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Kaina Stoicheia (Καινα στοιχεια) or “New Elements” is the title of several manuscript drafts of a document that Charles Sanders Peirce wrote circa 1904, intended as a preface to a book on the foundations of mathematics. It presents a consummate integration of his ideas on the interrelations of logic, mathematics, and semeiotic, or the theory of signs.
Logic is the study of the essential nature of signs.
A sign is something that exists in replicas. Whether the sign "it is raining" or "all pairs of particles of matter have component accelerations toward one another inversely proportional to the square of the distance" happens to have a replica in writing, in oral speech, or in silent thought, is a distinction of the very minutest interest to logic, which is a study, not of replicas, but of signs.
But this is not the only, nor the most serious error involved in making logic treat of "judgments" in place of propositions. It involves confounding two things which must be distinguished if a real comprehension of logic is to be attained.
A proposition, as I have just intimated, is not to be understood as the lingual expression of a judgment. It is, on the contrary, that sign of which the judgment is one replica and the lingual expression another. But a judgment is distinctly more than the mere mental replica of a proposition. It not merely expresses the proposition, but it goes further and accepts it.
I grant that the normal use of a proposition is to affirm it; and its chief logical properties relate to what would result in reference to its affirmation. It is, therefore, convenient in logic to express propositions in most cases in the indicative mood. But the proposition in the sentence, "Socrates est sapiens", strictly expressed, is "Socratem sapientum esse". The defence of this position is that in this way we distinguish between a proposition and the assertion of it; and without such distinction it is impossible to get a distinct notion of the nature of the proposition.
One and the same proposition may be affirmed, denied, judged, doubted, inwardly inquired into, put as a question, wished, asked for, effectively commanded, taught, or merely expressed, and does not thereby become a different proposition. What is the nature of these operations? The only one that need detain us is affirmation, including judgment, or affirmation to oneself.
As an aid in dissecting the constitution of affirmation I shall employ a certain logical magnifying-glass that I have often found efficient in such business. Imagine, then, that I write a proposition on a piece of paper, perhaps a number of times, simply as a calligraphic exercise. It is not likely to prove a dangerous amusement. But suppose I afterwards carry the paper before a notary public and make affidavit to its contents. That may prove to be a horse of another color. The reason is that this affidavit may be used to determine an assent to the proposition it contains in the minds of judge and jury; — an effect that the paper would not have had if I had not sworn to it. For certain penalties here and hereafter are attached to swearing to a false proposition; and consequently the fact that I have sworn to it will be taken as a negative index that it is not false. This assent in judge and jury's minds may effect in the minds of sheriff and posse a determination to an act of force to the detriment of some innocent man's liberty or property. Now certain ideas of justice and good order are so powerful that the ultimate result may be very bad for me.
This is the way that affirmation looks under the microscope; for the only difference between swearing to a proposition and an ordinary affirmation of it, such as logic contemplates, is that in the latter case the penalties are less and even less certain than those of the law. The reason there are any penalties is, as before, that the affirmation may determine a judgment to the same effect in the mind of the interpreter to his cost. It cannot be that the sole cause of his believing it is that there are such penalties, since two events cannot cause one another, unless they are simultaneous. There must have been, and we well know that there is, a sort of hypnotic disposition to believe what one is told with an air [of] command. It is Grimes's credenciveness, which is the essence of hypnotism. This disposition produced belief; belief produced the penalties; and the knowledge of these strengthens the disposition to believe.
C.S. Peirce, [“Kaina Stoicheia”], NEM 4, 248–249
Peirce, C.S. (c. 1904), Καινα στοιχεια (“New Elements”), MS 517, pp. 235–263 in Carolyn Eisele (ed.), The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy, Mouton, The Hague, 1976. Cf. “New Elements”, pp. 300–324 in The Essential Peirce, Volume 2 (1893–1913), Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1998. Online.
- Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
- Peirce, C.S, The New Elements of Mathematics by Charles S. Peirce, 4 volumes in 5, Carolyn Eisele (ed.), Mouton Publishers, The Hague, Netherlands, 1976. Humanities Press, Atlantic Highlands, NJ, 1976.
- Peirce, C.S., The Essential Peirce, Selected Philosophical Writings, Volume 1 (1867–1893), Nathan Houser and Christian Kloesel (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1992.
- Peirce, C.S., The Essential Peirce, Selected Philosophical Writings, Volume 2 (1893–1913), Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1998.