Who invented syllogism

The distinction, however, did not escape Aristotle, who saw that a progressive syllogism can be reversed thus: - 2. Yet such is the passion for one type that from Aristotle's time till now constant attempts have been made to reduce induction to syllogism.

Aristotle himself invented an inductive syllogism in which the major P is to be referred to the middle M by means of the minor S , thus: A, B, C magnets S attract iron P. Whately, on the other hand, proposed an inductive syllogism with the major suppressed, that is, instead of the minor premise above, he supposed a major premise, " Whatever belongs to A, B, C magnets belongs to all.

Reduction he defines as " the framing of possible premises for given propositions, or the construction of a syllogism when the conclusion and one premise is given. Lastly, Wundt's view is an interesting piece of eclecticism, for he supposes that induction begins in the form of Aristotle's inductive syllogism , S-P, S-M, M-P, and becomes an inductive method in the form of Jevons's inverse deduction, or hypothetical deduction, or analysis, M-P, S-M, S-P.

In detail, he supposes that, while an " inference by comparison," which he erroneously calls an affirmative syllogism in the second figure, is preliminary to induction, a second " inference by connexion," which he erroneously calls a syllogism in the third figure with an indeterminate conclusion, is the inductive syllogism itself.

This is like Aristotle's inductive syllogism in the arrangement of terms; but, while on the one hand Aristotle did not, like Wundt, confuse it with the third figure, on the other hand Wundt does not, like Aristotle, suppose it to be practicable to get inductive data so wide as the convertible premise, " All S is M, and all M is S," which would at once establish the conclusion, " All M is P. Wundt's point is that the conclusion of the inductive syllogism is neither so much as all, nor so little as some, but rather the indeterminate "M and P are connected.

He agrees with Jevons in calling this second syllogism analytical deduction, and with Jevons and Sigwart in calling it hypothetical deduction. Hence induction cannot be reduced to Aristotle's inductive syllogism , because experience cannot give the convertible premise, " Every S is M, and every M is S "; that "All A, B, C are magnets " is, but that " All magnets are A, B, C " is not, a fact of experience.

The fact is that the uniformity of nature stands to induction as the axioms of syllogism do to syllogism ; they are not premises, but conditions of inference, which ordinary men use spontaneously, as was pointed out in Physical Realism, and afterwards in Venn's Empirical Logic.

The axiom of contradiction is not a major premise of a judgment: the dictum de omni et nullo is not a major premise of a syllogism : the principle of uniformity is not a major premise of an induction. It is not syllogism in the form of Aristotle's or Wundt's inductive syllogism , because, though starting only from some particulars, it concludes with a universal; it is not syllogism in the form called inverse deduction by Jevons, reduction by Sigwart, inductive method by Wundt, because it often uses particular facts of causation to infer universal laws of causation; it is not syllogism in the form of Mill's syllogism from a belief in uniformity of nature, because few men have believed in uniformity, but all have induced from particulars to universals.

Bacon alone was right in altogether opposing induction to syllogism , and in finding inductive rules for the inductive process from particular instances of presence, absence in similar circumstances, and comparison.

There are, as we have seen ad init. Bradley seems to suppose that the major premise of a syllogism must be explicit, or else is nothing at all. But it is often thought without being expressed, and to judge the syllogism by its mere explicit expression is to commit an ignoratio elenchi; for it has been known all along that we express less than we think, and the very purpose of syllogistic logic is to analyse the whole thought necessary to the conclusion.

Aristotle, however, treats it as a dialectical rival to syllogism , and it influenced Galilei and Bacon in their views of inference after the Renaissance. Plato's division is nevertheless neither syllogism nor exclusiva. It is not syllogism because it is based on the disjunctive, not on the hypothetical relation, and so extends horizontally where syllogism strikes vertically downward.

Again it is not syllogism because it is necessarily and finally dialectical. This is embodied in the group of treatises later known as the Organon 7 and culminates in the theory of syllogism and of. In the well-known sentences with which the Organon closes 8 Aristotle has been supposed to lay claim to the discovery of the principle of syllogism.

In the course of inquiry into the formal consequences from probable premises, the principle of mediation or linking was so laid bare that the advance to the analytic determination of the species and varieties of syllogism was natural.

It is in the Topics, further, that we clearly have a first treatment of syllogism as formal implication, with the suggestion that advance must be made to a view of its use for material implication from true and necessary principles. In any case, however, definition, syllogism , induction all invited further determination, especially if they were to take their place in a doctrine of truth or knowledge.

Again, so long as we keep to the syllogism as complete in itself and without reference to its place in the great structure of knowledge, the nerve of proof cannot be conceived in other than a formal manner.

The forms of syllogism , however, are tracked successfully through their figures, i. Syllogism already defined 1 becomes through exhibition in its valid forms clear in its principle. Syllogism must indeed be objective, i. Syllogism as formula for the exhibition of truth attained, and construction or what not as the instrumental process by which we reach the truth, have with writers since Hegel and Herbart tended to fall apart.

In the Posterior Analytics the syllogism is brought into decisive connexion with the real by being set within a system in which its function is that of material implication Posterior from principles which are primary, immediate and Analytics.

A formula such as the equality of the interior angles of a triangle to two right angles is only scientifically known when it is not of isosceles or scalene triangle that it is known, nor even of all the several types of triangle collectively, but as a predicate of triangle recognized as the widest class-concept of which it is true, the first stage in the progressive differentiation of figure at which it can be asserted.

Of a non-self-subsistent or attributive conception definition in its highest attainable form is a recasting of the syllogism , in which it was shown that the attribute was grounded in the substance or self-subsistent subject of which it is.

In the scientific syllogism the interposition of the earth is the middle term, the cause or " because " Sairt. In the aporematic treatment of the relation of definition and syllogism identical as to one form and in one view, distinct as to another form and in another view, much of Aristotle's discussion consists.

The advance from syllogism as formal implication is a notable one. The planets are near, and we know it by their not twinkling, 2 but science must conceive their nearness as the cause of their not twinkling and make the Arius in the real order the middle term of its syllogism.

Thirdly we have the limiting cases of this in the inductive syllogism 5ui 7riu'mw, 7 a syllogism in the third figure concluding universally, and yet valid because the copula expresses equivalence, and in analogy 8 in which, it has been well said, instances are weighed and not counted. The Aristotelian theory of the universal of science as secure from dependence on its instances and the theory of linking in syllogism remain a heritage for all later logic, whether accepted in precisely Aristotle's formula or no.

The major premise of syllogism , says the Pyrrhonist, is established inductively from the particular ' 'Errt4 opcc. Experience is appealed to as fruitful where the formal employment of syllogism is barren.

Rather a scientific process, which as experiential may be called inductive, but which is in other regards deductive as syllogism , is set up in constrast to syllogism YvI. The syllogism is ineffective, belonging to argumentation, and constraining assent where what we want is control of things. It is in virtue of this view of derived or mediate knowledge that Descartes speaks of the subsumptive syllogism as " of avail rather in the communication of what we already know.

But once we have established this general rule, we can move on to the second step in our argument, using this conclusion as a premise in an enthymeme. We can argue that all people asking for a bodyguard are scheming to make themselves despots, that Dionysius is someone asking for a bodyguard, and that therefore, Dionysius must be scheming to make himself despot.

Nonetheless, we can, in this way, induce probable conclusions and then use them to deduce probable consequences. Although these arguments are intended to be persuasive or plausible rather than scientific, but the reasoning strategy mimics the inductive-deductive movement of science without compelling, of course, the same degree of belief. We should point out that Aristotle does not restrict himself to a consideration of purely formal issues in his discussion of rhetoric.

He famously distinguishes, for example, between three means of persuasion: ethos , pathos , and logos. Secondly, persuasion may come through the hearers, when the speech stirs their emotions. Thirdly, persuasion is effected through the speech itself when we have proved [the point] by means of the persuasive arguments suitable to the case in question.

Aristotle concludes that effective arguers must 1 understand morality and be able to convince an audience that they themselves are good, trustworthy people worth listening to ethos ; 2 know the general causes of emotion and be able to elicit them from specific audience pathos ; and 3 be able to use logical techniques to make convincing not necessarily sound arguments logos.

Aristotle broaches many other issues we cannot enter into here. He acknowledges that the goal of rhetoric is persuasion, not truth. Such techniques may be bent to immoral or dishonest ends. Nonetheless, he insists that it is in the public interest to provide a comprehensive and systematic survey of the field. Unfortunately, Aristotle never explicitly explains what a topos is. The technical term derives from a Greek word referring to a physical location. Some scholars suggest a link to ancient mnemonic techniques that superimposed lists on familiar physical locations as a memory aid.

In relevant discussions in the Topics and the Rhetoric Aristotle offers helpful advice about finding or remembering suitable premises, about verbally out-manoeuvring an opponent, about finding forceful analogies, and so on. Examples of specific topoi would include discussions about how to argue which is the better of two alternatives, how to substitute terms effectively, how to address issues about genus and property, how to argue about cause and effect, how to conceive of sameness and difference, and so on.

Some commentators suggest that different topoi may have been used in a classroom situation in conjunction with student exercises and standardized texts, or with written lists of endoxa , or even with ready-made arguments that students were expected to memorize. An aporia is a common device in Greek philosophy. The Greek word aporia plural, aporiai refers to a physical location blocked off by obstacles where there is no way out; by extension, it means, in philosophy, a mental perplexity, an impasse, a paradox or puzzle that stoutly resists solution.

Aristotle famously suggests that philosophers begin with aporiai and complete their task by resolving the apparent paradoxes. An attentive reader will uncover many aporiai in Aristotle who begins many of his treatises with a diaporia , a survey of the puzzles that occupied previous thinkers.

Note that aporiai cannot be solved through some mechanical rearrangement of symbolic terms. Solving puzzles requires intelligence and discernment; it requires some creative insight into what is at stake. In a short work entitled Sophistical Refutations , Aristotle introduces a theory of logical fallacies that has been remarkably influential.

His treatment is abbreviated and somewhat obscure, and there is inevitably scholarly disagreement about precise exegesis. Aristotle thinks of fallacies as instances of specious reasoning; they are not merely errors but hidden errors. A fallacy is an incorrect reasoning strategy that gives the illusion of being sound or somehow conceals the underlying problem. Aristotle divides fallacies into two broad categories: those which depend on language sometimes called verbal fallacies and those that are independent of language sometimes called material fallacies.

There is some scholarly disagreement about particular fallacies, but traditional English names and familiar descriptions follow. Linguistic fallacies include: homonymy verbal equivocation , ambiguity amphiboly or grammatical equivocation , composition confusing parts with a whole , division confusing a whole with parts , accent equivocation that arises out of mispronunciation or misplaced emphasis and figure of speech ambiguity resulting from the form of an expression.

Independent fallacies include accident overlooking exceptions , converse accident hasty generalization or improper qualification , irrelevant conclusion, affirming the consequent assuming an effect guarantees the presence of one possible cause , begging the question assuming the point , false cause, and complex question disguising two or more questions as one.

Logicians, influenced by scholastic logic, often gave these characteristic mistakes Latin names: compositio for composition, divisio for division, secundum quid et simpliciter for converse accident, ignoranti enlenchi for nonrelevant conclusion, and petitio principii for begging the question.

Obviously, from a Greek perspective, one ought to obey both. The problem is that the question has been worded in such a way that anyone who answers will be forced to reject one moral duty in order to embrace the other. In fact, there are two separate questions here—Should one obey the wise? The same effect may have several causes.

But the interest here is theoretical: figuring out where an obviously-incorrect argument or proposition went wrong.

We should note that much of this text, which deals with natural language argumentation, does not presuppose the syllogistic form. Aristotle does spend a good bit of time considering how fallacies are related to one another. Fallacy theory , it is worth adding, is a thriving area of research in contemporary argumentation theory. Some of these issues are hotly debated. In the modern world, many philosophers have argued that morality is a matter of feelings, not reason.

Although Aristotle recognizes the connative or emotional side of morality, he takes a decidedly different tack. As a virtue ethicist , he does not focus on moral law but views morality through the lens of character. An ethical person develops a capacity for habitual decision-making that aims at good, reliable traits such as honesty, generosity, high-mindedness, and courage.

To modern ears, this may not sound like reason-at-work, but Aristotle argues that only human beings—that is, rational animals—are able to tell the difference between right and wrong. The operation of practical wisdom, which is more about doing than thinking, displays an inductive-deductive pattern similar to science as represented in Figure 3.

It depends crucially on intuition or nous. One induces the idea of specific virtues largely, through an exercise of non-discursive reason and then deduces how to apply these ideas to particular circumstances. We can distinguish then between moral induction and moral deduction.

In moral induction, we induce an idea of courage, honesty, loyalty, and so on. We do this over time, beginning in our childhood, through habit and upbringing. Once this intuitive capacity for moral discernment has been sufficiently developed—once the moral eye is able to see the difference between right and wrong,—we can apply moral norms to the concrete circumstances of our own lives.

In moral deduction, we go on to apply the idea of a specific virtue to a particular situation. We do not do this by formulating moral arguments inside our heads, but by making reasonable decisions, by doing what is morally required given the circumstances. Consider a somewhat simplified example. Suppose I induce the idea of promise-keeping as a virtue and then apply it to question of whether I should pay back the money I borrowed from my brother. The corresponding theoretical syllogism would be: Promise-keeping is good; giving back the money I owe my brother is an instance of promise-keeping; so giving the back the money I owe my brother is good.

The physical exchange of money counts as the conclusion. One induces a general principle and deduces a corresponding action. Aristotle does believe that moral reasoning is a less rigorous form of reasoning than science, but chiefly because scientific demonstrations deal with universals whereas the practical syllogism ends a single act that must be fitted to contingent circumstances.

There is never any suggestion that morality is somehow arbitrary or subjective. One could set out the moral reasoning process using the moral equivalent of an inductive syllogism and a scientific demonstration. Although Aristotle provides a logical blueprint for the kind of reasoning that is going on in ethical decision-making, he obviously does not view moral decision-making as any kind of mechanical or algorithmic procedure. Moral induction and deduction represent, in simplified form , what is going on.

Throughout his ethics, Aristotle emphasizes the importance of context. The practice of morality depends then on a faculty of keen discernment that notices, distinguishes, analyzes, appreciates, generalizes, evaluates, and ultimately decides.

In the Nicomachean Ethics , he includes practical wisdom in his list of five intellectual virtues. Scholarly commentators variously explicate the relationship between the moral and the intellectual virtues. Aristotle also discusses minor moral virtues such as good deliberation eubulia , theoretical moral understanding sunesis , and experienced moral judgement gnome.

And he equates moral failure with chronic ignorance or, in the case of weakness of will akrasia , with intermittent ignorance. Louis F. Groarke Email: lgroarke stfx. Francis Xavier University Canada. Aristotle: Logic Aristotelian logic, after a great and early triumph, consolidated its position of influence to rule over the philosophical world throughout the Middle Ages up until the 19 th Century. Categories The world, as Aristotle describes it in his Categories , is composed of substances—separate, individual things—to which various characterizations or properties can be ascribed.

From Words into Propositions Aristotle does not believe that all reasoning deals with words. Kinds of Propositions Aristotle suggests that all propositions must either affirm or deny something. Square of Opposition Aristotle examines the way in which these four different categorical propositions are related to one another.

Figure 1 The Traditional Square of Opposition As it turns out, we can use a square with crossed interior diagonals Fig. Inductive Syllogism Understanding what Aristotle means by inductive syllogism is a matter of serious scholarly dispute.

Science Aristotle wants to construct a logic that provides a working language for rigorous science as he understands it. A simple diagram of how science operates follows Figure 2. Non-Discursive Reasoning The distinction Aristotle draws between discursive knowledge that is, knowledge through argument and non-discursive knowledge that is, knowledge through nous is akin to the medieval distinction between ratio argument and intellectus direct intellection.

Rhetoric Argumentation theorists less aptly characterized as informal logicians have critiqued the ascendancy of formal logic, complaining that the contemporary penchant for symbolic logic leaves one with an abstract mathematics of empty signs that cannot be applied in any useful way to larger issues.

Fallacies In a short work entitled Sophistical Refutations , Aristotle introduces a theory of logical fallacies that has been remarkably influential. Moral Reasoning In the modern world, many philosophers have argued that morality is a matter of feelings, not reason. References and Further Reading a. Primary Sources Complete Works of Aristotle.

Edited by Jonathan Barnes. Princeton, N. The standard scholarly collection of translations. Aristotle in 23 Volumes. Cambridge, M. A scholarly, bilingual edition. Secondary Sources This list is intended as a window on a diversity of approaches and problems. Barnes, Jonathan, Aristotle Posterior Analytics.

Biondi, Paolo. Aristotle: Posterior Analytics II. Quebec, Q. Leiden: Brill, Engberg-Pedersen, Troels. Englebretsen, George. Assen, Netherlands: Van Gorcum, See also Sommers, below. Garrett, Dan, and Edward Barbanell. Encyclopedia of Empiricism. Westport, Conn.

Govier, Trudy. Problems in Argument Analysis and Evaluation. Providence, R. Groarke, Louis. Hamlyn, D. Oxford: Clarendon Press, Irwin, Terence. Keyt, David. Oxford University Press, McKirahan, Richard Jr.

Parry, William, and Edward Hacker. Aristotelian Logic. Peters, F. Rijk, Lambertus Marie de. Aristotle: Semantics and Ontology. Boston, M. Smith, Robin. Aristotle, Prior Analytics. Indianapolis, IN: Hackett, E, Zalta. Stanford, CA. Aldershot UK: Ashgate, His most thorough treatment of the theory of the syllogism can be found in the Dialectica , though he occasionally discusses it in other works as well, such as the Logica ingredientibus Minio-Paluello It is only in the Dialectica , however, that the theory is outlined in full.

But neither of these discussions is very extensive. Taken together, they are shorter than the discussion of topical inferences, which indicates that Abelard was most interested in developing a logic for sentences Green-Pedersen and Martin His presentation of syllogistic is condensed but highly original.

He must have seen it, but he cannot have had access to a copy himself. Abelard gives the four standard figures and shows how the second, third, and fourth he treats the fourth figure as part of the first figure with the terms in the conclusion converted can be reduced to the first in the standard ways using conversion rules and proofs through impossibility, but to clarify and simply the theory he also presents rules showing the validity of the different moods.

If we allow that the conjuncts in the antecedent of these conditional statements can switch places, and that a universal implies a particular, these rules exhaust the 24 valid syllogisms. These rules are based on the transitivity of class inclusion and were the standard way in which later medieval logicians explained how the first figure moods are perfect or evident.

It was elegant of Abelard to lay out these rules that entail the valid moods, but then again, the theory of the syllogism is an elegant and simple system. I will therefore not treat it in this overview, since it belongs to the history of sentential logic rather than syllogistic. Abelard is also associated with the history of modal logic. He is famous as the philosopher who introduced the distinction between de dicto and de re modal sentences.

Minio-Paluello Abelard concentrates his analysis on the logical structure of modal sentences, introducing some new distinctions and concepts that were later commonly used by medieval logicians. According to Abelard, modal terms are strictly speaking adverbs expressing how something said of the subject is actualized, e. Adverbs that do not modify an actual inherence, e. Abelard also noticed that in De interpretatione 12—13, Aristotle operates with nominal rather than adverbial modes, e.

He seems to have assumed that Aristotle did this because the nominal modes lead to many more problems than simple adverbial modes.

He calls these two alternatives de re necessity sentences and de sensu or de dicto necessity sentences, respectively.

Abelard seems to be the first to employ this terminology. A de re modal sentence expresses the mode through which the predicate belongs to the subject. The mode is, therefore, associated with a thing, whereas the mode in the de dicto case as he also calls it is said of what is expressed by a non-modal sentence. After Abelard, equipollence and other relations between modal sentences were commonly presented with the help of the square of opposition, which Abelard mentions though it does not appear as such in his works.

The square can be taken to refer to de dicto modal sentences or to singular de re modal sentences. Although the distinction between de dicto and de re modal sentences was common in logical treatises on the properties of the terms, syncategorematic terms, and the solution of sophisms, twelfth- and thirteenth-century logicians were mainly interested in the logical properties of singular de re modal sentences.

There is no detailed theory of quantified de re modal sentences from this period, and the first movements in this direction by Abelard and his followers were rather confused. A satisfactory theory of de re modal sentences did not appear until the fourteenth century, when the various relations between such sentences was presented by John Buridan in his octagon of opposition. Medieval logicians generally assumed that Aristotle dealt with de dicto modal sentences in the De Interpretatione and de re modal sentences in the Prior Analytics.

One reason may be that the only theory available concentrated on singular de re modal sentences, which are not part of modal syllogistic as developed by Aristotle. The structure of a composite modal sentence can be represented as follows:. A composite modal sentence corresponds to a de dicto modal sentence. The structure of a divided modal sentence can be represented as follows:. This type of modal sentence was characterized as de re because what is modified is how things res are related to each other, rather than the truth of what is said by the sentence dictum see Lagerlund 35—39, and the entry on medieval theories of modality for further details.

He says very little about it in his logical works, however. In less than five pages in the Dialectica — he treats modal, oblique, and temporal syllogistic logic. Earlier in the same work, he says a little about conversion rules.

He argues in both the Dialectica — and the Logica 15—16 that the conversion rules can be defended even on a de re reading, but the conversions he discusses are not modal conversions since the mode must be attached to the predicate and follow the term in the conversion, making the conversion into the conversion of an assertoric sentence. The conversions of de re modal sentences, as Abelard has defined them, do not hold, as Paul Thom has convincingly shown.

Thom 57— He also shows that uniform modal syllogisms are not generally valid, so that MMM is not valid unless the middle term in the major premise is read with the mode attached to it, as in:. MMM is consequently reduced to M—M. Abelard was therefore not attempting an interpretation of Aristotle, but must be seen as developing a new system based on his reading of de re sentences.

But this project must overcome several problems, particularly since Abelard cannot use the conversion rule. He dates it to c. By the time Kilwardby wrote his commentary, however, a Latin translation of a commentary by Averroes on the Prior Analytics was becoming known in the West.

In addition to these, a major commentary on the Posterior Analytics also became available. Nevertheless, his commentaries played an indispensable role throughout the later Middle Ages in the teaching and study of these difficult texts. One thing Averroes does do in these commentaries, however, is to build a strong connection between logic and a realist metaphysics, which had a clear influence on thirteenth-century logicians in the Latin West Lagerlund , In his commentary on the Prior Analytics , he pursues a line of interpretation which is more developed in the Quaesitum , a short treatise on mixed syllogisms see Uckelman and Lagerlund In the Quaesitum , Averroes focuses on modal syllogistics and develops an interpretation based on the metaphysical nature of the terms involved in different syllogisms.

It has been claimed that this short work is the final result of his inquiries into modal syllogistics Elamrani-Jamal , p. The Quaesitum has been studied by scholars in detail insofar as it clearly influenced Robert Kilwardby c. Although Kilwardby added nothing of substance to the theory of the assertoric syllogism, his interpretation of modal syllogistic is quite remarkable. It was also very influential in the thirteenth and early fourteenth centuries.

Albert the Great, Simon of Faversham, and Radulphus Brito — in other words, all of the major thirteenth-century commentators on the Prior Analytics — followed Kilwardby in their interpretations.

He begins by considering a counterexample to the accidental conversion of necessity sentences:. As we have seen, this is a common issue for de re readings of the modal sentences. He proceeds to give two separate solutions to this puzzle. The first is based on a distinction between different readings of According to Kilwardby, the meaning of the original subject term is changed when it no longer stands for the suppositum literate being , but for the abstract quality of being literate, and it is this change that blocks the conversion.

Kilwardby, however, preferred another solution to these difficulties for the conversion rules of necessity sentences. The second solution is based on a distinction between sentences that are necessary per se and those that are necessary per accidens. He writes I, fol. Kilwardby implies that the relation between the subject and predicate terms must be of a special kind if a sentence is to be called necessary per se.

Kilwardby thinks that sentences per se should be understood following An. Aristotle says that the first type of per se predication per se primo modo occurs when the definition of the subject includes the predicate. The second type of per se predication per se secundo modo occurs when the definition of the predicate includes the subject. A sentence is per se necessary if it involves either of these two predications, according to Kilwardby.

Necessity per accidens belongs to all other necessity sentences, which lack this intrinsic relation between subject and predicate. He seems to assume that in a per se necessity sentence, the subject term is not an accidental term but an essential or necessary term, and that the subject is essentially per se linked to the predicate rather than merely through the weaker relation of inseparability.

Consequently, if the subject term is necessary and the link is necessary, it follows that the predicate term cannot be merely a contingent accidental term. It must be necessary as well. The Aristotelian theory of necessity syllogistic is thus limited to a special class of terms, all of which stand for substances.

The same terminology is also used to explain syllogistic for contingency sentences, which suggests that Kilwardby was trying to develop a uniform and highly original interpretation of the theory.

A number of recent scholars have offered similar interpretations of Aristotle see van Rijen , Patterson , Thom , and Nortmann In the mixed syllogism L—L L represents a necessity sentence , the assertoric minor premise cannot be any kind of assertoric sentence because then the terms could merely be accidentally connected.

Kilwardby therefore introduced a distinction between absolutely simpliciter and as-of-now ut nunc assertoric sentences. The origins of this distinction can be found in Aristotle An. An absolutely assertoric sentence involves a per se predication whereas an as-of-now assertoric sentence involves a per accidens predication. In this way, he can guarantee that an essential connection between the terms in a valid L—L syllogisms is preserved through to the conclusion. This is not unproblematic see Lagerlund , 39—42 , though the distinction between different assertoric sentences needed somehow to be made and remained a problem throughout the later Middle Ages.

In the end, Kilwardby did not arrive at just the moods accepted by Aristotle. There are also some other moods he does not succeed in validating and others still he grants but which are not accepted by Aristotle. See Knuuttila , Lagerlund , and Thom and An important figure in the history of syllogistic logic is Richard of Campsall c.

Sometime before he wrote his Questions on the Books of the Prior Analytics Questiones super librum Priorum Analeticorum , a commentary on the first book of the Prior Analytics that devotes 14 of its 20 questions to modal syllogistic. He seems to think that there is nothing to add to the theory of assertoric syllogistic and his presentation of it is fairly standard, but he has lots of interesting things to say about modal syllogistic.

Campsall seems to have held that the system of modal syllogisms presented in the Prior Analytics was intended for divided modal sentences, and so he tries to prove that what Aristotle said is basically correct when modal sentences are understood in this way. But this turns out to be a very cumbersome task.

It is no surprise that he does not quite succeed, as he occasionally admits. In his reply to one of the questions in his commentary, he makes a brief remark about the difference between composite and divided modal sentences.

The universal negative modal sentence is singular when it is taken in the composite sense, that is, when it is read so that the modality is predicated of what a non-modal proposition expresses dictum or, as Campsall says, when it is predicated of the inherence.

He goes on to explain that a necessity sentence in the composite sense signifies that the corresponding non-modal sentence is necessarily true. When the universal negative necessity proposition is taken in the divided sense, it is universal. The modality does not qualify the dictum as a whole, but only the mode of removal of whatever is under the predicate term from whatever is under the subject term. Both the conversion rules and the syllogisms for modal sentences in the composite sense are validated by a small number of consequences, such as:.

The corresponding non-modal sentence is here assumed to be valid. Similar consequences can be formed for possibility and contingency sentences. These exhaust the theory of syllogism for composite modal sentences and Campsall accordingly spends little time elaborating it. Therefore, he must show how the conversion rules can be made to hold on such a reading of modal sentences. In his attempt to give Aristotelian modal syllogistic a consistent interpretation, Campsall is forced to adopt a very artificial reading of divided modal sentences.

He is clearly influenced by the suppositum approach suggested by Kilwardby, but he thinks that both subject and predicate terms should be taken in this way. Furthermore, he states that the terms in divided modal sentences should be taken as standing for that which is now under them. He believes that with these conditions, the conversion rules and almost all of the moods accepted by Aristotle can be shown to be valid.

Campsall takes the terms to signify how things actually are now. If that which is Socrates can be one of those that are white now, it is one of them; otherwise, Socrates could not have been that particular white being in the first place. Campsall thinks that Socrates can be this white being B 1 or that white being B 2 or …, that is, B 1 , B 2 , …, B n , and if Socrates is not actually B 1 now, he is B 2 now, etc.

This is not as crazy as it might first seem. Consider the following schema in quantified modal logic:. Given his interpretation of divided modal sentences and consequences like , Campsall manages to prove the conversion rules. His concept of contingency allows for simultaneous alternatives, such that if something exists, it is possible for it not to exist at that very same moment.

Campsall thus abandons the fundamental Aristotelian principle of the necessity of the present see Knuuttila and the entry on medieval theories of modality for discussion of criticisms of this principle in the late thirteenth century. In other words, he denies the necessity of the present for affirmative sentences and accepts it for negative ones. Accepting simultaneous alternatives and denying the necessity of the present are typical of modal semantics and modal logic after Campsall, especially in the work of figures such as William of Ockham and John Buridan.

It is historically interesting that Campsall employs these principles in his work, even though they are embedded in a theory whose elements point in another direction, towards Kilwardby. In many respects, he paves the way for the next generation of logicians. His complicated interpretation also shows that no matter how hard one might try, there is no way to give a consistent interpretation of what Aristotle says in the Prior Analytics for discussion, see Lagerlund , Thom and Knuuttila Around the time William of Ockham c.

More emphasis was placed on the theory of consequences than the theory of syllogisms. A theory of consequences was developed by Abelard in the course of his discussion of topical inferences and hypothetical syllogisms, and during the thirteenth century the basic idea was further developed in treatments of the topics, but in the fourteenth century works devoted solely to consequences began to appear Green-Pedersen Like Campsall, Ockham has nothing to add to the theory of assertoric syllogisms, which was by then well understood.

Ockham uses this method frequently, though not as frequently as Buridan later did. The method is used to prove the third figure moods. These sentences can be easily analyzed with the technical machinery of modern logic but only by accepting that they can fit into nonsyllogistic arguments.

The first and the third examples of noncategorical sentences just given contain more than two terms and so cannot fit into a syllogism. Logical deductions can be made from them in combination with other premises, but the conclusion may take many more than two steps to reach. SparkTeach Teacher's Handbook. Page 1 Page 2.