must begin with the utilisation of the laws of deductive, for-mal logic.
2.21 By “formal logic” we mean the deductive method of testing in-ferences constructed from the law of contradiction (the con-stant ~ ), together with any one of the following concon-stants: V (disjunction), • (conjunction), ⊃ (implication).
2.211 The primitives V, •, ⊃ are relatively primitive, since they are in-ter-definable; the contradictory function, however, is abso-lutely primitive, since it cannot be defined by means of any selection of V, •, ⊃; the law of contradiction is thus the funda-ment of deductive logic.
2.212 Whether Whitehead and Russell, in their Principia Mathemat-ica, are correct that mathematics is a special case of deductive logic, or Gödel is right, by way of his incompleteness theorem, that one cannot derive mathematics from logic since a formal axiomatic system only partially characterises the concepts ex-pressed therein, the deductive affinities between logic and mathematics are so close that we may refer to formal logic also as mathematical logic.
2.22 In any case, the word “logic” properly has no plural.
2.221 “Sometimes people suppose that ‘symbolic’ logic and so-called
‘classical’ or Aristotelian logic differ from each other in the sense of being logics that are not identical with respect to their subject matters. This, however, is completely mistaken. There is only one subject matter of both ‘symbolic’ and ‘classical’
logic, namely, formal concepts. Aside from the use of a nota-tion which allows for both precise expression and ease in de-duction, the only difference between the ‘logics’ consists in the more accurate and extended analyses symbolic logicians have made of their common subject matter” (Ambrose and Lazerow-itz).
2.2211 To denigrate logic as “Aristotelian”—and to imply that the Protestant Reformers, in opposing medieval Aristotelian scho-lasticism, depreciate formal logic because they reject Aristo-tle’s metaphysics—is absurd.
2.22111 Aristotle was the first thinker in the West to provide a system-atic classification of zoological phenomena (Historia Animalium, etc.); should we therefore reject taxonomy as
“Aristotelian?”
2.222 Non-Western “logics” are in reality varieties of that universal system of inference based on the law of contradiction which we in the West designate as “classical” or “symbolic” logic.
2.2221 This is true even for “Indian logic” which developed in a reli-gious atmosphere where paradox and the conjunctio opposito-rum were basic motifs: “The most interesting thing about this [Indian] variety of logic is that in quite different circumstances and without being influenced by the West, it developed in many respects the same problems and reached the same solu-tions. Examples are the syllogism of the Tarka-Samgraha and Mathurãnãtha’s definition of number” (Bochenski).
2.223 Hegel’s “dialectic logic,” while the necessary base for his own metaphysic, for the Marxist interpretation of history, for all process philosophies and theologies, and for an influential death-of-God position (Altizer), is really no “logic” at all.
2.2231 If, as Hegel supposes, truth and falsity are not sharply defined
scious certainty of being all reality,” then the necessity of knowing everything before understanding anything would pre-clude the possibility of knowledge as such.
2.2232 “Hegel thought that, if enough was known about a thing to dis-tinguish it from all other things, then all its properties could be inferred by logic. This was a mistake, and from this mistake arose the whole imposing edifice of his system. This illustrates an important truth, namely, that the worse your logic, the more interesting the consequences to which it gives rise” (Russell).
2.2233 Hegel was wrong to assert that logic is the same thing as meta-physics, but he was quite right in identifying his own “logic”
with his metaphysics.
2.23 Why should we commit ourselves to the primitives of formal logic, thereby presupposing the validity of its inferential sys-tem?
2.231 Not because a “revelation” establishes or employs the laws of logic; for without the prior acceptance of the law of contradic-tion we would not be able to understand what a “revelacontradic-tion”
was saying, much less distinguish a genuine revelation from a pseudo-revelation.
2.2312 Apart from the law of contradiction, a revelation would in prin-ciple be impossible, for “revelation” would blend with “ob-scuration” and mankind would remain in darkness as before.
2.232 Employment of formal logic is justified by the best of all “rea-sons”: sheer necessity.
2.2321 To reject the law of contradiction is to destroy the possibility of knowledge as such, since from a tautology or a contradiction anything whatever follows.
2.2322 To argue against formal logic is to employ formal logic al-ready.
2.2323 Of logic one must say what Emerson said of Brahma: “When me they fly, I am the wings.”
2.23231 In the present work we have thus been airborne on the wings of logic since proposition 1; owing to the necessitarian character of the flight, we saw no need to describe the craft until now.
2.2324 “Why should I be logical?” You needn’t be, if that is your pref-erence; no police force exists to enforce the laws of logic. A man is permitted to be as nonsensical as he wishes; all he loses is the ability to communicate with his fellows and the possibil-ity of discovering truth.
2.23241 If, however, one’s refusal to employ the laws of logic is carried out consistently in society, the laws of the land concerning in-sanity (however primitive they may be) will officially separate the individual from the society in which he can no longer mean-ingfully function.
2.2325 Strictly speaking, the question, “How can I know that the laws of logic are valid?” has no meaning, since an answer to it would require the existence and employment of a higher-level logic (L2) to justify mathematical logic as we know it (L); L2 would then require justification by L3, and so on. Generalizing, Ln+1 would always have to be appealed to to justify Ln –to the limit of L∞-l needing justification by L∞. But such an infinite series, even if it existed (and we could not know if it did), would still stand without justification of its last term and thus without grounding for any of its other terms.
2.2326 If worlds exist in which other “logics” prevail, we cannot, qua human beings, know their characteristics, though we might on evidential grounds come to know of their existence.
2.23261 Wittgenstein is right that “we could not say what an ‘illogical’
world would look like,” but this does not limit God to the cre-ation of worlds in which formal logic as we know it is norma-tive.
2.24 Deductive logic is entirely formal, i.e., it does not instruct us as to the content of the world, but shows us how validly to inter-relate the facts of the world.
2.2401 To utilize logic, then, is not to commit oneself in any way reli-giously; it is merely to admit that one is a human being.
2.241 In another sense, however, logic does tell us something about the world: “The propositions of logic describe the scaffolding of the world, or rather they represent it. They have no ‘subject matter.’ It is clear that something about the world must be
indi-essence involves the possession of a determinate character—
are tautologies. This contains the decisive point” (Wittgen-stein).
2.25 The assertions of formal logic, like the propositions of pure mathematics, are absolutely certain; but their certainty, rather than stemming from an accord with the facts of the world, de-rives from their tautological character—from their analytical separation from the world of fact.
2.26 To argue, then, as did the Deists of the 18th century “Enlight-enment,” that one’s religious position follows from Reason alone, is to misunderstand the character of logic.
2.261 If one’s religion did actually derive from logic alone, it would be perfectly certain; but it would gain such certainty at the ex-pense of losing all factual content.
2.262 Logic cannot prescribe the content of religious truth; it can only tell you that if A is true, then it has implications B, C, D, ... N, and that ~A, ~B, ~C, ... ~N must be rejected. But, in performing this analytical function, logic renders an invaluable service.
2.263 Those who claim that their religion follows from logic or rea-son are really saying that the content of their religious beliefs seems logical or reasonable to them; but whether their religious views are indeed consistent with the laws of logic must be as-certained, and the content of their convictions must be exam-ined against the facts of the world.
2.3 Deductive logic cannot pass judgment on the factual nature