarily Sein. So when we designate this entity with the term ‘ Dasein ‘, we are expressing not its “what” (as if it were a table, house or tree) but its Sein .

—Being and Time , Martin Heidegger, 1927

What is the nothing ? Our very first approach to this question has something unusual about it. In our asking we posit the nothing in advance as something that “is” such and such; we posit it as a being. But that is exactly what it is distinguished from. Interrogating the nothing–asking what and how it, the Nothing, is–turns what is interrogated into its opposite. The question deprives itself of its own object. Accordingly, every answer to this question is also impossible from the start. For it necessarily assumes the form, the nothing “is” this or that. With regard to the nothing, question and answer alike are inherently absurd.

—What Is Metaphysics?, Martin Heidegger, 1929 AD

Whenever we attempt to express the matter of immediate experience, we find that its understanding leads us beyond itself, to its contemporaries, to its past, to its future, and to the universals in terms of which its definiteness is exhibited. But such universals, by their very character of universality, embody the potentiality of other facts with varying types of definiteness. Thus the understanding of the immediate brute fact requires its metaphysical interpretation as an item in the world with some systematic relation to it. When thought comes upon the scene, it finds the interperations as matters of practice. Philosophy does not initiate interpretations. Its search for a rationalistic scheme is the search for more adequate criticism, and for more adequate justifications of the interpretations which we perforce employ. Our habitual experience is a complex of failure and success in the enterprise of interpretation. If we desire a record of uninterpreted experience, we must ask a stone to record its autobiography. Every scientific memoir in its records of the “facts” is shot through and through with interpretation. The methodology of rational interpretation is the product of the fitful vagueness of consciounsess. Elements which shine with immediate distinctness, in some circumstances, retire into pneumbral shadow in other circumstances, and into black darkness on other occasions. And yet all occasions proclaim themselves as actualities within the flux of a solid world, demanding a unity of interpretation.

—Process and Reality , Alfred Whitehead, 1929 AD

For the sake of greater perspicuity, we shall use the symbol ‘c’ as a typographical abbreviations of the expression ‘the sentence printed on this page, line 5 from the top’. Consider now the following sentence:

c is not a true sentence (Note: this sentence is typeset on line 5)

Having regard to the meaning of the symbol ‘c’, we can establish empirically:

1. ‘c is not a true sentence’ is identical with c

For the quotation-mark name of the sentence c (or for any other of its names), we can set up an explanation of type:

2. ‘c is not a true sentence’ is a true sentence if and only c is not a true sentence.

The premises 1 and 2 together at once give a contradiction:

c is a true sentence if and only if c is not a true sentence.

• The Concept of Truth in Formalized Languages, Alfred Tarski, 1931 AD

The main source of the difficulties met with seems to lie in the following: it has not always been kept in mind that the semantical concepts have a relative character, that they must always be related to a particular language. People have not been aware that the language about which we speak need by no means coincide with the language in which we speak. They have carried out the semantics of a language in that language itself and, generally speaking, they have proceeded as though there was only one language in the world. The analysis of the antimonies mentioned shows, on the contrary, that the semantical concepts simply have no place in the language to which they relate, that the language which contains its own semantics, and within which the usual logical laws hold, must inevitably be inconsistent.

—The Concept of Truth in Formalized Languages, Alfred Tarski, 1931 AD

For this reason, when we investigate the language of a formalized deductive science, we must always distinguish clearly between the language about which we speak and the language in which we speak, as well as between the science which is the object of our investigation and the science in which the investigation is being carried out. The names of the expressions of the first language, and of the relations between them, belong to the second language, called the meta language (which may contain the first as a part). The description of these expressions, the definition of the complicated concepts, especially of those connected with the construction of a deductive theory (like the concept of consequence, of provable sentence, possibly of true sentence), the determiniation of the properties of these concepts, is the task of the second theory which we shall call the metatheory.

—The Concept of Truth in Formalized Languages, Alfred Tarski, 1931 AD

To say what rules of grammar make up a propositional game would require giving the characteristics of propositions, their grammar. We are thus led to the question, What is a proposition? I shall not try to give a general definition of “proposition”, as it is impossible to do so. This is no more possible than it is to give a definition of the word “game”. For any line we might draw would be arbitrary. Our way of talking about propositions is always in terms of specific examples, for we cannot talk about these more generally than about specific games. We could begin by giving examples such as the proposition “There is a circle on the blackboard 2 inches from the top and 5 inches from the side”. Let us represent this as “(2,5)”. Now let us construct something that would be said to make no sense, “(2,5,7)”. This would have to be explained (and you could give it sense), or else you could say it is a mistake or a joke. But if you say it makes no sense, you can explain why by explaining the game in which it has no use. Nonsense can look less and less like a sentence, less and less like a part of language. “Goodness is red” and “Mr. S came to today’s redness” would be called nonsense, whereas we would never say a whistle was nonsense. An arrangement of chairs could be taken as a language, so that certain arrangements would be nonsense. Theoretically you could always say of a symbol that it makes sense, but if you did so you would be called upon to explain its sense, that is, to show the use you give it, how you operate with it. The words “nonsense’ and “sense” get their meaning only in particular cases and may vary from case to case. We can still talk of sense without giving a clear meaning to “sense”, just as we talk of winning or losing without the meaning of our terms being absolutely clear.

—Wittgenstein Lectures, Ludwig Wittgenstein, 1932 AD

Presence to self, on the contrary, supposes that an impalpable fissure has slipped into being. If being is present to itself, it is because it is not wholly itself. Presence is an immediate deterioration of coincidence, for it supposes separation. But if we ask ourselves at this point “what it is” which separates the subject from himself, we are forced to admit it is “nothing”. Ordinarily what separates is a distance in space, a lapse in time, a psychological difference, or simply the individuality of two co-presents–in short, a “qualified” reality. But in the case which concerns us, “nothing” can separate the consciousness of belief from belief, since belief is “nothing other” than the consciousness of belief.

—Being and Nothingness, Jean-Paul Sartre, 1943 AD

If you do know that “here is one hand”, we’ll grant you all the rest.

—On Certainty, Ludwig Wittgenstein, 1951 AD

Ask yourself whether our language is complete-—whether it was so before the symbolism of chemistry and the notation of the infinitesimal calculus were incorporated in it; for these are, so to speak, suburbs of our language. (And how many houses or streets does it take before a town begins to be a town?) Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular streets and uniform houses.

—Philosophical Investigations, Ludwig Wittenstein, 1953 AD

To imagine a language is to imagine a form of life.

—Philosophical Investigations, Ludwig Wittgenstein, 1953 AD

Naturally, because the void is indiscernible as a term (because it is not-one), its inaugural appearance is a pure act of nomination. This name cannot be specific; it cannot place the void under anything that would subsume it–this would reestablish the one. The name cannot indicate that the void is this or that. The act of nomination, being a-specific, consumes itself, indicating nothing other than the unpresentable as such. In ontology, however, the unpresentable occurs within a presentative forcing which disposes it as the nothing from which everything proceeds. The consequence is that the name of the void is a pure proper name, which indicates itself, which does not bestow any index of difference within what it refers to, and which auto-declares itself in the form of the multiple, despite there being nothing which is numbered by it.

—Being and Event, Alain Badiou, 1988 AD