Which of aristotle syllogisms are not valid

The premise containing the major term is the major premise , and the premise containing the minor term is the minor premise. Aristotle then systematically investigates all possible combinations of two premises in each of the three figures. For each combination, he either demonstrates that some conclusion necessarily follows or demonstrates that no conclusion follows. The results he states are correct. The precise interpretation of this distinction is debatable, but it is at any rate clear that Aristotle regards the perfect deductions as not in need of proof in some sense.

For imperfect deductions, Aristotle does give proofs, which invariably depend on the perfect deductions. Thus, with some reservations, we might compare the perfect deductions to the axioms or primitive rules of a deductive system.

A direct deduction is a series of steps leading from the premises to the conclusion, each of which is either a conversion of a previous step or an inference from two previous steps relying on a first-figure deduction.

Conversion, in turn, is inferring from a proposition another which has the subject and predicate interchanged. Specifically, Aristotle argues that three such conversions are sound:. He undertakes to justify these in An. From a modern standpoint, the third is sometimes regarded with suspicion.

Using it we can get Some monsters are chimeras from the apparently true All chimeras are monsters ; but the former is often construed as implying in turn There is something which is a monster and a chimera , and thus that there are monsters and there are chimeras. For further discussion of this point, see the entry on the square of opposition. He says:.

An example is his proof of Baroco in 27a36—b Aristotle proves invalidity by constructing counterexamples. This is very much in the spirit of modern logical theory: all that it takes to show that a certain form is invalid is a single instance of that form with true premises and a false conclusion. In Prior Analytics I. Having established which deductions in the figures are possible, Aristotle draws a number of metatheoretical conclusions, including:.

His proof of this is elegant. First, he shows that the two particular deductions of the first figure can be reduced, by proof through impossibility, to the universal deductions in the second figure:. He then observes that since he has already shown how to reduce all the particular deductions in the other figures except Baroco and Bocardo to Darii and Ferio , these deductions can thus be reduced to Barbara and Celarent.

This proof is strikingly similar both in structure and in subject to modern proofs of the redundancy of axioms in a system. Many more metatheoretical results, some of them quite sophisticated, are proved in Prior Analytics I. In contrast to the syllogistic itself or, as commentators like to call it, the assertoric syllogistic , this modal syllogistic appears to be much less satisfactory and is certainly far more difficult to interpret.

Aristotle gives these same equivalences in On Interpretation. However, in Prior Analytics , he makes a distinction between two notions of possibility. He then acknowledges an alternative definition of possibility according to the modern equivalence, but this plays only a secondary role in his system. Aristotle builds his treatment of modal syllogisms on his account of non-modal assertoric syllogisms: he works his way through the syllogisms he has already proved and considers the consequences of adding a modal qualification to one or both premises.

A premise can have one of three modalities: it can be necessary, possible, or assertoric. Aristotle works through the combinations of these in order:. Though he generally considers only premise combinations which syllogize in their assertoric forms, he does sometimes extend this; similarly, he sometimes considers conclusions in addition to those which would follow from purely assertoric premises.

As in the case of assertoric syllogisms, Aristotle makes use of conversion rules to prove validity. The conversion rules for necessary premises are exactly analogous to those for assertoric premises:.

Possible premises behave differently, however. Aristotle generalizes this to the case of categorical sentences as follows:. This leads to a further complication. Such propositions do occur in his system, but only in exactly this way, i. Such propositions appear only as premises, never as conclusions. He does not treat this as a trivial consequence but instead offers proofs; in all but two cases, these are parallel to those offered for the assertoric case.

Malink , however, offers a reconstruction that reproduces everything Aristotle says, although the resulting model introduces a high degree of complexity. This subject quickly becomes too complex for summarizing in this brief article. From a modern perspective, we might think that this subject moves outside of logic to epistemology.

However, readers should not be misled by the use of that word. The remainder of Posterior Analytics I is largely concerned with two tasks: spelling out the nature of demonstration and demonstrative science and answering an important challenge to its very possibility. Aristotle first tells us that a demonstration is a deduction in which the premises are:.

The interpretation of all these conditions except the first has been the subject of much controversy. Aristotle clearly thinks that science is knowledge of causes and that in a demonstration, knowledge of the premises is what brings about knowledge of the conclusion.

The fourth condition shows that the knower of a demonstration must be in some better epistemic condition towards them, and so modern interpreters often suppose that Aristotle has defined a kind of epistemic justification here. However, as noted above, Aristotle is defining a special variety of knowledge.

Comparisons with discussions of justification in modern epistemology may therefore be misleading. In Posterior Analytics I. Instead, they maintained:. Aristotle does not give us much information about how circular demonstration was supposed to work, but the most plausible interpretation would be supposing that at least for some set of fundamental principles, each principle could be deduced from the others.

Some modern interpreters have compared this position to a coherence theory of knowledge. Aristotle rejects circular demonstration as an incoherent notion on the grounds that the premises of any demonstration must be prior in an appropriate sense to the conclusion, whereas a circular demonstration would make the same premises both prior and posterior to one another and indeed every premise prior and posterior to itself.

However, he thinks both the agnostics and the circular demonstrators are wrong in maintaining that scientific knowledge is only possible by demonstration from premises scientifically known: instead, he claims, there is another form of knowledge possible for the first premises, and this provides the starting points for demonstrations.

To solve this problem, Aristotle needs to do something quite specific. It will not be enough for him to establish that we can have knowledge of some propositions without demonstrating them: unless it is in turn possible to deduce all the other propositions of a science from them, we shall not have solved the regress problem. Moreover and obviously , it is no solution to this problem for Aristotle simply to assert that we have knowledge without demonstration of some appropriate starting points.

He does indeed say that it is his position that we have such knowledge An. There is wide disagreement among commentators about the interpretation of his account of how this state is reached; I will offer one possible interpretation. What he is presenting, then, is not a method of discovery but a process of becoming wise. The kind of knowledge in question turns out to be a capacity or power dunamis which Aristotle compares to the capacity for sense-perception: since our senses are innate, i.

Likewise, Aristotle holds, our minds have by nature the capacity to recognize the starting points of the sciences. In the case of sensation, the capacity for perception in the sense organ is actualized by the operation on it of the perceptible object. So, although we cannot come to know the first premises without the necessary experience, just as we cannot see colors without the presence of colored objects, our minds are already so constituted as to be able to recognize the right objects, just as our eyes are already so constituted as to be able to perceive the colors that exist.

It is considerably less clear what these objects are and how it is that experience actualizes the relevant potentialities in the soul. Aristotle describes a series of stages of cognition. First is what is common to all animals: perception of what is present. Next is memory, which he regards as a retention of a sensation: only some animals have this capacity. Even fewer have the next capacity, the capacity to form a single experience empeiria from many repetitions of the same memory.

Finally, many experiences repeated give rise to knowledge of a single universal katholou. This last capacity is present only in humans. The definition horos , horismos was an important matter for Plato and for the Early Academy. External sources sometimes the satirical remarks of comedians also reflect this Academic concern with definitions.

Aristotle himself traces the quest for definitions back to Socrates. Since a definition defines an essence, only what has an essence can be defined. What has an essence, then? A species is defined by giving its genus genos and its differentia diaphora : the genus is the kind under which the species falls, and the differentia tells what characterizes the species within that genus.

As an example, human might be defined as animal the genus having the capacity to reason the differentia. However, not everything essentially predicated is a definition. Such a predicate non-essential but counterpredicating is a peculiar property or proprium idion. Aristotle sometimes treats genus, peculiar property, definition, and accident as including all possible predications e. Topics I. Later commentators listed these four and the differentia as the five predicables , and as such they were of great importance to late ancient and to medieval philosophy e.

Just what that doctrine was, and indeed just what a category is, are considerably more vexing questions. They also quickly take us outside his logic and into his metaphysics. We can answer this question by listing the categories. Here are two passages containing such lists:. Of things said without any combination, each signifies either substance or quantity or quality or a relative or where or when or being-in-a-position or having or doing or undergoing.

To give a rough idea, examples of substance are man, horse; of quantity: four-foot, five-foot; of quality: white, literate; of a relative: double, half, larger; of where: in the Lyceum, in the market-place; of when: yesterday, last year; of being-in-a-position: is-lying, is-sitting; of having: has-shoes-on, has-armor-on; of doing: cutting, burning; of undergoing: being-cut, being-burned. Categories 4, 1b25—2a4, tr. Ackrill, slightly modified. These two passages give ten-item lists, identical except for their first members.

Here are three ways they might be interpreted:. Which of these interpretations fits best with the two passages above? The answer appears to be different in the two cases. This is most evident if we take note of point in which they differ: the Categories lists substance ousia in first place, while the Topics list what-it-is ti esti. A substance, for Aristotle, is a type of entity, suggesting that the Categories list is a list of types of entity.

As Aristotle explains, if I say that Socrates is a man, then I have said what Socrates is and signified a substance; if I say that white is a color, then I have said what white is and signified a quality; if I say that some length is a foot long, then I have said what it is and signified a quantity; and so on for the other categories.

What-it-is, then, here designates a kind of predication, not a kind of entity. This might lead us to conclude that the categories in the Topics are only to be interpreted as kinds of predicate or predication, those in the Categories as kinds of being. Even so, we would still want to ask what the relationship is between these two nearly-identical lists of terms, given these distinct interpretations.

However, the situation is much more complicated. First, there are dozens of other passages in which the categories appear. These latter expressions are closely associated with, but not synonymous with, substance. Moreover, substances are for Aristotle fundamental for predication as well as metaphysically fundamental. He tells us that everything that exists exists because substances exist: if there were no substances, there would not be anything else.

He also conceives of predication as reflecting a metaphysical relationship or perhaps more than one, depending on the type of predication.

For reasons explained above, I have treated the first item in the list quite differently, since an example of a substance and an example of a what-it-is are necessarily as one might put it in different categories. His attitude towards it, however, is complex.

In Posterior Analytics II. However, Aristotle is strongly critical of the Platonic view of Division as a method for establishing definitions. He also charges that the partisans of Division failed to understand what their own method was capable of proving. Closely related to this is the discussion, in Posterior Analytics II.

Since the definitions Aristotle is interested in are statements of essences, knowing a definition is knowing, of some existing thing, what it is. His reply is complex:. He sees this as a compressed and rearranged form of this demonstration:. As with his criticisms of Division, Aristotle is arguing for the superiority of his own concept of science to the Platonic concept.

Knowledge is composed of demonstrations, even if it may also include definitions; the method of science is demonstrative, even if it may also include the process of defining. Aristotle often contrasts dialectical arguments with demonstrations. The difference, he tells us, is in the character of their premises, not in their logical structure: whether an argument is a sullogismos is only a matter of whether its conclusion results of necessity from its premises.

The premises of demonstrations must be true and primary , that is, not only true but also prior to their conclusions in the way explained in the Posterior Analytics. The premises of dialectical deductions, by contrast, must be accepted endoxos. Recent scholars have proposed different interpretations of the term endoxos.

On one understanding, descended from the work of G. Anyone arguing in this manner will, in order to be successful, have to ask for premises which the interlocutor is liable to accept, and the best way to be successful at that is to have an inventory of acceptable premises, i.

In fact, we can discern in the Topics and the Rhetoric , which Aristotle says depends on the art explained in the Topics an art of dialectic for use in such arguments.

My reconstruction of this art which would not be accepted by all scholars is as follows. Given the above picture of dialectical argument, the dialectical art will consist of two elements. One will be a method for discovering premises from which a given conclusion follows, while the other will be a method for determining which premises a given interlocutor will be likely to concede. The first task is accomplished by developing a system for classifying premises according to their logical structure.

The second task is accomplished by developing lists of the premises which are acceptable to various types of interlocutor. Then, once one knows what sort of person one is dealing with, one can choose premises accordingly.

We find enumerations of arguments involving these terms in a similar order several times. Typically, they include:. The four types of opposites are the best represented. Each designates a type of term pair, i. Contraries are polar opposites or opposed extremes such as hot and cold, dry and wet, good and bad. A pair of contradictories consists of a term and its negation: good, not good. A possession or condition and privation are illustrated by sight and blindness. Relatives are relative terms in the modern sense: a pair consists of a term and its correlative, e.

Unfortunately, though it is clear that he intends most of the Topics Books II—VI as a collection of these, he never explicitly defines this term. Interpreters have consequently disagreed considerably about just what a topos is. Discussions may be found in Brunschwig , Slomkowski , Primavesi , and Smith An art of dialectic will be useful wherever dialectical argument is useful. Aristotle mentions three such uses; each merits some comment. In these exchanges, one participant took the role of answerer, the other the role of questioner.

The questioner was limited to questions that could be answered by yes or no; generally, the answerer could only respond with yes or no, though in some cases answerers could object to the form of a question. Answerers might undertake to answer in accordance with the views of a particular type of person or a particular person e. There appear to have been judges or scorekeepers for the process. Gymnastic dialectical contests were sometimes, as the name suggests, for the sake of exercise in developing argumentative skill, but they may also have been pursued as a part of a process of inquiry.

Its function is to examine the claims of those who say they have some knowledge, and it can be practiced by someone who does not possess the knowledge in question. The examination is a matter of refutation, based on the principle that whoever knows a subject must have consistent beliefs about it: so, if you can show me that my beliefs about something lead to a contradiction, then you have shown that I do not have knowledge about it. In fact, Aristotle often indicates that dialectical argument is by nature refutative.

Dialectical refutation cannot of itself establish any proposition except perhaps the proposition that some set of propositions is inconsistent. More to the point, though deducing a contradiction from my beliefs may show that they do not constitute knowledge, failure to deduce a contradiction from them is no proof that they are true.

In Topics I. One reason he gives for this follows closely on the refutative function: if we have subjected our opinions and the opinions of our fellows, and of the wise to a thorough refutative examination, we will be in a much better position to judge what is most likely true and false.

He adds a second use that is both more difficult to understand and more intriguing. The Posterior Analytics argues that if anything can be proved, then not everything that is known is known as a result of proof. What alternative means is there whereby the first principles of sciences are known? Against this background, the following passage in Topics I. Further discussion of this issue would take us far beyond the subject of this article the fullest development is in Irwin ; see also Nussbaum and Bolton ; for criticism, Hamlyn , Smith Aristotle says that rhetoric, i.

The correspondence with dialectical method is straightforward: rhetorical speeches, like dialectical arguments, seek to persuade others to accept certain conclusions on the basis of premises they already accept. A fallacy of equivocation occurs when a term is used in a different way within the course of an argument. So, for example. The middle term of a valid syllogism is distributed in at least one of the premises. The fallacy of the undistributed middle occurs when this doesn't happen.

For instance, the middle term furry animals in this syllogism. All dogs are furry animals Some furry animals are cats Therefore, dogs are cats.

Rule 3: If a term is distributed in the conclusion, it must be distributed in the premises. A conclusion that states something about a whole class must be supported by a premise that does the same thing. For example:. The fallacy of illicit major occurs as above when the major term is distributed in the conclusion, but not in the major premise.

The fallacy of illicit minor occurs when the minor term is distributed in the conclusion, but not in the minor premise. The fallacy of exclusive premises occurs when a syllogism has two premises that are negative.