Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes in Postulate 2 that a straight-line segment can be extended as far as necessary; he uses this fact, but only to find a larger finite length for example in Book I, **Propositions** 11, 16, and 20. For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension.

— Morris Kline,

**Propositions** show what they say: tautologies and contradictions show that they say nothing.

— Ludwig Wittgenstein, in Joachim Schulte

The central **Propositions** [of Descartes]?are these: There is a path
that leads to the truth so surely that any one who will follow
it must needs reach the goal? And there is one
guiding rule by which a man mayalways find this path?give
unqualified assent to no **Propositions** but those the truth
of which is so clear and distinct that they cannot be
doubted.

— 1870 Lay Sermons, Addresses, and Reviews.

If an angel were ever to tell us anything of his philosophy I believe many
**Propositions** would sound like 2 times 2 equals13.

— c.1770 Aphorisms, Notebook B (translated by R J Hollingdale, 1990).

Literature is based not on life but on **Propositions** about life, of which this
is one.

— 1957 Opus Posthumous, Aphorisms,'Adagia'.

To make our position clearer, we may formulate it in another way. Let us call a proposition which records an actual or possible observation an experiential proposition. Then we may say that it is the mark of a genuine factual proposition, not that it should be equivalent to an experiential proposition, or any finite number of experiential **Propositions**, but simply that some experiential **Propositions** can be deduced from it in conjunction with certain other premises without being deducible from those other premises alone.

— p. 20 (Language, Truth, and Logic (1936))

The principles of logic and mathematics are true simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and mathematics are analytic **Propositions** or tautologies.

— p. 77 (Language, Truth, and Logic (1936))

I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such **Propositions**, which one could neither prove nor dispose of.

— A reply to Olbers' 1816 attempt to entice him to work on Fermat's Theorem. As quoted in

A great part of its theories derives an additional charm from the peculiarity that important **Propositions**, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.

— On higher arithmetic.

I was trying to show colour, but I realized at the private view that the public were prisoners of a preconceived point of view and that, confronted with all these surfaces of different colours, they responded far more to the inter-relationship of the different **Propositions**, they reconstituted the elements of a decorative polychromy.

— 1956; in: "Yves Klein, 1928 – 1962, Selected Writings", ed. J & J, the Tate Gallery, London 1974, p. 30

A theorem is a proposition which is a strict logical consequence of certain definitions and other **Propositions**

— Rapoport, Anatol. "Various meanings of “theory”."

LOGIC :; The principle governing human intellection. Its nature may be deduced from examining the following **Propositions**, both of which are held by human beings to be true and often by the same people: “I can’t so you musn’t,” and “I can but you musn’t.”

— the happening world (15) “Equal and Opposite”

Notwithstanding their attacks on the basic conception of rationalism, on synthetic a priori judgments, that is, material **Propositions** that cannot be contradicted by any experience, the empiricist posits the forms of being as constant.

— p. 146 ("The Latest Attack on Metaphysics" (1937))

Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional **Propositions** a priori, and it will be your task to find out the false deduction in each case.

—

The progressive historical role of capitalism may be summed up in two brief **Propositions**: increase in the productive forces of social labour, and the socialisation of that labour. But both these facts manifest themselves in extremely diverse processes in different branches of the national economy.

— Lenin, Vladimir Ilich, "The Development of Capitalism in Russia, “The Mission of Capitalism” (1899)", Marxists .

In obedience to the feeling of reality, we shall insist that, in the analysis of **Propositions**, nothing "unreal" is to be admitted. But, after all, if there is nothing unreal, how, it may be asked, could we admit anything unreal? The reply is that, in dealing with **Propositions**, we are dealing in the first instance with symbols, and if we attribute significance to groups of symbols which have no significance, we shall fall into the error of admitting unrealities, in the only sense in which this is possible, namely, as objects described.

— Ch. 16: Descriptions.

The last word remains with Christ and Handel ; and this must stand as the best defence of Tolerance until a better man than I makes a better job of it. Put shortly and undramatically the case is that a civilization cannot progress without criticism, and must therefore, to save itself from stagnation and putrefaction, declare impunity for criticism. This means impunity not only for **Propositions** which, however novel, seem interesting, statesmanlike, and respectable, but for **Propositions** that shock the uncritical as obscene, seditious, blasphemous, heretical, and revolutionary.

— Preface, The Sacredness Of Criticism

Life consists Of **Propositions** about life. The human Revery is a solitude in which We compose these **Propositions**, torn by dreams, By the terrible incantations of defeats And by the fear that the defeats and the dreams are one. The whole race is a poet that writes down The eccentric **Propositions** of its fate.

— "Men Made Out of Words"

There are numerous theorems in economics that rely upon mathematically fallacious **Propositions**.

— Chapter 12, Don't Shoot Me, I'm Only The Piano, p. 259

No doubt, there are those who believe that judges-and particularly dissenting judges-write to hear themselves say, as it were, I I I. And no doubt, there are also those who believe that judges are, like Joan Didion, primarily engaged in the writing of fiction. I cannot agree with either of those **Propositions**.

— William J. Brennan, Jr.,

Intuitive cognition is such that when some things are cognized, of which one inheres in the other, or one is spatially distant from the other, or exists in some relation to the other, immediately in virtue of that non-propositional cognition of those things, it is known if the thing inheres or does not inhere, if it is spatially distant or not, and the same for other true contingent **Propositions**, unless that cognition is flawed or there is some impediment.

—

It is as easy to count atomies as to resolve the **Propositions** of a lover.

— Act III, scene 2, line 245.

A great part of its theories derives an additional charm from the peculiarity that important **Propositions**, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.

— Carl Friedrich Gauss On higher arithmetic.