HISTORY OF LOGIC

The Pre-Socratics.
The earliest beginnings of Logic are found in Pre-Socratic philosophy (the period of Greek philosophy from the middle of the 7th century to the middle of the 5th century B.C.), in the dialectical arguments of Eleatic School (the group of early Greek philosophers of the 6th and 5th century B.C., centering in Elea), who saw in the laws of identity, contradiction, and excluded middle the nature of reality. The Greek philosopher Parmenides of Elea (c.515 B.C.- c.450 B.C.) held that thought and being are identical and one need only follow out the principles of logical consistency to gain the truth about reality. Sense experience, on the other hand, is full of contradictions and hence the world of sense experience is not real. Thought seeks the common, universal and unchanging. What is common to all things that exist is being. Being exists and non-being does not exist. Hence, being is and non-being is not. The logic of Parmenides' arguments depends upon taking the verb to-be as meaning identity and existence. Now if being is and non-being is not, then being cannot come into existence from non-being. Hence, being is eternal. If anything changes, something that was not comes to be; and since non-being does not exist, change is impossible. Since a being occupies space and non-being does not, then empty space does not exist. If anything moves, it must occupy a space where it was not; and since empty space does not exist, motion is impossible. If things are separated from each other, they are separated by empty space; but since empty space does not exist, things cannot be separated from each other. Thus being must be one, not many. Parmenides is the earliest defenders of this quantitative monism. The Milesian philosophers, Thales (640-546 B.C.), who taught that the nature all things is water and Anaximenes (588-524 B.C.), who argued that all things come from, and return to air, were qualitative monist. Finally, being is homogeneous, not having parts; for if it has parts, then it would be many; but since being is one, it has not parts and thus is it homogeneous. From this Parmenides argued that being is finite; like the surface of a sphere, it is "perfected on every side," equally distance from its center at every point. Parmenides' student, Mellisus of Samos (5th century B.C.), rejected his teacher's conclusion that being is finite; he argued that being must be spatially and temporally infinite. For if being is finite, then beyond it there would be empty space; but since empty space is nothing, non-being, it does not exist and thus being occupies all space. Thus being is spatially infinite. Mellisus, like Parmenides, rejected a void or vacuum, empty space, as impossible and non-existent, "for what is empty is nothing. What is nothing cannot be." Zeno of Elea (c.490-430 B.C.), who was a firm adherent to Parmenides' ideas, tried to show logically that they are true by showing their opposite are absurd (reductiones ad absurdum) and impossible. By this form of argument he attempted to show logically that pluralism (that reality is numerically many, not one), empty space, and motion are impossible.
Socrates and Plato.
The Sophists drew skeptical conclusions concerning thought and reason from the ambiguities in concepts. But Socrates (470-400 B.C.), as recorded in the Dialogues of Plato (427-347 B.C.), opposed this skepticism by an inductive inquiry into the basic meaning of concepts. In his earliest dialogues Socrates is presented as using an inductive argument consisting of a examination of individual cases in order to discover what they have in common, the characteristics under which they are to be classed, that is, by their appropriate definitions. In Euthyphro Socrates deals with the nature of piety, in the Charmides with temperance, and in Lysis with friendship. Socrates' typical method in these dialogues was to challenge the use of a word which led to a consideration of cases; and although definitive conclusions are not reached in the dialogues mentioned, the Socratic effort was to detach the appropriate universal definitions from these cases. Thus is Socrates distinguished from the Sophists, for the latters' relativism and skepticism consisted in a large part in utilizing the ambiguities of ordinary language. In his later dialogues Plato came to the conclusion that by dichotomous division thought might proceed from the inclusive and universal concept "Being." to the definition of specific classes, by arranging the content of the realm of thought into a classification system. In Book I of the Republic definitions are to be discredited if they can be reduced to absurdity, or shown to be either vague or inconsistent.
Aristotle.
Aristotle (384-322 B.C.) credits Socrates for contributing to philosophy "inductive arguments and universal definitions." Aristotle carried these contributions into a larger systematic development. In his writings on logic, the Categories, Topics, On Interpretation, Prior Analytics, and Posterior Analytics, he developed an analysis of propositions, their interrelations, their quantification, and their use in inference. He was the first philosopher to have a clear grasp of the many aspects implicit in the obtaining knowledge. He not only discerned the role of definition, induction, and deduction in the development of science, he also saw the differences between the sciences, dividing them into theoretical sciences which aim at truth, practical sciences that aimed at action, and productive sciences that aimed at making. Truth is correspondence with reality, saying "of what is that is it is and of what is not that it is not." The theoretical sciences seek knowledge of the truth. The components of knowledge relate differently in these different disciplines, and the more abstract disciplines, such as logic and metaphysics, are more completely ordered to the universal and necessary. For Aristotle it seemed clear that not only metaphysics but also logic, as the most abstract discipline, have a special relationship to the other fields, such as physics, astronomy, and biology. The science now called logic, that was called by him the "analytic", is a discipline preliminary to all the others, since its purpose is to set forth the conditions that must be observed by all thing which has truth as its aim.
The inner relationship of these fields of knowledge can be seen by examining the role that intuition plays in the production of knowledge. Intuition has two principal roles:
(1) We intuit the particulars of sense directly. And from sense particulars we form definitions, and make inductions, providing the premises for deductions.
(2) At the other pole of our experience we also intuit the most general principles of explanation, including logical principles themselves.
And between these two poles the basic structuring of theories goes on. Theories involve definitions, axioms, postulates, and hypotheses.
Axioms are the indemonstrable primary premises of demonstration, that is, notions commonly held to be true;
postulates relate to the subject whose attributes are to be examined. According to Aristotle, unlike axioms, postulates are demonstrable, but are used in a given inquiry without demonstration.
Hypotheses are like postulates in that they are capable of proof, but are accepted for the purpose of the examination without proof. They differ from postulates in ranging more widely, and including more than the primary premises of the demonstration.
Modern mathematics and logic do not recognize Aristotle's distinction between axioms and postulates, treating postulates as equivalent to axioms, as the assumed without proof premises of demonstration.
Aristotle believed in real definitions; that is, definitions are either true, or not true, of the thing defined. And a true definition expresses the essence of the thing defined. To do so, it is necessary to relate the term being defined to the larger class of which it is a member, that is, its genus, and at the same time one must state how the sub-class of the thing being defined (the species) differs from other members of the genus class. When "man" is defined as a "rational animal," this is a definition by genus and difference, the genus being "animal" and the differentia being "rationality." The definition of man gives the essence of species man as a "rational animal."
This description of definition uses three of the five predicables: species, genius, and difference. A predicable (from the Latin praedicare "to affirm") is a type of predicate that may be affirmed or denied of something. The other two predicables are "property" and "accident." While the definition of a thing is expected to state the essence of the thing, a "property," while not stating its essence, yet it does belongs only to that thing. For example, to be capable of learning grammar is a property, and yet this capacity is not part of the definition of "man." An "accident," on the other hand, is a predicate with only a contingent relation to its subject. It just happens to characterize the subject in question. The red color of red house is an accident in that the house might have been some other color, and might be another color tomorrow. Aristotle did not identify "species" as a predicable, although the concept entered into his discussion and his list of predicables is: definition, genus, differentia, property, and accident.
When we make inductions from the particulars of sense, we get generalizations and abstractions. Aristotle was confident that intuition, or direct seeing, gives the nature of what is sensed before him. In the comparison of any sense data, one can generalize about their nature, seeing what they have in common, and how they differ. In On Memory and Recollection (Ch. II, par. 611), Aristotle advanced the associative principles of similarity, contrast, and contiguity to apply to this process. Induction separates or abstracts the sense datum from some of the conditions of its existence in the world making generalization possible. When induction is done correctly, we are both thinking the form in, and abstracting the form from, the images or phantasmata of sense. Logic represents a higher level of abstraction, an abstraction of the universal essence from the particular existence of things in the world.
According to Aristotle, science, in the strict sense of the word, is "demonstrated knowledge of the causes of things." Such demonstrated knowledge is obtained by syllogistic deduction from premises that in themselves are true. Thus in Aristotle's view, science is a matter of syllogistic demonstrations from certain or true premises. Therefore, at the center of Aristotle's logic is the syllogism, that is, that form of reasoning whereby, given two propositions called the premises, a third proposition called the conclusion follows necessarily from them. Aristotle was the first philosopher to formulate the theory of the syllogism, and his minute analysis of its various forms was definitive, so far as subject-predicate propositions are concerned; so that to this part of deductive logic little has been added since his day. According to Aristotle, logic is the science of predicating properties of subjects and deductively drawing conclusions from them as premises by means of the syllogism. The basis of syllogistic reasoning is the presence of a term common to both premises (the middle term) so related as subject or predicate of each of the other terms of the premises that a conclusion may be drawn regarding the relation of these terms to one another. Since this syllogistic reasoning depends upon the "middle term," Aristotle believed that the advance of knowledge depended upon the discovery of "middle terms."
Aristotle gave clear expression to two of the three Laws of Thought. He says of the principle of contradiction: "The firmest of all principles is that it is impossible for the same thing to belong and not to belong to the same thing at the same time in the same respect." And of the principle of the excluded middle he says: "It is not possible that there should be anything between the two parts of a contradiction, but it is necessary either to affirm or deny one thing of any one thing." Aristotle nowhere states the principle of identity, although it is presupposed.
Aristotle also recognized the modality of propositions, that is, the alternate ways of classifying propositions with respect to their relations to existence. Today there are three widely recognized modes; they are the modes of possibility, actuality, and necessity. If we let "p" stand for any proposition, the mode of possibility is expressed by "It is possible that p is true." The mode of actuality expressed simply by "p is true." And the mode of necessity is expressed by "It is necessary that p is true." These propositions and their negations, the impossible, not actual, and not necessary, are the modal propositions. Aristotle speaks of four modes of propositions: those which are possible, impossible, contingent, and necessary. The contingent modality speaks of that which may or may not be actual. Hence, the mode of contingency presupposes the modality of the possible. The necessary, in turn, has the modality of both the possible and the actual within it.
Aristotle also held that when one's premise are not certain but only probable, a shift from science to dialectic has occurred. There is also a further shift to eristic where one's goal is not knowledge but merely victory in disputation.
Megarian School.
After Aristotle, his followers, the Peripatetic School, continued to develop Aristotelian logic. Its rival was the Megarian and later the Stoic schools of logic. The Megarian school, perhaps because of its connection with the dialectic of Zeno of Elea, was primarily interested in developing a propositional logic. Concentrating on "if ... then" and "either ... or" logical relationships, the Megarian school worked out the valid inference forms for these relations. Among the Megarians were Euclid (450-374 B.C.) of Megara (not to be confused with Euclid of Alexandria, the founder of Euclidean geometry), who was a disciple of Socrates (470-400 B.C.) and the founder of the school; Euclid's disciple, Ichtias; Eubulides of Miletus, who became the second director of the school and developed the Paradox of the Liar; his disciple, Philo of Megara; Stilpo of Megara and Diodorus Cronus, both of whom seem to have been teachers of Zeno (334-264 B.C.) of Citium who founded Stoicism. Philo of Megara (4th century B.C.) anticipated the Propositional Calculus in his definition of what now known as material implication: that is, a conditional proposition is true in all cases except where the antecedent is true and the consequent is false. In modal logic he introduced the definition of the possible as that which can be true by virtue of its internal nature, that is, the possible is self-consistent.
Stoic School.
Zeno of Citium is regarded as the founder of the Stoic school of philosophy whose views were refined and unified by his followers: Cleanthes and Chrysippus. Chrysippus (c.280-206 B.C.), who was born in Cilicia, studied in Athens under Zeno and Cleanthes, and became the third leader of the Stoics from 232 to 208 B.C., not only organized the Stoic philosophy into a system, but he defended this new system against the criticism of the Academy by creating Stoic logic. He defended it so well that the biographer, Diogenes Laertius (3rd century A.D.), said that without Chrysippus, Stoicism would not have existed. Of his logic Diogenes Laertius also said: "If the gods use dialectic, they can use none other than that of Chrysippus." Chrysippus went beyond Aristotle in developing the logic of propositions. He originated or at least formulated the five valid inference forms of the Propositional Calculus. They are:
1. If the first, then the second; but the first; therefore the second.
  (Modus Ponens).
2. If the first, then the second; but not the second; therefore not the first.
  (Modus Tollens).
3. Not both the first and the second; but the first; therefore not the second.
  (Disjunctive Syllogism).
4. Either the first or the second; but the first; therefore not the second.
  (Disjunctive Syllogism).
5. Either the first or the second; but not the second; therefore the first.
  (Disjunctive Syllogism).
The Stoics called these form the "indemonstrables" (from the Greek apodeiktikos), because these forms were thought to be self-evident, not requiring and not capable of demonstration.
Chrysippus also contributed to definitions of modality. According to Cicero, Chrysippus followed Philo of Megara's definition of the possible as the self-consistent. According to Diogenes Laertius, Chrysippus defined the possible as that which will be true when external circumstances don't prevent its being so, and the necessary as that which is true, and cannot become false either in itself or through external circumstances.
Epicurean School.
A second, although less formidable, rival to the logic of the Peripatetic School was the logic of the Epicurean School of philosophy, where the claim was made that all logical connections are derived empirically, and thus rest on induction or analogy. The school stressed its empirical approach to logic, finding the Stoic logic excessively rational in its approach. The philosophers of the Epicurean school likewise criticized the Peripatetics and the Skeptics on the grounds of using vacuous, rational arguments. The Epicurean philosopher, Philodemus of Gadara (1st century B.C.), who was born in Syria and studied in Athens under Zeno of Sidon and Demetrius of Laconia, as co-head of a school of Epicurean philosophy in Naples, engaged in polemic against the Stoics, holding that the relation between a sign and the things it signified is established in every case, either by induction or analogy. The Stoics position, on the other hand, based such connections on a relationship of logical necessity.
Porphyry.
The Neoplatonic philosopher Porphyry (c. 232-304 A.D.) modified Aristotle's list of five predicables by replacing "definition" with "species". Aristotle did not identify "species" as a predicable, although the concept entered into his discussion. Prophyry's list of predicables is: species, genus, differentia, property, and accident. He diagrammed the relation of the predicables genus, differentia, species, and individuals in illustrative graph that became known as the Tree of Prophyry (Arbor Porphrii). He presented this in the chapter on "species" in his work Isagoge. Taking the category of substance as his illustration, Porphyry diagrammed the relationship between this basic category, which is a genus and not a species, and other concepts of lesser generality but are also genus and species. The differentiae control these terms, down to the most narrow species which contains individuals as its members. At the right of the diagram he places the name of the each level of the diagram; for example, the highest level of the diagram that specifies the basic category "substance" is named "Summum Genus". Porphyry seems to have combined Plato's method of division with Aristotle's theory of predicables. The tree came into Latin literature through the translation by Boethius, and exercised great influence during the Middle Ages. Prophyry suggests that it is beyond the power of man to know whether genus and species exist as subsistent entities or just as concepts. Boethius' speculation on the manner of existence of genus and species sparked the medieval discussion of universals.
Boethius.
Boethius (480-525 A.D.) was a Roman philosopher who attempted a synthesis of Hellenistic, Roman and Christian thought, "saving" the best of the old, and using it to engender the new. He has been called the last Roman and the first Scholastic. He was Consul of Rome and minister to Theodoric, king of the Ostrogoths. He was accused of treason, imprisoned, and executed. While awaiting execution he composed his famous work, On the Consolation of Philosophy. In addition to numerous works in logic and theology, he translated Aristotle's works on logic, as well as commentaries by Prophyry, Cicero, and Victorinus. His development of philosophical terminology in Latin, prepared the ground for the appearance of Scholasticism. By equating "substance" with "hypostasis" Boethius defined person as an individual substance of rational nature. Boethius, in his commentary on the Isagoge of Porphyry, quotes Porphyry as remarking that at present he refuses to state whether genera and species are subsistent entities or whether they consist in concepts alone; if subsisting, whether they are material or immaterial and, further, whether they are separate from sensible objects or not, on the grounds that such exalted matters cannot be treated in an introduction. However, Boethius himself goes on to treat the subject, first of all remarking on the difficulty of the question and the need of care in considering it. He then points out that there are two ways in which an idea may be formed: by abstraction and by composition. If by composition, for example, one joins together arbitrarily man and horse to form the idea of a centaur, joining together objects which nature does not allow to be joined together, such arbitrarily constructed ideas are "false." On the other hand, if we form the idea of a line, as for example the line as considered by a geometer, although such a line does not exist by itself in extramental reality, the idea is not "false", since bodies involve lines and all we have done is to isolate the line and consider it in abstraction. Composition (as in the composition of horse and man to form the centaur) produces a false idea, whereas abstraction produces an idea which is true, even though the thing conceived does not exist extramentally but in the state of abstraction or separation. Thus he held that the ideas of genus and species are true ideas, since they are formed by abstraction, while the idea formed by composition (for example, a centaur) is false. The former "subsist in sensible things, but are understood without bodies". Consequently, "genera and species are in individuals, but, as thought, are universals". His discussion of this problem, including his comments on Porphyry, initiated the medieval discussion of universals.
Pseudo-Dionysius.
Boethius is one of the two channels that the philosophy of the ancient world was passed on the Middle Ages. The other channel was the writings of the Pseudo-Dionysius. This is the name given to unknown author who, at the end of the 4th or the beginning of the 5th century A.D., wrote a series of treatises on God and mystical theology. He used the name of Dionysius the Areopagite, who was mentioned in Acts 17:34 as one of the Athenian converts of the Apostle Paul. The high esteem and respect that these writings enjoyed among the theologians and philosophers of the Middle Ages was in great part due to the author's use of pseudonym: "Dionysius the Presbyter, to his fellow presbyter Timothy", which was mistaken as the name of the author. The writings were translated from the Greek into Latin by John Scotus Eringena among others and was the subject of commentaries by Hugo of Saint Victor, Robert Grosseteste, Albertus Magnus, and Thomas Aquinas. All these authors accepted the authenticity of the writings; but in time it became clear that they embodied important elements taken from developed neo-Platonism and they constituted in fact an attempt to reconcile neo-Platonism and Christianity, so that they would have to be attributed to an author of a much later date than the historic Dionysius the Areopagite. The classical schools of neo-Platonism existed at Athens from about 380 to 529 A.D. and at Alexandria in Egypt from about 430 A.D. until the Mohammedan conquest of Alexandria in 642 A.D. According to these writings, there are two ways of approaching God, who is the center of all speculation, a positive way (kataphatike) and a negative way (apophatike). In the positive way or method, the mind begins "with the most universal statements, and then through intermediate terms [proceeds] to particular titles", thus beginning with "the highest category". The affirmative way means ascribing to God the perfections found in creatures, that is, the perfections which are compatible with the spiritual Nature of God. In his work Divine Names (De divinis Nominibus) the Psuedo-Dionyius pursues the affirmative way showing how names such as Goodness, Life, Wisdom, Power, are applicable to God in a transcendental manner and how they apply to creatures only in virtue of their derivation from God and their varying degrees of participation in those qualities which are found in God not as inhering qualities but in substantial unity. Thus he begins with the idea or name of goodness, which is the most universal name, inasmuch as all things, existent or possible, share in goodness to some degree, but which at the same time expresses the Nature of God. God is the Good, and, as the Good, God is the overflowing source of creation and is its final end, and "from the Good comes the light which is an image of Goodness, so that the Good is described by the name of 'Light', being the archetype of that which is revealed in the image". Here the neo-Platonic influence is evident. The Pseudo-Dionysius dependence on neo-Platonism is particularly clear in his use of other names of God. In chapter 13 of the Divine Names, where the Pseudo-Dionysius speaks of "One" as "the most important title of all", he is clearly is echoing the Plotinian doctrine of the ultimate Principle as the One.
The negative way, which is the method of exclusion from God of the imperfections of creatures, is pursued in his Mystical Theology (De mystica Theologia). The Pseudo-Dionysius preferred the negative way. As God is utterly transcendent, we praise Him best "by denying or removing all things that are - just as men who, carving a statue out of marble, remove all the impediments that hinder the clear perception of the latent image and by this mere removal display the hidden statue itself in its hidden beauty". Human beings are inclined to form anthropomorphic concepts of God, and it is necessary to strip away these human, all-too-human conceptions by the via remotionis. But the Pseudo-Dionysius does not mean that from this process there results a clear view of what God is in Himself: the analogy of the statue making must not mislead us. When the mind has stripped away from its idea of God the human modes of thought, it enters into the "Darkness of Unknowing", wherein it "renounces all the apprehension of the understanding and is wrapped in that which is wholly intangible and invisible... united ... to Him that is wholly unknowable"; this is mysticism. This "Darkness of Unknowing" is not due to the unintelligibility of the Object considered in itself, but to the finiteness of the human mind, which is blinded by excess of light. This doctrine is doubtless influenced by neo-Platonism.
John Scotus Eringena.
The Irish churchman and philosopher, John Scotus Eringena (810-877 A.D.), was a follower of St. Augustine and Pseudo-Dionysius. He was born in Ireland and educated in an Irish monastery. By 850 he was in France at the court of Charles the Bald, and teaching in the Palatine School. By 855 he had begun his translation of the writings of the Pseudo-Dionysius from Greek into Latin, and produced a commentary on them. Scotus elaborated a system of ideas that attempts to integrate a philosophic interpretation of the universe with his Christianity. Scotus in his On the Division of Nature elaborates a hierarchical structure of nature. The term "nature" means to Scotus, not only the natural world, but also God and the supernatural sphere; it denotes all Reality. When he asserts that all of nature is divided into four species, namely,
1. nature which creates and is not created,
2. nature which is created and creates,
3. nature which is created and does not create, and
4. nature that neither creates nor is created,
he is apparently making God and creatures a species of nature, and seems to be asserting a monistic doctrine of reality and a pantheistic view of God. But at the beginning of Book 2 he makes clear that it is not his intention to assert that creatures are actually part of God or that God is genus of which creatures are species, although he retains his fourfold division of nature and says that God and creatures may be looked at as forming together a universitas, a "universe" or totality. "Nature which creates and is not created" is clearly God Himself, who is the cause of all things but is Himself without cause. He is the beginning or first principle, since all creatures proceed from Him, the "middle" (medium) since it is in Him and through Him that creatures subsist and move; and the end or final cause, since He is the terminus of the creature's movement of self-development and perfection. He is the first cause, which brought creatures into existence from a state of non-existence, out of nothing (ex nihilo). This doctrine of God is clearly in accordance with Christian theology and contains a clear statement of the divine transcendence and self-existence. But he also held that God is also nature which neither creates nor created. This seems to make God part of nature; yet this must be in very special sense since Scotus clearly asserts that God is above nature. Scotus borrowed, as he himself plainly affirms, from the Pseudo-Dionysius, the twofold method of theology, that is, that knowledge of God can be obtained by the affirmative (kataphatike) way and the negative (apophatike) way. When using the negative method one denies that the divine essence or substance is any of those things, "which are", as understood by us; when using the positive, one predicates of God those things "which are", in the sense that the cause is manifested in the effect. Scotus also takes from the same writer the idea should not be called Truth or Wisdom or Essence, but rather super-Truth, super Wisdom or super-Essence, since no names borrowed from creatures can be applied to God in their strict and usual sense; they can be applied to God metaphorically (metaphorice or translative). Also Scotus attempts to show that the use of the affirmative method does not contradict the doctrine of the ineffable and incomprehensible character of the Godhead and that the negative method is the fundamental one. By the affirmative method we take some significant property of existence and attribute this property to God, for example, "God is wise." But by the negative method, God is not wise. This seems to be a contradiction, but in fact both assertions are correct. God is wise and not-wise, because He is super-wise. And so too with other properties; God is super-substance, super-essence, super-goodness. Although human categories cannot be predicated of God, the use of these two methods allows Scotus to relate God to all that is. The ten categories of Aristotle, the basic category of substance of which the others are attributes or ways in which substance exists: in a certain quantity, of a certain quality, with a certain relation, in a certain place, at a certain time, in a certain position, in a certain state, and in certain state of action (active), or being acted upon (passive), do not apply to God. Since He is the creator of all substances, God is infinitely more than substance. The categories are founded upon and apply to created things and are strictly inapplicable to God. God transcends every human category and, though we can learn from the creatures that God is, we cannot learn what He is. But though this doctrine of the inapplicability of the categories to God would seem to place the transcendence of God and the clear distinction between Him and creatures beyond all doubt, his consideration of the category of making (facere) and action seems to leads Scotus to a very different conclusion. Scotus argued that since making involves motion and motion cannot be attributed to God, then neither can making be attributed to God. Now, if making cannot be attributed to God, then how can we explain the Scriptural doctrine that God made all things? Scotus explanation is that God's making must be co-eternal with Himself. For if God existed before He made the world, then God would not only be in time but also His making would be an accident accruing to Him. Since both suppositions are impossible, then making is eternal and identical with God, and not an accident of God. "When we hear that God makes all things, we should understand nothing else but that God is in all things, i.e. is the essence of all things. For He alone truly is, and everything which is truly said to be in those things which are, is God alone." Such statements would seem to come very near to pantheism, to the doctrine of Spinoza. In spite of this pantheistic passage quoted, he goes on to reaffirm creation out of nothing, and it is clear when he refuses to say that God makes or made the world, he is not intending to deny creation but rather to deny of God making in the only sense in which making is understood, namely as an accident, as falling under a particular category.
Abelard.
Peter Abelard (1079-1142 A.D.), famed throughout Europe for his skill in dialectics, composed four treatises on logic. He believed that he had discovered the laws governing all human thought. Thus believing he had an instrument of universal application, he applied it with equal vigor and unequal success to logical and theological problems. In logic Abelard abandoned the realism of the Platonic tradition. Marshaling a variety of arguments against his contemporary realists, he concluded with the assertion that universality, the characteristic by reason of which the same term can be predicated of several things, can in no wise be anything real but must of necessity belong to the term, the word which is so predicated. This word signifies no essence, nothing real, but it does make known a status, a state, for example, "to be man." Here we have a denial of Plato, not in the name of Aristotle, but in the name of a logic conceived as the unique method of knowing all objects, whatever they may be. But this denial had a curious consequence. Since the object of the universal idea is nothing real, it follows that the most genuine and valid knowledge that can be had is not of the universal, the abstracted form of Aristotle, the verum ens of Platonism, but is of particular things. So much for Aristotelian science. In logic he regarded the verb "to be" as the copula of every categorical proposition (although he was not the first to do so), and held that the copula does not predicate existence. His claim that an affirmative categorical proposition is true only when the subject and predicate terms stand for the same things led to the theory of Suppositio terminorum ("substitution of terms"). He dealt with some problems of negation and modality. He entered the argument concerning the paradoxes of implication by holding that the antecedent of a true conditional statement must require the consequent intrinsically in order to rule out the truth of such conditionals as "If Socrates is a stone, then he is a ass." Among valid rules for argument he included early versions of modus ponendo ponens and modus tollendo tollens, transitivity, negation and the interrelations of modal statements. He abandoned the position of Stoic logicians that disjunction is to be interpreted strongly.
"New Logic"
After Abelard died, the remaining books of Aristotle's Organon became available in Western Europe in translation from the Arabic and Greek. These were quickly followed by a translation of Aristotle's Metaphysics, Physics, and De Anima, along with works by the Arabic philosophers Avicenna and Averroes. The newly received part of the Organon came to be called the "new logic" (ars nova), to distinguish them from the previously known parts called the "old logic" (ars vetus). In the the new part of the Organon, the De Sophisticis Elenchis had immediate appeal because it had not been contained in the Boethian writings, and because the detection and resolution of fallacious arguments was very useful to the scholatic method of the "disputed question," then becoming established in the schools. With the formation of the new universities of Paris and Oxford at the beginning of 13th century, the teaching of logic fell within the providence of the lower faculty of arts, but because the philosophical works of Aristotle and the Arabic writers were considered dangerous to orthodox teaching, the study of them were reserved for the higher faculty of theology. As a result the development of logic within the arts faculty continued along the formal and linguistic lines that had been established in the 12th century, enriched with materials drawn from the new translations of the Prior Analytics, Topics, and De Sophisticis Elenchis. The teachers of the arts faculty disregarded the Posterior Analytics, as it seem to them to be irrelevant to the study of formal logic. But the theologians studied the Posterior Analytics, and the whole of the Organon, in the context of Aristotle's metaphysics and theory of knowledge. Utilizing the works of Avicenna and Averroes, they wrote literal commentaries on the Aristotelian works in an effort to recover the "original Aristotle" in authentic form. Such commentaries were written by Robert Grosseteste (1175-1253), Thomas Aquinas (1224-1274), Robert Kilwardby (died 1279), Giles of Rome (c.1247-1316). Albert the Great (died 1279) wrote paraphrases of each book of the Organon as a part of his encylopedic enterprise to write corresponding treatises on all the works of Aristotle in order to recover the "original Aristotle". With respect to logic there arose in the 13th century an Aristotelian "purism" promoted by the theologians, occuring at the same time with the development of new methods and problems in logic of the arts faculty. This Aristotelian purism came to be called teh logica antiqua and the logic of the arts faculty the logica moderna.
William of Sherwood.
Among the representatives of the logic in the arts faculty of the University of Paris was William of Sherwood and Peter of Spain. In the 13th century the English philosopher, William of Sherwood (c.1210-1270), produced a manual of logic, Introductiones in Logicam (Introduction to Logic), in which the valid moods in each figure of the syllogism is presented by mnemonic verses with the names Barbara, Celarent, Cesare, Festino, etc. The sequence of vowels in these names indicate the mood; "Barbara", for example, is the name for the mood AAA, so that the valid syllogism AAA is called a syllogism in Barbara. Medieval logicians had classified the valid forms of the syllogism; they called the universal affirmative proposition ("All men are mortal.") an "A" proposition, and the particular affirmative proposition ("Some Athenians are virtuous.") an "I" proposition (from the first two vowels of the Latin affirmo, meaning "to affirm"). They called the universal negative proposition ("No men are immortal.") an "E" proposition, and the particular negative proposition ("Some Athenians will not betray a trust.") an "O" proposition (from the first two vowels of the Latin nego, meaning "to deny"). Thus the syllogism --
All men are mortal;
All partriarchs are men;
Therefore, All partriarchs are mortal.
-- is in mood AAA (naming in order the major premise, minor premise, and conclusion) and is a syllogism in Barbara. And the syllogism --
No virtuous beings betray a trust;
Some Athenians are virtuous;
Therefore, Some Athenians will not betray a trust.
-- is in mood EIO and is a syllogism in Festino.
William studied and taught at Oxford and in Paris from where he influenced Peter of Spain, Lambert of Auxerre, Albert Magnus, and Thomas Aquinas. He was one of the founders of the 13th century movements called terminist logic in which the nature of language was emphasized; the word was treated first as a physical entity, and only secondarily as significant term. This expressed the understanding in early medieval schools of logic as an "art of language," closely associated with grammar and rhetoric, and useful in the interpretation of the texts of the Bible and of the Church Fathers and in the reconciliation of the apparent condradictions found in them. This shaped the medieval conception of logic as a discipline concerned with the syntax and semantics of natural language and with the validity of inference forms. William considered the conjunctive, disjunctive, and conditional propositions as they are used in Modern logic. In his work Syncategoremata, he analyzed what is now called logical constants: "and," "or," "not," "if," "every," "some," etc., and gave to them truth-functional definitions.
In the same century, Peter of Spain (1226-1277), later Pope John XXI (1276), composed his Summulae Logicales, a logic textbook widely used in the later Middle Ages and up to the 17th century, passing through 150 printed editions. The point of view is very similar to that of William of Sherwood.