User:Wvbailey/Law of Excluded Middle

This page is User:Wvbailey's holding place for background information about the LoEM (Law of Excluded Middle). Also to be found here will be background about the so-called Laws of Thought.

General definitions

Hamilton 1860:59-60

Par. XVI. Law of Excluded Middle.

¶ XVI. The principle of Excluded Third or Middle – viz., between two contradictories (principium Exclusi Medii vel Tertii') enounes that condition of thought which compels us, of two repugnant notions, which cannot both coëxist, to think either the one or the other as existing. Hence arises the general axiom -- Of contradictory attributions, we can only affirm one of a thing; and if one be explicitly affirmed, the other is implicitly denied. A either is or is not. A either is or is not B.⁴

By the laws of Identity and Contradiction, I am warranted to

⁴ See Schulze, Logik, §19.-ED.

conclude from the truth of one contradictory proposition to the falsehood of the other, and by the law of Excluded Middle, I am warranted to conclude from the falsehood of one contradictory proposition to the truth of the other.

Logical significance of this law: And in this lies the peculiar force and import of this last principle. For the logical significance of the law of Excluded Mliddle consists in this, that it limits or shuts in the sphere of the thinkable in relation to affirmation; for it determines, that, of the two forms given in the laws of Identity and Contradiction, and by these laws affirmed as those exclusively possible, the one or the other must be affirmed as necessary.

The principle of Disjunctive Judgments: The law of Excluded Middle is the principle of Disjunctive Judgments, that is, of judgments in which a plurality of judgments are contained, and which stand such a reciprocal relation that the affirmation of one is the denial of the other.

Plato, Strobraeus

from Hamilton LECT. V. LOGIC. 62:

The principles of Contradiction and Excluded Middle can be traced back to Plato: The principles of Contradiction and of Excluded Middle can both be traced back to Plato, by whom they were enounced and frequently applied; though it was not till long after, that either of them obtained a distinctive appellation. To take the principle of Contradiction first. This law Plato frequently employs, but the most remarkable passages are found in the Phœdo, in the Sophista, and in the fourth and seventh books of the Republic.²

² See Phœdo, p. 103; Sophista, p.252; Republic, iv. p. 436; vii. p. 525. – ED.

from Hamilton LECT. V. LOGIC. 65:

Law of Excluded Middle: The law of Excluded Middle between two contradictories remounts, as I have said, also to Plato, though the Second Alcibiades, the dialogue in which it is most clearly expressed, must be admitted to be spurious.¹ It is also in the fragments of Pseudo-Archytas, to be found in Stobraeus.²

¹ Second Alcibiades, p. 139. See also Sophista, p. 250 – ED.

² Eclogœ.. 1. ii. c. 2, p. 158, Ed. Antwerp, 1575; Part ii. tom. 1, p. 22, ed. Heeren. Cf. Simplicius, In Arist. Categ., pp. 97, 103, ed. Basil, 1551. –ED.

---

from GBWW Dialogues of Plato:

• I. Plato: Euthydemus, pp: 72d-73b
• II. Republic, BK IV, pp: 350d-351b

I. Euthydemus p. 72d - 73a: This has to do with the existance, and nature of, contradiction and also the existence of falsehood, as opposed to the teachings of Dionysodurus and Protagoras.

"[Ctesippus is arguing with Dionysodorus, Socrates is narrating:] ... And here is Dionysodorus fancying that I am angry with him, when really I am not angry at all; I do but contradict him when I think that he is speaking imporoperly to me: and you must not confound abuse and contradiction, O illustrious Dionysodurus; for they are quite different things. / Contradiction! said Dionysodorus; why, there never was such a thing. / Certainly there is, he replied; there can be not question of that. Do you, Dionysodorus, maintain that there is not? / You will never prove to me, he said, that you have heard any one contradicting any on else. / Indeed, said Ctesippus; then now you may hear me contradicting Dionysodours. / Are you preparted to make that good? / Certainly, he said. / Well, have not all things words expressive of them? / Yes. / Of their existence and their non-existence? / Of their existence. / Yes, Ctesippus, and we just now proved, as you may remember, that no man could affirm a negative; for no one could affirm that which is not. / And what does that signify? said Ctesippus; you and I may contradict all the same for that. / But can we contradict one another, said Dionysodours, when both of us are describing the same thing? Then we must surely be speaking the same thing? Or when neither of us is speaking of the same thing? For then neither of us says a word about the thing at all? / [etc] / Here Ctesippus was silent; and I in my astonishment said: What do you mean Dionysodours? I have often heard, and have been amazed to hear, this thesis of yours, which is maintained and employed by the disciples of Protagoras, and others before them, and which to me appears to be quite wonderful, and sucidal as well as destructive, and I think that I am most likely to hear the truth about it from you. The dictum is that there is no such thing as a falsehood; a man must either say what is true or say nothing. Is not that your position? / He assented. / But if he cannot speak falsely, may he not think falsely? / No, he cannot, he said. / Then there is no such thing as false opinion? / No, he said. / Then there is no such thing as ignorance, or men who are ignorant: for is not ignorance, if there be such a thing, a mistake of fact? / Certainly, he said. / And that is impossible? / Impossible, he replied. / Are you saying this as a paradox, Dionysodorus; or do you seriously maintain no man to be ignorant? / Refute me, he said. But how can I refute you, if, as you say, to tell a falsehood is impossible? / Very true, said Euthydemus. / Neither did I tell you just now to refute me, said Dionysodours; for how can I tell you to do that which is not? [etc]"

II. The Republic IV after lines 436, pages 350d-351b. An expression of the law of (non-)contradiction--

"The same thing clearly cannot act or be acted upon in the same part or in relation to the same thing at the same time, in contrary ways; and therefore whenever this contradiction occurs in things apparently the same, we know that they are really not the same, but different. / Good / For example, I said, can the same thing be at rest and in motion a the same time in the same part? / Impossible. / [examples of things apparently at motion and at rest simultaneiously, such as a spinning top] / Then none of these objections will confuse us, or incline us to believe that the same thing at the same time, [437] in the same part or in relation to the same thing, can act or be acted upon in contrary ways. / Certainly not, according to my way of thinking. / [etc]"

Aristotle

from Hamilton LECT. V. LOGIC. 65, re Law of Excluded Middle:

Explicitly enounced by Aristotle: It is explicitly and emphatically enounced by Aristotle in many passages both of his Metaphysics (l. iii. (iv.) c.7.) and of his Analytics, both Prior (l. i. c. 2) and Posterior (1. i. c. 4). In the first of these, he says: "It is impossible that there should exist any medium between contradictory opposites, but it is necessary either to affirm or to deny everything of everything." And his expressions are similar in the other books.

GBWW Metaphysics Book IV, CH 3-8 pp.524b-532d; BK XI, CH1 [1059a 23-24] 587a; CH 4-6 589d-592b:

Cicero

from Hamilton LECT. V. LOGIC. 65, re Law of Excluded Middle:

Cicero: Cicero says" that the foundation of Dialectic is, that whatever is enounced is either true or false." This is from his Academics (1. ii. c. xxix.), and there are parallel passages in his Topics (c. xiv.) and his De Oratore (I. ii. c. XXL).

Middle Ages

from Hamilton LECT. V. LOGIC. 65, re Law of Excluded Middle:

This law, though universally recognized as a principle in the Greek Peripatetic school, and in the schools of the middle ages, only received the distinctive appellation by which it is now known at a comparatively modern date.³

³ Lex contradictoriarum, principium contradicentium (sc. propositionum), as used in the schools, included the law of Contradiction and the law of Excluded Middle. See Molinaeus Elementa Logica, 1. ii. c. 14, [p. 172, ed. 1608. “Contradicentium usus explicatur uno axiomate: -- Contradicentia non possunt de eodem simul esse vera; et necessarium est contradicentium alterum cuilibet rei convenire, alterum non convenire.” – ED.]

Thomas Aquinas

The following speaks only to the Law of (Non-)Contradiction and tacitly to the Law of "being" and self-evidence. "Truth" is treated in article 4.

From GBWW Thomas Aquinas Vol. 2: Question XCIV OF THE NATURAL LAW (in Six Articles)

The six articles are: (1) What is the natural law? (2) What are the precepts of the natural law? (3) Whether all acts of virtue are prescribed by the natural law? (4) Whether the natural law is the same in all? (5) Whether it is changeable? (6) Whether it can be abolished from the heart of man.

The pertainent article is Article 2: Whether the Natural Law Contains Severl Precepts, or One Only? pages 221-222:

"The precepts of the natural law in man stand in relation to practical matters as the first principles to matters of demonstration. But there are several indemonstrable principles. Therefore there are also several precepts of the natural law. ... Now a thing is said to be self-evident in two ways: first, in itself; secondly in relation to us. Any propostion is said to be self-evident in itself if its predicate is contained in the notion of the subject, although one who does not know the definjition of the subject it happens that such a propostion is not self-evident. ... Hence it is as Boetheus says (De Hebdom.),¹ certain axioms or propostions are universally self-evident to all; and such are those propositions whose terms are know to all, as, Every whole is greater than its part, and, Things equal to one and the same are equal to one another. But some propostions are self-evident only to the wise, who understand the meaning of terms of such propositions [etc]...

"Now a certain order is to be found in those things that are apprehended by man. For that which, before anything else falls under apprehension, is being, the understanding of which is included in all things whatsoever a man apprehends. Therefore the sfirst indemonstrable principle is that the same thing cannot be affirmed and denied at the same time, which is based on the notion of being and not-being; and on this principle all others are based, as is stated in the Metaphysics². Now as being is the first thing that falls under the apprehension absolutely, so good is the first thing that falls under the apprehension of the practical reason [etc]"

¹ 1 PL 64, 1311

² Aristotle, IV, 3 (1005^b 29)

Hamilton lectures: ON THE FUNDAMENTAL LAWS OF THOUGHT-THEIR CONTENTS AND HISTORY (1830's)

William Hamilton (Henry L. Mansel and John Veitch, ed.), 1860 Lectures on Metaphysics and Logic, in Two Volumes. Vol. II. Logic, Boston: Gould and Lincoln.

Hamilton died in 1856, so this is an effort of his editors Mansel and Veitch. Most of the footnotes are additions and emendations by Mansel and Veitch -- see the preface for background information.

PREFACE

THE Lectures comprised in the present Volume form the second and concluding portion of the Biennial Course on Metaphysics and Logic, which was commenced by Sir William Hamilton on his election to the Professorial Chair in 1836, and repeated, with but slight alterations, till his decease in 1856. The Appendix contains various papers, composed for the most part during this period, which, though portions of their contents were publicly taught at least as early as 1840, were only to a very small extent incorporated into the text of the Lectures. The Lectures on Logic, like those on Metaphysics, were chiefly composed during. the session in which they were first delivered (1837-8); and the statements made in the Preface to the previous volume, as regards the circumstances and manner of their composition, are equally applicable to the present course. In this, as in the preceding series, the Author has largely availed himself of the labors of previous writers, many of whom are but little known in this country. To the works of the German logicians of the present century, particularly to those of Krug and Esser, these Lectures are under special obligations.

In the compilation of the Appendix, some responsibility rests with the Editors; and a few words of explanation may be necessary as regards the manner in which they have attempted to perform this portion of their task. In publishing the papers of. a deceased writer, composed at various intervals during a long period of years, and treating of difficult and controverted questions, there are two opposite dangers to be guarded against. On the one hand, there is the danger of compromising the Author's reputation by the publication of documents which his maturer judgment might not have sanctioned; and, on the other hand, there is the danger of committing an opposite injury to him and to the public, by withholding writings of interest and value. Had Sir William Hamilton, at any period of his life, published a systematic treatise on . Logic, or had his projected New Analytic of Logical Forms been left in a state at all approaching to completeness, the Editors might probably have obtained a criterion by which to distinguish between those speculations which would have received the final imprimatur of their Author, and those which would not. In the absence of any such criterion, they have thought it better to run the risk of giving too much than too little; - to publish whatever appeared to have any philosophical or historical interest, without being influenced by its coincidence with their own opinions, or by its coherence with other parts of the Author's writings. It is possible that, among the papers thus published, may be found some which are to be considered rather as experimental exercises than as approved results; but no papers have been intentionally omitted, except such as were either too fragmentary to be intelligible; or manifestly imperfect sketches of what has been published here or elsewhere in a more matured form.

The Notes, in this as in the previous volume, are divided into three classes. Those printed from the manuscript of the Lectures appear without any distinctive mark; those supplied from the Author's Commonplace- Book and other papers are enclosed within square brackets without signature; and those added by the Editors are marked by the signature" ED." These last, as in the Lectures on Metaphysics, are chiefly confined to occasional explanations of the text and verifications of references. In conclusion, the Editors desire to express their acknowledgments to those friends from whom they have received assistance in tracing the numerous quotations and allusions scattered. through this and the preceding volume.

In particular, their thanks are due to Hubert Hamilton, Esq., whose researches among his father's books and papers have supplied them with many valuable materials; and to H. W. Chandler, Esq., Fellow of Pembroke College, Oxford, who has aided them from the resources of a philosophical learning cognate in many respects to that of Sir William Hamilton himself.∃∀ℵ

LECTURE V. PURE LOGIC. PART I. - STOICHEIOLOGY. SECTION I. NOETIC. - ON THE FUNDAMENTAL LAWS OF THOUGHT-THEIR CONTENTS AND HISTORY.

LECT. V. LOGIC. p. 52

HAVING terminated our consideration of the various questions of which the Introduction to Logic is composed, we proceed to the doctrines which make up the science itself; and commence the First Great Division of PURE LOGIC -- that which treats of its elementary or constituent processes, Stoicheiology. But Stoicheiology was again divided into two parts, -- into a part which considered the Fundamental Laws of Thought in general, and into a part which considered these laws as applied to and regulating the special function of Thought in its various gradations of Conception, Judgment, and Reasoning. The title, therefore, of the part of Logic on which we are about to enter is, Pure Logic, Part I. Stoicheiology - Section I Noetic. On the Fundamental Laws of Thought.

The character of Thought in general: Before, however, descending to the consideration of these laws, it is necessary to make one or two preliminary statements touching the character of that thought of which they are the necessary conditions; and, on this point, I give, in the first place, the following paragraph:

¶ X. Logic considers Thought, not as the operation of thinking, but as its product; it does not treat of Conception, Judgment, and Reasoning, but of Concepts, Judgments, and Reasonings.

Thought as the object of Logic: I have already endeavored to give you a general knowledge of what is meant by thought. You are aware that this term is, in relation to Logic, employed in its strictest and most limited signification, -- viz., as the act or product of the Discursive Faculty, or Faculty of

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Relations; but it is now proper to consider, somewhat more closely, the determinate nature of this process, and the special point of view in which it is regarded by the logician.

The subject, form, and matter of thought: In an act of thinking, there are three things which we can discriminate in consciousness, - 1°, There is the thinking subject, that is, the mind or ego, which exerts or manifests the thought; 2°, There is the object about which we think, which is called the matter of thought; and, 3°, There is a relation between subject and object of which we are conscious, -- a relation always manifested in some determinate mode or manner; -- this is the form of thought.

Thought as the object respecitvely of Psychology and of Logic: Now, of these three, Logic does not consider either the first or the second. It takes no account, at least no direct account, of the real subject, or of the real object, of thought, but is limited exclusively to the form of thought. This has been already stated. But, again, this form of thought is considered by Logic only in a certain aspect. The form of thought may be viewed on two sides or in two relations. It holds, as has been said, a relation both to its subject and to its object, and it may accordingly be viewed either in the one of these relations or in the other. In so far as the form of thought is considered in reference to the thinking mind,- to the mind by which it is exerted,- it is considered as an act, or operation, or energy; and in this relation it belongs to Phaenomenal Psychology. Whereas, in so far as this form is considered in reference to what thought is about, it is considered as the product of such an act, and, in this relation, it belongs to Logic. Thus Phaenomenal Psychology treats of thought proper as conception, judgment, reasoning; Logic, or the Nomology of the understanding, treats of thought proper as a concept, as a judgment, as a reasoning. Whately, I have already shown you, among other errors in his determination of the object-matter of Logic, confounds or reverses this; for he proposes to Logic, not thought considered as a product, but reasoning alone; and that, too, considered as a producing operation. He thus confounds Logic with Phaenomenal Psychology.

Be it, therefore, observed, that Logic, in treating of the formal laws of thought, treats of these in reference to thought considered as a product; that is, as a concept, a judgment, a reasoning; whereas Psychology, as the Phaenomenology of mind, considers thought as the producing act, that is, as conception, judgment, reasoning. (You here see, by the way, the utility of distinguishing concept and conception. It is unfortunate that we cannot also distinguish more

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precisely judgment and reasoning as producing acts, from a judgment and a reasoning as products.)

Thought a mediate and complex cognition: ¶ XI. Thought, as the knowledge of one thing in relation to another, is a mediate and complex cognition.

Explication: The distinctive peculiarity of thinking in general is, that it involves the cognition of one thing by the cognition of another. All thinking is, therefore, a mediate cognition; and is thus distinguished from our knowledge in perception, external and internal, and in imagination; in both of which acts we are immediately cognitive of the object, external or internal, presented in the one, and of the object, external or internal represented in the other. In the Presentative and Representative Faculties, our knowledge is of something considered directly and in itself; in thought, on the contrary, we know one object only through the knowledge of another. Thus in perception, of either kind, and in imagination, the object known is always a single determinate object; whereas in thought, - in thought proper, - as one object is only known through another, there must always be a plurality of objects in every single thought. Let us take an example of this in regard to the simplest act of thought. When I see an individual, -- say Bucephalus or Highflyer, -- or when I represent him in imagination, I have a direct and immediate apprehension of a certain object in and through itself, without reference to aught else. But when I pronounce the term Horse, I am unable either to perceive in nature, or to represent in imagination, any one determinate object corresponding to the word. I obtain the notion corresponding to this word, only as the result of a comparison of many perceptions or imaginations of Bucephalus, Highflyer, Dobbin, and other individual horses; it, therefore, contains many representations under it, has reference to many objects, out of relation to which it cannot possibly be realized in thought; and it is in consequence of this necessity of representing (potentially at least) a plurality of individual objects under the notion horse, that it obtains the denomination concept, that is, something taken up or apprehended in connection with something else. This, however, requires a further explication. When we perform an act of thought, of positive thought, this is done by thinking something, and we can think anything only by thinking it as existing; while, again, we cannot think a thing to exist except in certain determinate modes of existence. On the other hand, when we perform an act of negative thought, this is

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done by thinking something as not existing in this or that determinate mode, and when we think it as existing in no determinate mode, we cease to think it at all; it becomes a nothing, a logical nonentity (non-ens Logicum).

It being thus understood that thought can only be realized by thinking something; it being further understood that this something, as it is thought, must be thought as existing; and it being still further understood that we can think a thing as existing only by thinking it as existing in this, that, and the other determinate manner of existence, and that whenever we cease to think something, something existing, something existing in a determinate manner of existence, we cease to think at all; -- this, I say, being understood, it is here proper to make you, once for all, acquainted with the various terms by which logicians designate the modes or manners of cogitable existence. I shall therefore comprise these in the following paragraph:

The various terms by which the modes of cogitable existence are designated: ¶ XII. When we think a thing, this is done by conceiving it as possessed of certain modes of being, or qualities, and the sum of these qualities constitutes its concept or notion (Greek words, conceptum, conceptus, notio). As these qualities or modes (Greek words, qualitates, modi) are only identified with the thing by a mental attribution, they are called attributes (Greek word, attributa); as it is only in or through them that we say or enounce aught of a thing, they are called predicates, predicables, and predicaments, or categories, these words being here used in their more extensive signification (Greek words, proedicata, proedicabilia, proedicamenta); as it is only in and through them that we recognize a thing for what it is, they are called notes, signs, marks, characters (notoe, signa, characteres, discrimina); finally, as it is only in and through them that we become aware that a thing is possessed of a peculiar and determinate existence, they are called properties, differences, determinations (proprietates, determinationes). As consequent on, or resulting from, the existence of a thing, they have likewise obtained the name of consequent (Greek word, consequentia, etc.). What in reality has no qualities, has no existence in thought, -- it is a logical nonentity; hence, e converso, the scholastio aphorism, -- non-entis nulla sunt proedicata. What, again, has no qualities attributed

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to it, though attributable, is said to be indetermined (Greek word, indeterminatum); it is only a possible object of thought.¹

Explication. what is involved in thinking an object.: This paragraph, which I have dictated that you might be made once for all acquainted with the relative terms in use among logicians, requires but little explacation. I may state, however, that the mind only thinks an object by separating it from others; that is, by marking it out or characterizing it; and in so far as it does this, it encloses it within certain fixed limits, that is, determines it. But if this discriminative act be expressed in words, I predicate the marks, notes, characters, or determinations of the thing; and if, again, these be comprehended in one total thought, they constitute its concept or notion. If, for example, I think of Socrates as son of Sophroniscus, as Athenian, as philosopher, as pug-nosed, these are only so many characters, limitations, or determinations, which I predicate of Socrates, which distinguish him from all other men, and together make up my notion or concept of him.

The attribution involved in thought is regulated by laws: But as thought, in all its gradations of conception, judgment, and reasoning, is only realized by the attribution of certain qualities or characters to the objects of, or about which we think; so this attribution is regulated by laws, which render a great part of this process absolutely necessary.

What is meant by a law as applicable to free intelligence: But when I speak of laws and of their absolute necessity in relation to thought, you must not suppose that these laws and that necessity are the same in the world of mind as in the world of matter. For free intelligences, a law is an ideal necessity given in the form of a precept, which we ought to follow, but which we may also violate if we please; whereas, for the existences which constitute the universe of nature, a law is only another name for those causes which operate blindly and universally in producing certain inevitable results. By law of thought, or by logical necessity, we do not, therefore, mean a physical law, such as the law of gravitation, but a general precept which we are able certainly to violate, but which if we do not obey, our whole process of thinking is suicidal, or absolutely null. These laws are, consequently, the primary conditions of the possibility of valid thought, and as the whole of Pure Logic is only an articulate development of the various modes in which they are applied, their consideration in general constitutes the first chapter in an orderly

1 [Schulze, Logik, § 18. Rösling. P. 63.] [Die Lehren der reinen Logik, Ulm, 1826. CF. Krug, Logik, § 16. – ED.]

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system of the science.

Order of consideration of the fundamental laws of thought: Now, in explaining to you this subject, the method I shall pursue is the following: I shall, first of all, state in general the number and significance of the laws as commonly received; I shall then more particularly consider each of these by itself and in relation to the others; then detail to you their history; and, finally, state to you my own views in regard to their deduction, number, and arrangement.

¶ XIII. The Fundamental Laws of Thought, or the conditions of the thinkable, as commonly received, are four: -- 1. The Law of Identity; 2. The Law of Contradiction; 3. The Law of Exclusion or of Excluded Middle; and, 4. The Law of Reason and Consequent, or of Sufficient Reason.

Of these in their order.

Par. XIII. Fundamental Laws of Thought:

¶ XIV. The principle of Identity (principium Identitatis) expresses the relation of total sameness in which a concept stands to all, and the relation of partial sameness in which it stands to each, of its constituent characters. In other words, it declares the impossibility of thinking the concept and its characters as reciprocally unlike. It is expressed in the formula A is A, or A=A; and by A is denoted every logical thing, every product of our thinking faculty, - concept, judgment, reasoning, etc.¹

Explication: The principle of Identity is an application of the principle of the absolute equivalence of a whole and of all its parts taken together, to the thinking of a thing by the attribution of constituent qualities or characters. The concept of the thing is a whole, the characters are the parts of that whole.² This law may, therefore, be also thus enounced, - Everything is equal to itself, - for in a logical relation the thing and its concept coincide; as, in Logic, we abstract altogether from the reality of the thing which the concept represents. It is, therefore, the same whether we say that the concept is equal to all its characters, or that the thing is equal to itself.³

The law has, likewise, been expressed by the formula - In the

¹ [Shultze, Logik, § 17. Gerlach, Logik, § 37.] Cf. Krug, Logik, § 17. – ED.

² See Schulze, Logik. p. 82-3. – ED.

³ See Krug, Logik, p. 40. – ED.

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predicate, the whole is contained explicitly, which in the subject is contained implicitly. It is also involved in the axiom –- Nota notoe est nota rei ipsuius.¹

Its logical importance -- The principle of all logical affirmation and definition: The logical importance of the law of identity lies in this – that it is the principle of all logical affirmation and definition. An example or two may be given to illustrate this.

This illustrated: 1. In a concept, which we may call Z, the characters a, b, and c, are thought as its constituents; consequently, the concept, as a unity, is equal to the characters taken together –- Z = (a + b + c). If the former be affirmed, so also is the latter; therefore, Z being (a + b + c) is a, is b, is c. To take a concrete example: The concept man is a complement made up of the characters, 1°, substance, 2°, material, 3°, organized, 4°, animated, 5°, rational, 6°, of this earth; in other words man is substance, is material, is organized, is animated, is rational. Being, as entering into every attribution, may be discharged as affording no distinction.

2. Again, suppose that, in the example given, the character a is made up of the characters l, m, n, it follows, by the same law of Identity, that Z = a = (l, m, n) is l, is m, is n. The concept man contains in it the character animal, and the character animal contains in it the characters corporeal, organized, living, etc.

The second law is the principle of Contradiction or Non-contradiction, in relation to which I shall dictate the following paragraph:

The law of Contradiction:

¶ XV. When an object is determined by the affirmation of a certain character, this object cannot be thought to be the same when such character is denied of it. The impossibility of this is enounced in what is called the principle of Contradiction (principium Contradictionis seu Repugnantiœ). Assertions concerning a thing are mutually contradictory, when the one asserts that the thing possesses the character which the other asserts that it does not. This law is logically expressed in the formula - What is contradictory is unthinkable. A = not A = 0, or A – A = 0.

Its proper name: Now, in the first place, in regard to the name of this law, it may be observed that, as it enjoins the absence of contradiction as the indispensable condition of

¹ See Kant, Logik, p. to. - ED.

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thought, it ought to be called, not the Law of Contradiction, but the Law of Non-contradiction, or of non-repugnantia.¹

How enounced: This law has frequently been enounced in the formula – It is impossible that the same thing can at once be and not be; but this is exposed to sundry objections. It is vague, and therefore useless. It does not indicate whether a real or a notional existence is meant; and if it mean the former, then is it not a logical but a metaphysical axiom. But even as a metaphysical axiom it is imperfect; for to the expression at once (simul) must be added, in the same place, in the same respect, etc.²

This law has likewise been expressed by the formula – Contradictory attributes cannot be united in one act of consciousness. But this is also obnoxious to objection. For a judgment expressed as good a unity of consciousness as a concept. But when I judge that round and square are contradictory attributes, there are found in this judgment contradictory attributes, but yet a unity of consciousness. The formula is, therefore, vaguely and inaccurately expressed.

The logical import of this law lies in its being the principle of all logical negation and distinction.

The law of Identity and the law of Contradiction are coördinate and reciprocally relative, and neither can be educed as second from the other as first; for in every such attempt at derivation, the supposed secondary law is, in fact, always necessarily presupposed.3 These are, in fact, one and the same law, differing only by a positive and negative expression.

In relation to the third law, take the following paragraph:

Par. XVI. Law of Excluded Middle.

¶ XVI. The principle of Excluded Third or Middle – viz., between two contradictories (principium Exclusi Medii vel Tertii') enounes that condition of thought which compels us, of two repugnant notions, which cannot both coëxist, to think either the one or the other as existing. Hence arises the general axiom -- Of contradictory attributions, we can only affirm one of a thing; and if one be explicitly affirmed, the other is implicitly denied. A either is or is not. A either is or is not B.⁴

By the laws of Identity and Contradiction, I am warranted to

¹ Compare Krug, Logik, §18.-ED.

² Compare the criticism of Kant, Kritik d.r. V, p. 134, ed. Rosenkranz, - ED.

³ This is shown in more in detail by Hoffbauer, Anfangsgriinde der Logik, § 23 – ED.

⁴ See Schulze, Logik, §19.-ED.

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conclude from the truth of one contradictory proposition to the falsehood of the other, and by the law of Excluded Middle, I am warranted to conclude from the falsehood of one contradictory proposition to the truth of the other.

Logical significance of this law: And in this lies the peculiar force and import of this last principle. For the logical significance of the law of Excluded Mliddle consists in this, that it limits or shuts in the sphere of the thinkable in relation to affirmation; for it determines, that, of the two forms given in the laws of Identity and Contradiction, and by these laws affirmed as those exclusively possible, the one or the other must be affirmed as necessary.

The principle of Disjunctive Judgments: The law of Excluded Middle is the principle of Disjunctive Judgments, that is, of judgments in which a plurality of judgments are contained, and which stand such a reciprocal relation that the affirmation of one is the denial of the other.

I now go on to the fourth law.

Par. XVII. Law of Sufficient Reason, or of Reason and Consequent:

¶ XVII. The thinking of an object, as actually characterized by positive or by negative attributes, is not left to the caprice of Understanding – the faculty of thought; but that faculty must be necessitated to this or that determinate act of thinking by a knowledge of something different from, and independent of; the process of thinking itself. This condition of our understanding is expressed by the law, as it is called, of Sufficient Reason (principium Rationis Sufficientis); but it is more properly denominated the law of Reason and Consequent (principium Rationis et Consecutionis). That knowledge by which the mind is necessitated to affirm or posit something else, is called the logical reason ground, or antecedent; that something else which the mind is necessitated to affirm or posit, is called the logical consequent; and the relation between the reason and consequent, is called the logical connection or consequence. This law is expressed in the formula - Infer nothing without a ground or reason.¹

Relations between Resaon and Consequent: The relations between Reason and Consequent, when comprehended in a pure thought, are the following:

1. When a reason is explicitly or implicitly given, then there must

¹ See Schulze, Logik, §19, and Krug, Logik, §20, - ED.

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exist a consequent; and, vice versa, when a consequent is given, there must also exist a reason.

2. Where there is no reason there can be no consequent; and, vice versa, where there is no consequent (either implicitly or explicitly) there can be no reason. That is, the concepts of reason and of consequent, as reciprocally relative, involve and suppose each other.

The logical significance of this law: The logical significance of the law of Reason and Consequent lies in this, - That in virtue of it, thought is constituted into a series of acts all indissolubly connected; each necessarily inferring the other. Thus it is that the distinction and opposition of possible, actual and necessary matter, which has been introduced into Logic, is a doctrine wholly extraneous to this science.

Reason and Consequent, and Cause and Effect: I may observe that “Reason is something different from Cause, and Consequent something different from Effect; though cause and effect, in so far as they are conceived in thought, stand to each other in the relation of reason and consequent. Cause is thus thought of as a real object, which affords the reason of the existence of another real object, the effect; and effect is thought of as a real object, which is the consequent of another real object, the cause. Accordingly, every cause is recognized in thought as a reason, and every effect is recognized in thought as a consequent; but the converse is not true, that every reason is really considered a cause, and every consequent really considered an effect. We must, therefore, carefully distinguish mere reason and mere consequent, that is, ideal or logical reason and consequent, from the reason which is a cause and the consequent which is an effect, that is, real or metaphysical reason and consequent.

Logical and Metaphysical Reason and Consequent: "The expression logical reason and consequent refers to the mere synthesis of thoughts; whereas the expression metaphysical reason and consequent denotes the real connection of existences. Hence the axiom of Causality, as a metaphysical principle, is essentially different from the axiom of Reason and Consequent, as a logical principle. Both, however, are frequently confounded with each other; and the law of Reason and Consequent, indeed, formerly found its place in the systems of Metaphysic, while it was not, at least explicitly, considered in those of Logic.

Generality of the terms Condition and Conditioned: The two terms condition and conditioned happily express at once the relations both of reason and consequent, and of cause and effect. A condition is a thing which determines (negatively at least) the

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existence of another; the conditioned is a thing whose existence is determined in and by another. If used in an ideal or logical signification, condition and conditioned import only the reason in conjunction with its consequent; if used in a real or metaphysical sense, they express the cause in connection with its effect."¹

History of the development of the fundamental Laws of Thought: I have now, in the prosecution of our inquiry into the fundamental laws of logical thinking, to say a few words in regard to their History, - their history being the narration of the order in which, and of the philosophers by whom, they were articulately developed.

The law of Identity last developed in the order of time: Of the first three laws, which, from their intimate cognition, may not unreasonably be regarded as only the three sides or phases of a single law, the law of ldentity, which stands first in the order of nature, was indeed that last developed in the order of time; the axioms of Contradiction and of Excluded Middle having been long enounced, ere that of Identity had been discriminated and raised to the rank of a coordinate principle. I shall not, therefore, now follow the order in which I detailed to you these laws, but the order in which they were chronologically generalized.

The principles of Contradiction and Excluded Middle can be traced back to Plato: The principles of Contradiction and of Excluded Middle can both be traced back to Plato, by whom they were enounced and frequently applied; though it was not till long after, that either of them obtained a distinctive appellation. To take the principle of Contradiction first. This law Plato frequently employs, but the most remarkable passages are found in the Phœdo, in the Sophista, and in the fourth and seventh books of the Republic.²

Law of Contradiction emphatically enounced by Aristotle: This law was, however, more distinctively and emphatically enounced by Aristotle. In one place,³ he says: "It is manifest that no one can conceive to himself that the same thing can at once be and not be, for thus he would hold repugnant opinions,

¹ Krug, Logik, pp. 62, 68. This exposition of the law of Reason and Consequent does not represent the Author’s latest view. In a note to the Discussions, p. 160 (where as similar doctrine had been maintained in the article as originally published), he says: “The logical relation of Reason and Consequent, as more than a mere corollary of the law of Noncontradiction in its three phases, is, I am confident of proving, erroneous.” And again, in the same work, p. 608: “The principle of Sufficient Reason should be excluded from logic. For, in as much as this principle is not material, it is only a derivation of the three formal laws; and in as much as it is material, it coincides with the principle of Causality, and is extra-logical.” The Laws of Thought, properly so called, are thus reduced to three – those of Identity, Contradiction, and Excluded Middle. – ED.

² See Phœdo, p. 103; Sophista, p.252; Republic, iv. p. 436; vii. p. 525. – ED.

³ Metaph., 1. iii. (iv.) c. 3.

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and subvert the reality of truth. Wherefore, all who attempt to demonstrate, reduce everything to this as the ultimate doctrine; for this is by nature the principle of all other axioms." And in several passages of his Metaphysics,¹ in his Prior Analytics,² and in his Posterior Analytics,³ he observes that "some had attempted to demonstrate this principle, - an attempt which betrayed an ignorance of those things whereof we ought to require a demonstration, and of those things whereof we ought not: for it is impossible to demonstrate everything; as in this case, we must regress and regress to infinity, and all demonstration would, on that supposition, be impossible."

With the Peripatetics the highest principle of knowledge. Obtained its name from the Greek Aristotelians: Following Aristotle, the Peripatetics established this law as the highest principle of knowledge. From the Greek Aristotelians it obtained the name by which it has subsequently been denominated, the principle, or law, or axiom, of contradiction, (--Greek words--). This name, at least, is found in the Commentaries of Ammonius and Philoponus, where it is said to be "the criterion which divides truth from falsehood throughout the universe of existence.” The schoolmen general, taught the same doctrine; and Suarez even says, that the law of contradiction holds the same supremacy among the principles of existence.⁵

Controversies respecting the truth and character of this law: After the decline of the Aristotelian philosophy, many controversies arose touching the truth, and still more touching the primitive or axiomatic character, of this law. Some maintained that it was indemonstrable; others that it could be proved, but proved only indirectly by a reductio ad absurdum; while others, again, held that this could be directly done, and that, consequently, the law of Contradiction was not entitled to the dignity of a first principle.⁶

¹ L.iii. c. 4.

² L. ii. c. 2.

³ L. i. c. 2.

⁴ For the name, see Ammonius, In De Interpret., Comment., p. 153 b, ed. Ald. Venet. 1546. Philoponus, In Anal. Pr., p. 13 b, 38 b, ed. Venet. 1536. In Anal. Post., p. 30 b, ed. Ald. Venet. 1534. The language quoted in the text is nearly a translation of Ammonius In Categ., p. 140 a. [--quotation in Greek--]. Ammonius is followed by Philoponnus, who says, -- [--quotation in Greek--]. In Anal. Post., 1, i. c. xi. f. 30 b. – ED. [Cf Augustinus Niphus Suessanus, In Anal. Post., p. 88, ed. Paris, 1540.]

⁵ See [Alstedius, Artium Liberalium Systema (8vo), p. 174. "Cognitio a priori est principiorum; inter quae agmen ducit hoc, impossibile est idem esse et non esse. . ... Consule Metaph., Suarexil: -- 'Hoc, inquam, tenet primatum inter principia cognoscendi, sicut Deus inter principia essendi.'"]

⁶ Cf. Suares. Disputationes Meaphysicae, Disp. iii. § 3/ -- ED. [Alstedius, Encyclopaedia, 1. iii., Archelogia, c. vii. p. 80.]

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Locke. Leibnitz.: In like manner, its employment was made a further matter of controversy. Finally, it was disputed whether it were an immediate, native, or a priori datum of intelligence; or whether it were an a posteriori and adventitious generalization from experience. The latter alternative, that it was only an induction, was maintained by Locke.¹ This opinion was, however, validly refuted by Leibnitz, who showed that it is admitted the moment the terms of its enunciation are understood, and that we implicitly follow it even when we are not explicitly conscious of its dictate.² Leibnitz, in some parts of his works, seems to identify the principles of Identity and Contradiction; in others, he distinguishes them, but educes the law of Identity out of the law of Contradiction.³

Its truth denied by modern absolutists: It is needless to pursue the subsequent history of this principle, which in latter times has found none to gainsay the necessity and universality of its truth, except among those philosophers who, in Germany, have dreamt that man is competent to a cognition of the absolute: and as a cognition of the absolute can only be established through positions repugnant, and, therefore, on logical principles, mutually exclusive, they have found it necessary to start with a denial of the fundamental laws of thought; and so, in their effort to soar to a philosophy above logic and intelligence, they have subverted the conditions of human philosophy altogether. Thus Schelling and Hegel prudently repudiated the principles of Contradiction and Excluded Middle as having any application to the absolute;⁴ while again those philosophers (as Cousin) who attempt a cognition of the absolute without a preliminary repudiation of the laws of Logic, at once involve themselves in contradictions, the cogency of which they do not deny, and from which they are wholly unable to extricate themselves.⁵

¹ Essay, B. i. ch. ii. § 4. – ED.

² Nouveaux Essais, B. i. ch. i. § 4. – ED.

³ Compare Théodicée, § 44, Monadologie, § 31, with Nouveaux Essais, 1. i. ch. i. § 10; l. iv. ch. II. § 1. - ED.

⁴ See Schelling, Vom Ich als Princip der Philosophie, § 10; Hegel, Logik, b. II. c. 2; Encyklopadie § 115, 119. Schelling endeavors to abrogate the principle of Contradiction In relation to the higher philosophy, by assuming that of Identity; the empirical antagonism between ego and non-ego being merged in the identity of the absolute ego. Hegel regards both principles alike as valid only for the finite Understanding, and as inapplicable to the higher processes of the Reason. This difference between the two philosophers is pointed out by the latter in his Geschichte der Philosphie, (Werke, xv. p. 598) – ED. [On rejection of the Logical Laws by Schelling, Hegel, etc. See Bachmann, Über die Philosophie, meiner zeit, p. 218, ed. Jena, 1816. Bolzano, Wissenschafislehre, iv. Logik, § 718. Sigwart, Logik, § 58, p. 42, ed. 1835. Herbart, De Principio Logico Exclusi Medii inter Contradictoria non negligendo, Gotting, 1833. Hartenstein, De Methodo Philosophioe Logicoe Legibus adstringenda, finibus non terminanada, Lipsieae, 1835. On the logical and metaphysical significance of the principle of Contradiction, see Platner, Phil. Aph., I. § 678, and Kant, Kritik d. reinen Vermunft, p. 191, ed. 1790.]

⁵ See the Author’s criticism of Cousin, Discussions, p. 1 et seq. – ED.

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But this by the way, and on a subject which at present you cannot all be supposed to understand.

Law of Excluded Middle: The law of Excluded Middle between two contradictories remounts, as I have said, also to Plato, though the Second Alcibiades, the dialogue in which it is most clearly expressed, must be admitted to be spurious.¹ It is also in the fragments of Pseudo-Archytas, to be found in Stobraeus.²

Explicitly enounced by Aristotle: It is explicitly and emphatically enounced by Aristotle in many passages both of his Metaphysics (l. iii. (iv.) c.7.) and of his Analytics, both Prior (l. i. c. 2) and Posterior (1. i. c. 4). In the first of these, he says: "It is impossible that there should exist any medium between contradictory opposites, but it is necessary either to affirm or to deny everything of everything." And his expressions are similar in the other books.

Cicero: Cicero says" that the foundation of Dialectic is, that whatever is enounced is either true or false." This is from his Academics (1. ii. c. xxix.), and there are parallel passages in his Topics (c. xiv.) and his De Oratore (I. ii. c. XXL). This law, though universally recognized as a principle in the Greek Peripatetic school, and in the schools of the middle ages, only received the distinctive appellation by which it is now known at a comparatively modern date.³

Baumgarten: I do not recollect having met with the term principium exclusi medii in any author older than the Leibnitzian Baumgarten,⁴ though Wolf⁵ speaks of the exclusio medii inter contradictoria.

Law of Identity. Antonius Andreas: The law of Identity, I stated, was not explicated as a coordinate principle till a comparatively recent period. The earliest author in whom I have found this done, is Antonius Andreas, a scholar of Scotus, who flourished at the end of the thirteenth and beginning of the fourteenth century. The schoolman, in the fourth book of his Commentary of Aristotle's Metaphysics,⁶ - a commentary which is full of the most ingenious and original views, - not only asserts to the law of Identity a coördinate dignity with the law of Contradiction,

¹ Second Alcibiades, p. 139. See also Sophista, p. 250 – ED.

² Eclogœ.. 1. ii. c. 2, p. 158, Ed. Antwerp, 1575; Part ii. tom. 1, p. 22, ed. Heeren. Cf. Simplicius, In Arist. Categ., pp. 97, 103, ed. Basil, 1551. –ED.

³ Lex contradictoriarum, principium contradicentium (sc. propositionum), as used in the schools, included the law of Contradiction and the law of Excluded Middle. See Molinaeus Elementa Logica, 1. ii. c. 14, [p. 172, ed. 1608. “Contradicentium usus explicatur uno axiomate: -- Contradicentia non possunt de eodem simul esse vera; et necessarium est contradicentium alterum cuilibet rei convenire, alterum non convenire.” – ED.]

⁴ Metaphysica, § 10. ED.

⁵ Ontologia, §§ 52, 53.

⁶ Quaestio v. p. 21, a, ed. Venet., 1513. – ED.

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but, against Aristotle, he maintains that the principle of Identity, and not the principle of Contradiction, is the one absolutely first. The formula in which Andreas expressed it was Ens est ens. Subsequently to this author, the question concerning the relative priority of the two laws of Identity and of Contradiction became one much agitated in the schools; though there were also found some who asserted to the law of Excluded Middle this supreme rank.¹

Leibnitz, Wolf, Baumgarten: Leibnitz, as I have said, did not always distinguish the principles of Identity and of Contradiction. By Wolf the former was styled the principle of Certainty,(principium Certitudinis);² but he, no more than Leibnitz himself, sufficiently discriminated between it and the law of Contradiction. This was, however, done by Baumgarten, another distinguished follower of Leibnitz,³ and from him it received the name of the principle of Position, that is, of Affirmation or Identity,(principium Positionis sive Identitatis), - the name by which it is now universally known.

Fichte and Schelling, Hegel.: This principle has found greater favor, in the eyes of the absolutist philosophers, than those of Contradiction and Excluded Middle. By Fichte and Schelling it has been placed as the primary principle of all philosophy.⁴ Hegel alone subjects it, along with the other laws of thought, to a rigid but fallacious criticism; and rejects it along with them, as belonging to that lower sphere of knowledge, which is conversant only with the relative and finite.⁵

Law of Reason and Consequent, Recognized by Plato and Aristotle: The fourth law, that of Reason and Consequent, which stands apart by itself from the other three, was, like the laws of Contradiction and Excluded Middle, recognized by Plato.4 He lays it down as a postulate of reason, to admit nothing without a cause; and the same is frequently done by his scholar Aristotle.⁷ Both, however, in reference to this principle, employ the ambiguous term cause ( -- Greek words --). Aristotle, indeed, distinguishes the law of Reason, as the ideal principle of knowledge (-- Greek words --,

¹ [Alex. de Ales, In Arist. Metaph., iv. t. 9.] Compare Suarez, Disp. Metaph., Disp. iii. § 3. Alexauder professes to agree with Aristotle in giving he first place to the principle of Contradiction, but, in fact, he identifies it with that of Excluded Middle, de quovis affirmatio vel negatio.—ED.

² Ontologia, § 55, 288. – ED.

³ Metaphysica, § 11. – ED.

⁴ See Fichte, Gurdlage der gesammien Wissenshafislehre, § 1. Schelling, Vom Ich, § 7. – ED.

⁵ See above, p. 64, note 4. – ED.

⁶ Philebus, p. 26. – ED.

⁷ E. g. Anal. Post., ii. 16; Phys., ii. 3; Metaph., i. 1. 3; Rhet., ii. 23. – ED.

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principium cognoscendi), from the real principle of Production, (Greek words, principium fiendi, - principium essendi).¹

Cicero, The Schoolmen: By Cicero, the axiom of reason and consequent was, in like manner, comprehended under the formula, nihil sine causa,³ a formula adopted by the schoolmen; although they, after Aristotle, distinguished under it the ratio essendi, and the ratio cognoscendi.

Liebniz called attention to Law of Sufficient Reason: In modern times, the attention of philosophers was called to this law of Leibnitz, who, on the two principles of Reason and of Contradiction, founded the whole edifice of his philosophy.³ Under the latter law, as I have mentioned, he comprehended, however, the principle of Identity; and in the former he did not sufficiently discriminate, in terms, the law of Causality, as a real principle, from the law of Reason, properly so called, as a formal or ideal principle. To this axiom he gave various denominations, now calling it the principle of Determining Reason, now the principle of Sufficient Reason, and now the principle of Convenience or Agreement (convenientia) ; making it, in its real relation, the ground of all existence; in its ideal, the ground of all positive knowledge. On this subject there was a celebrated controversy between Leibnitz and Dr. Samuel Clarke, - a controversy on this, as on other points, eminently worthy of your study. The documents in which this controversy is contained, were published in the English edition under the title, A collection of Papers which passed between the late learned Mr. Leibnitz and Dr. Clarke, in the years 1715 and 1716, relating to the Principles of Natural Philosophy and Religion, London, 1717.⁴

Wolf: Wolf, the most distinguished follower of Leibnitz, employs the formula – "Nothing is without a sufficient reason why it is, rather than why it is not; that is, if anything is supposed to be (ponitur esse), something also must be supposed, whence it may be understood why the same is rather than is not."⁵ He blames the schoolmen for confusing reason (ratio) with cause (causa): but his censure equally applies to his master Leibnitz, as to them and Aristotle; for all of these philosophers, though they did not confound the two principles, employed ambiguous terms to denote them.

¹ Metaph., iv. (v.) 1. - ED.

² De Divinatione, II. c. 28. - ED.

³ See Theodicée, § 44. Monadologie, §§ 81,32. -ED.

⁴ See especially, Leibnitz’s, Second Letter, p. 20, In which the principle of Contradiction or Identity is assumed as the foundation of all mathematics and that of Sufficient Reason as the foundation of natural philosophy – ED.

⁵ See Fischer’s Logik, [§ 59, p. 38, ed. 1838. Compare Wolf, Ontologia, §§ 70, 71 – ED.]

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Discussion regarding the Leibnitzian doctrine of the law of Sufficient Reason: The Leibnitian doctrine of the universality of the law of Sufficient Reason, both as a principle of existence and of thought, excited much discussion among the philosophers, more particularly of Germany. In the earlier half of the last century, some controverted the validity of the principle, others attempted to restrict it.¹ Among other arguments, it is alleged, by the advocates of the former opinion, if the principle be admitted, that everything must have a sufficient reason why it is, rather than why it is not, - on this hypothesis, error itself will have such a reason, and, therefore, must cease forthwith to be error.²

Many philosophers, as Wolf and Baumgarten, endeavored to demonstrate this principle by the principle of Contradiction; while others, with better success, showed that all such demonstrations were illogical.³

In the more recent systems of philosophy, the universality and necessity of the axiom of Reason has, with other logical laws, been controverted and rejected by speculators on the absolute.⁴

¹ As Feuerlin and Daries. See Bachmann, Logik, p. 56, Leipsig, 1823; Cf. Degerando, Hist. Comp. des Syst. de Phil., t. ii, p. 145, ed. 1804. – ED.

² See Bachmann, Logik, p. 56. With the foregoing history of the laws of Thought, compare the same author, Logik, § 18-31. – ED.

³ [Kiesewetter, Allgemeine Logik, P. i. p. 57]; compare Lectures on Metaphysics, ii. pp. 396, 397, notes. – ED.

⁴ [On principle of Double Negation as another law of Thought, see Fries, Logik, § 41, p. 190; Calker, Denklehre oder Logik und Dialektik, § 165, p. 453; Beneke, Lehrbuch der Logik, § 64, p. 41.

Venn 1881

Venn Symbolic Logic 1881:237 -- Logical statements or equations:

But since our alternatives are collectively exhaustive as well as mutually exclusive, it is a contradiction in terms to suppose them all to vanish; this, it will be noticed, being our generalized form corresponding to the so-called Law of Excluded Middle. Suppose for instance, just for illustration, that we write down such a form as this,

Axy +Bx~y +Ox~y +D~x~y = 0,

one or more of the four factors A, B, C, D, must be supposed = 0, in order ~ avoid contradiction.

Thompson 1860

WILLIAM THOMSON, 1860, AN OUTLINE OF THE NECESSARY LAWS OF THOUGHT: A TREATISE ON PURE AND APPLIED LOGIC,FROM THE FOURTH LONDON EDITION, NEW YORK: SHELDON AND COMPANY. BOSTON: GOULD & LINCOLN.

p. 248

Four principal criteria of truth have been in different forms advocated by logicians; the reader is now in a position to estimate their value. 1st CRITERION. The principle of Contradiction. "The same attribute cannot be at the same time affirmed and denied of the Same subject." Or" the same subject cannot have two contradictory attributes." Or" the attribute cannot be contradictory of the subject."• To illustrate this - at a particular time facts were observed as to the motions of the planets, which were inconsistent with the received theory that these motions were circular. The theory was consequently modified, first by the introduction of epicycles, and finally by the substitution of the

• The first mode of statement is Aristotle's [ -- Greek words -- ] Metaph. IV. (r.) Iii. The second is Aristotelian; the third is Kant's.

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theory of elliptic revolution; because otherwise the astronomer must have affirmed of the planets a circular and a non-circular motion, or in other words must have assigned to a subject, to which he had already given" circular motion," a predicate contradictory of this.

2d CRITERION. Tke principle of Identity. "Conceptions which agree can be united in thought, or affirmed of the same subject at the same time." This principle is the complement of the former.

3d CRITERION. The principle of the Middle being excluded (lex exelusi medii). "Either a given judgment must be true, or its contradictory; there is no middle course." So that the proof of a judgment forces us to abandon its contradictory entirely, as would the disproof of it force us upon a full acceptance of the contradictory. This law, among other uses, applies to the dialectical contrivance known to logicians as reductio per impossibile . 4th CRITERION. The principle of sufficient (or determinantt) reason. "Whatever exists, or is true, must have a sufficient reason why the thing or proposition should be as it is and not otherwise.":I: From this law are educed such applications as '*' This is the aVTt&eatr ~, oVlC tan ILeT~ lCao9' avrfrv, of Aristotle, (An. Post, I. i. lCafJ' ain1;v, "as appears pt:r Be from the nature of the assertion." Trend.) Compare Metaph. IV. (r.) 7, and Alexander', comment. t C. A. Crusius, in a tract on this subject, finds fault with the ambiguity of .. sufficient," which might seem "sufficient for this effect" without excluding it from the possibility of producing some other. According to him, this principle involves absolute necessity, and destroys morality. t Leibnitz, Theod. I. § 44. Upon this principle, and those of Contradiction and Identity, Leibnitz has based his Logic.

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these :-1. Granting the reason, we must grant what follows from it. On this depends syllogistic inference. 2. If we reject the consequent, we must reject the reason. If we admit the consequent, we do not of necessity admit the reason. Now the distinction between formal and material truth, or in other words between self-consistency in thinking, a.nd conformity with facts, assists materially in forming an estimate of the worth of these principles. A judgment may be formally true, and materially false; a.s in the inference" No men err, Socrates is a man, therefore he cannot err," which is correctly drawn, yet proves a falsehood from a falsehood: or it may be materially true,. yet formally false, as, " Socrates is a man, Socrates erred, therefore all men err; " where a true judgment has been drawn from two true judgments, yet not correctly. The four criteria in question are useful in securing formal truth, that is, in keeping our thoughts in harmony with each other; but for the discovery of material truth, for giving us thoughts that are true representations of facts, they are either useless, or only useful as principles subordinate to the higher criterion of which all applied Logic is but the expansion, that every proposition must rest upon sufficient evidence. The principle of contradiction has been already implied in the doctrine of privative conceptions in the theory of disjunctive judgments and inferences and in other places. The principle of the excluded middle is the canon of the inference from contradictory opposition upon which the refutation of a false conclusion must rest. The principle of the sufficient reason is implied in the syllogistic

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canon that every conclusion must follow from and depend on sufficient premisses; it is employed in other forms, in hypothetical reasonings in particular. And in these purely formal applications the criteria have their importance, but that not the highest. Viewed as instruments for judging of material truth, they sink into mere rules for the reception of evidence. The first is a caution against receiving into our notion of a subject any attribute that is irreconcilable with some other, already proved upon evidence we cannot doubt. The second is a permission to receive attributes that are not thus mutually opposed, or a hint to seek for such only. The third would compel us to reconsider the evidence of any proposition, when other evidence threatened to compel us to accept its contradictory. The fourth commands that we seek the causes and laws that have determined the existence of our subject, for the subject cannot be adequately known except in these. So that the vaunted criteria of truth are rules of evidence; and there is no one means of judging of truth, except what the whole science of Evidence affords. 

Jevons 1880

W. STANLEY JEVONS, 1880, ELEMENTARY LESSONS IN LOGIC: DEDUCTIVE AND INDUCTIVE, WITH COPIOUS QUESTIONS AND EXAMPLES, AND A VOCABULARY OF LOGICAL TERMS. MACMILLAN AND CO, London and New York

LESSON XIV: THE LAWS OF THOUGHT.

p. 117

BEFORE the reader proceeds to the lessons which treat of the most common forms of reasoning, known as the syllogism, it is desirable that he should give a careful attention to the very simple laws of thought on which all reasoning must ultimately depend. These laws describe the very simplest truths, in which all people must agree, and which at the same time apply to all notions which we can conceive. It is impossible to think correctly and avoid evident self contradiction unless we observe what are called the Three Primary Laws of Thought which may be stated as follows; I. The Law of Identity Whatever is, is. 2. The Law of Contradiction Nothing can both be and not be. 3. The Law of Excluded Middle Everything must either be or not be. Though these laws when thus stated may seem absurdly obvious, and were ridiculed by Locke and others on that account, I have found that students are seldom able to see at first their full meaning and importance. It will be pointed out in Lesson XXIII, that logicians have

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overlooked until recent years the very simple way in which all arguments may be explained when these self-evident laws are granted; and it is not too much to say that the whole of logic will be plain to those who will constantly use these laws as the key.

The first of the laws may be regarded as the best definition we can give of identity or sameness. Could anyone be ignorant of the meaning of the word Identity, it would be sufficient to inform him that everything is identical with itself.

The second law however is the one which requires more consideration. Its meaning is that nothing can have at the same time and at the same place contradictory and inconsistent qualities. A piece of paper may be blackened in one part, while it is white in other parts; or it may be white at one time, and afterwards become black; but we cannot conceive that it should be both white and black at the same place and time. A door after being open may be shut, but it cannot at once be shut and open. Water may feel warm to one hand and cold to another hand, but it cannot be both warm and cold to the same hand. No quality can both be present and absent at the same time; and this seems to be the most simple and general truth which we can assert of all things. It is the very nature of existence that a thing cannot be otherwise than it is; and it may be safely said that all fallacy and error arise from unwittingly reasoning in a way inconsistent with this law. All statements or inferences which imply a combination of contradictory qualities must be taken as impossible and false, and the breaking of this law is the mark of their being false. It can easily be shewn that if Iron be a metal, and every metal an element, Iron must be an element or it can be nothing at all, since it would combine qualities which are inconsistent (see Lesson XXIII).

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The Law of Excluded Middle is much less self-evident than either of the two preceding ones, and the reader will not perhaps see at the first moment that it is equally important and necessary with them. Its meaning may be best explained by saying that it is impossible to mention any thing and any quality or circumstance, without allowing that the quality or circumstance either belongs to the thing or does not belong. The name of the law expresses the fact that there is no third or middle course; the answer must be Yes or No. Let the thing be rock and the quality hard; then rock must be either hard or not-hard. Gold must be either white or not white; a line must be either straight or not straight; an action must be either virtuous or not virtuous. Indeed when we know nothing of the terms used we may nevertheless make assertions concerning them in accordance with this law. The reader may not know and in fact chemists may not really know with certainty, whether vanadium is a metal or not a metal, but anyone knows that it must be one or the other. Some readers may not know what a cycloid is or what an isochronous curve is; but they must know that a cycloid is either an isochronous curve or it is not an isochronous curve.

This law of excluded middle is not so evident but that plausible objections may be suggested to it. Rock, it may be urged, is not always either hard or soft, for it may be half way between, a little hard and a little soft at the same time. This objection points to a distinction which is of great logical importance, and when neglected often leads to fallacy. The law of excluded middle affirmed nothing about hard and soft, but only referred to hard and not-hard; if the reader chooses to substitute soft for not-hard he falls into a serious confusion between opposite terms and contradictory terms. It is quite possible that a thing may be neither hard nor soft, being half way

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between; but in that case it cannot be fairly called hard so that the law holds true. Similarly water must be either warm or not-warm, but it does not follow that it must be warm or cold. The alternative not-warm evidently includes all cases in which it is cold besides cases where it is of a medium temperature, so that we should call it neither warm nor cold. We must thus carefully distinguish questions of degree or quantity from those of simple logical fact. In cases where a thing or quality may exist to a greater or less extent there are many alternatives. Warm water, for, instance may have any temperature from 70^o perhaps up to 120^o. Exactly the same question occurs in cases of geometrical reasoning; for Euclid in his Elements frequently argues from the self-evident truth that any line must be either greater than, equal to, or less than any other line. While there are only two alternatives to choose from in logic there are three in Mathematics; thus one line, compared with another, may be-

{greater.................greater}

In Logic. not-greater...{...... equal Mathemnatics.

...... less

Another and even more plausible objection may be raised to the third law of thought in this way. Virtue being the thing proposed, and triangular the quality, the Law of Excluded Middle enables us at once to assert that virtue is either triangular or not-triangular. At first sight it might seem false and absurd to say that an immaterial notion such as virtue should be either triangular or not, because it has nothing in common with those material substances occupying space to which the notion of figure belongs. But the absurdity would arise, not from any falseness in the law, but from misinterpretation of the expression not-triangular. If in saying that a thing is

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"not triangular" we are taken to imply that it has some figure though not a triangular figure, then of course the expression cannot be applied to virtue or anything immaterial. In strict logic however no such implied meaning is to be allowed, and not-triangular will include both things which have figure other than triangular, as well as things which have not the properties of figure at all; and it is in the latter meaning that it is applicable to an immaterial thing.

These three laws then being universally and necessarily true to whatever things they are applied, become the foundation of reasoning. All acts of reasoning proceed from certain judgments, and the act of judgment consists in comparing two things or ideas together and discovering whether they agree or differ, toot is to say whether they are identical in any qualities. The laws of thought inform us of the very nature of this identity with which all thought is concerned. But in the operation of discourse or reasoning we need certain additional laws, or axioms, or self-evident truths, which may be thus stated:

• 1. Two terms agreeing with one and the same third term agree with each other.
• 2. Two terms of which one agrees and the other does not agree with one and the same third term, do not agree with each other.

These self-evident truths are commonly called the 'Canons or Fundamental Principles of Syllogism, and they are true whatever may be the kind of agreement in question. The example we formerly used (p. 3) of the agreement of the terms "Most useful metal" and "cheapest metal" with the third common term" Iron," was but an instance of the first Canon, and the agreement consisted in complete identity. In the case of the" Earth," the "Planets," and "bodies revolving in elliptic orbits,"

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the agreement was less complete, because the Earth is only one of many Planets, and the Planets only a small portion of all the heavenly bodies, such as Satellites, Comets, Meteors, and Double-Stars which revolve in such orbits.

The second of the Canons applies to cases where there is disagreement or difference, as in the following example:

Venus is a planet

Planets are not self-luminous.

Therefore Venus is not self-luminous.

The first of these propositions states a certain agreement to exist between Venus and planet, just as in the previous case of the Earth, but the second proposition states a disagreement between Planet and self-luminous bodies; hence we infer a disagreement between Venus and self-luminous body. But the reader will carefully observe that from two disagreements we can never infer anything. If the following were put forth as an argument it would be evidently absurd:-

Sirius is not a planet

Planets are not self-luminous.

Therefore Sirius is not self-luminous.

Both the premises or propositions given are true, and yet the conclusion is false, for all the fixed stars are self-luminous, or shine by their own light. We may, in fact, state as a third canon that-

3. Two terms both disagreeing with one and the same third term may or may not agree with each other.

Self-evident rules, of an exactly similar nature to these three Canons, are the basis of all mathematical reasoning, and are usually called axioms. Euclid's first axiom is that "Things which are equal to the same thing are equal to one another;" and whether we apply it to the length of lines, the magnitude of angles, areas, solids, numbers,

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degrees, or anything else which admits of being equal or unequal, it holds true. Thus if the lines A and B are each equal to C it is evident that each is equal to the other

• A------
• B------
• C------
• D---------
• E----

Euclid does not give axioms corresponding to the second and third Canons, but they are really used in Geometry. Thus if A is equal to B, but D is not equal to D, it follows that A is not equal to D, or things of which one is equal, but the qther unequal to the same third thing, are unequal to each other. Lastly, A and E are two lines both unequal to D and unequal to each other, whereas A and B are two lines both unequal to D but equal to each other; thus we plainly see that "two things both unequal to the same thing may or may not be equal to each other."

From what precedes it will be apparent that all reasoning requires that there should be one agreement at least; if there be two agreements we may reason to a third agreement; if there be one agreement and one difference we may reason to a second difference; but if there be two differences only we cannot reason to any conclusion whatever. These self-evident Principles will in the next Lesson serve to explain. Some of the rules of the Syllogism.

Logicians however have not confined themselves to the use of these Canons, but have often put the same truth into a different form in axioms known as the Dicta de omni et nullo of Aristotle. This celebrated Latin phrase means" Statements concerning all and none," and the axiom, or rather pair of axioms, is usually given in the following words:

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Whatever is predicated of a term distributed whether affirmatively or negatively, may be predicated in like manner of everything contained under it.

Or more briefly:

What pertains to the higher class pertains also to the low".

This merely means, in untechnical language, that what may be said of all the things of any sort or kind may be said of any one or any part of those things; and, secondly, what may be denied of all the things in a class may be denied of any one or any part of them. Whatever may be said of "All planets" may be said of Venus, the Earth, Jupiter, or any other planet; and, as they may all be said to revolve in elliptic orbits, it follows that this may be asserted of Venus, the Earth, Jupiter, or any other planet. Similarly, according to the negative part of the Dicta, we may deny that the planets are self-luminous, and knowing that Jupiter is a planet may deny that Jupiter is self-luminous. A little reflection would show that the affirmative Dictum is really the first of the Canons in a less complete and general form, and that the negative Dictum is similarly the second Canon. These Dicta in fact only apply to such cases of agreement between terms as consist in one being the name of a smaller class, and another of the larger class containing it. Logicians have for the most part strangely overlooked the important cases in which one term agrees with another to the extent of being identical with it; but this is a subject which we cannot fitly discuss here at any length. It is treated in my little work called The Substitution of Similars*.

Some logicians have held that in addition to the three laws which are called the Primary Laws of Thought,

*Macmillan and Co. 1869.

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there is a fourth called "The Principle or Law of Sufficient Reason." It was stated by Leibnitz in the following words:

Nothing happens without a reason why it should be so rather than otherwise.

For instance, if there be a pair of scales in every respect exactly alike on each side and with exactly equal weights in each scale, it must remain motionless and in equilibrium, because there is no reason why one side should go down more than the other. It is certainly a fundamental assumption in mechanical science that if a body is acted upon by two perfectly equal forces in different directions it will move equally between them, because there is no reason why it should move more to one side than the other. Mr Mansel, Sir W. Hamilton and others consider however that this law has no place in logic, even if it can be held self-evident at all; and the question which appears open to doubt need not be discussed here.

I have so freely used the word axiom in this lesson that it is desirable to clear up its meaning as far as possible. Philosophers do not perfectly agree about its derivation or exact meaning, but it certainly comes from the verb άξιόω which is rendered, to think worthy. It generally denotes a self-evident truth of so simple a character that it must be assumed to be true, and, as it cannot be proved by any simpler proposition, must itself be taken as the basis of reasoning. In mathematics it is clearly used in this sense.

See Hamilton's Lectures 0n Logic, Lectures 5 and 6.

William James

James diverges greatly from the traditional Aristotelian et. al. Laws of Thought. He closely examines as many aspects of cognition as he could write about given the science available at that time. This book is huge.

The following are cited by GBWW Volume 1 The Syntopicon Volume 1: Angel to Love p. 1044a:

"After discrimination, association!" p. 360

• 1 Psychology, p. 299b [footnote 1]
• 2 Psychology, p. 319b-320a
• 3 Psychology, p. 360b-361a
• 4 Psychology, p. 667-668a
• 5 Psychology, p. 868b-873a
• 6 Psychology, p. 878a-879b

• 1 Psychology, p. 299b [footnote 1]: Chapter XII CONCEPTION, The Sense of Sameness"

"There are two other "principles of identity" in philosophy. The ontological one asserts that every real thing is wht it is, that a is a, and b, b. The logical one says that what is once true of the subject of a judgment is always true of that subject. The ontological law is a tautological truism; the logical principle is already more, for it implies subjects unalterable by time. The psychological law also implies facts which might not be realized: there might be no succession of thoughts; or if there were, the later ones might not think of the eralier; or if they did, they might not recall the content thereof; or, recalling the content, they might not take it as "the same" with antying else.

• 2 Psychology, p. 319b-320a: Chapter XIII DISCRIMINATION AND COMPARISON, The Principle of Meidate Comparison

"... to put it briefly and universally, the more than the more is more than the less; such is the greate synthetic principle of mediate comparison which is involved in the possession by the human mind of the sense of serial increase. In Chapter XXVIII [NECESSARY TRUTHS AND THE EFFECTS OF EXPERIENCE, 851] we shall see the altogether overwhelming importance of this principle in the conduct of all our higher rational operations.

• 3 Psychology, p. 360b-361a : Chapter XIV ASSOCIATION

A discussion of a "law of [thought]-association" or "law of associative thinking"

• 4 Psychology, p. 667-668a : Chapter XXII REASONING, In Resaoning, We Pick Out Essential Qualities

A discussion of "reasoning".

• 5 Psychology, p. 868b-873a : Chapter XXVIII NECESSARY TRUTHS

Couturat 1914

LOUIS COUTURAT, translated by LYDIA GILLINGHAM ROBINSON, WITH A PREFACE BY PHILIP E. B. JOURDAIN 1914, THE ALGEBRA OF LOGIC, CHICAGO AND LONDON, THE OPEN COURT PUBLISHING COMPANY

pages 23 - 24 -- Excluded middle, Principle of

r6. The Principles of Contradiction and of Excluded Middle. By definition, a term and its negative verify the two formulas

aa' = 0, a + a' = 1,

which represent respectively the principle of contradiction and the principle of excluded middle.¹

C. L: 1. The classes a and a' have nothing in common; in other words, no element can be at the same time both a and not-a. 2. The classes a and a' combined form the whole; In other words, every element is either a or not-a.

P. 1.: I. The simultaneous affirmation of the propositions a and not-a is false; in other words, these two propositions cannot both be true at the same time. 2. The alternative affirmation of the propositions a and not-a is true; in other words, one of these two propositions must be true. Two propositions are said to be contradictory when one is the negative of the other; they cannot both be true or false at the same time. If one is true the other is false; if one is false the other is true. This is in agreement with the fact that the terms 0 and I are the negatives of each other; thus we have

OXI=O, 0+1=1.

Generally speaking, we say that two terms are contradictory when one is the negative of the other.

¹As Mrs. LADD•FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD-FRANKLIN proposes to call them respectively the principle of exclusion and the principle of exhaustion, inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are exhaustive (of the universe of discourse).