Aristotle held that there are three forms of deductive reasoning, scientific, dialectical, and rhetorical. Each is distinguished by the level of conviction that its premises command, and each is limited in the scope of its application to specific kinds of issues, although not to specific subjects.
Scientific reasoning, outlined and elaborated in the Prior and Posterior Analytics, is also known as syllogistic reasoning. Syllogistic reasoning is a form of inferential reasoning based on a self-evident and irrefutable truth that ends in a necessarily true and therefore irrefutable conclusion. E.g. All geometric entlities which have 180degrees are triangles. This shape has 180 degrees; therefore it is a triangle. Consider another example. All ruminants are cows. The animal before us is a ruminant; therefore it is a cow. If it were irrefutably true that all ruminants are cows (it is not), and the animal before us is in fact a ruminant, then the conclusion that it is a cow would be irrefutable. We would have scientifically proven (demonstrated) that this animal is a cow. If the premise is faulty, if the genera of ruminants contains species other than cows, as in fact it does, then our conclusion is faulty. It is also possible to make formally accurate but useless (not merely necessary but inevitable and therefore uninteresting) conclusions based on accurate inference from irrefutable premises: all cows are ruminants, this is a cow, it is a ruminant. And of course, it is possible to make faulty conclusions from refutable premises (called contentious reasoning): all ruminants are cows; this alpaca is a ruminant, therefore it is a cow. There are also sophistic games to be played here: all cows ruminate, I sat for hours last night pondering the complexities of life; therefore I am a cow. A conclusion reached by means of proper syllogism is said to have been demonstrated. Demonstrable proofs are the most secure forms of proof, but they are available only when an irrefutable premise is available. Without the presence of an irrefutable premise, demonstration is impossible. This is why ethics is not a science.
Dialectical reasoning, outlined in Topica, resembles syllogistic reasoning, but it relies on premises that are only probably true, not necessarily true. The phrase Aristotle uses is something like premises agreed to by the few or by the experts. The conclusions reached by inferential reasoning from probable opinions are nevertheless necessary conclusions. Because the premise is not necessarily true, the conclusion while necessary is not demonstrable. If we can deduce necessary truths by way of dialectic, we can use dialectic as a prolegomenon for scientifically valid (demonstrable) arguments; that is, we can take the conclusion of a dialectical argument and use it as the premise for a scientific argument. Aristotle also says that one can use dialectic to sharpen the mind and to amuse oneself and one's friends. E.g. your friend asserts: "All cows are ruminants. This is a ruminant; therefore it is a cow." You rapidly perform what is called conversion (switching the predicate for the subject) and thus you realize that while all cows are ruminants not all ruminants are cows, and therefore while this animal may in fact be a ruminant, it may not necessarily be a cow; and so you ask, "Do all cows have thick brown shaggy coats and beards?". Your respondent confidently replies, "Of course not." "Aha," you think, although you remain outwardly calm (at the end of Topica Aristotle advises never let your respondent/opponent see where you are going). "Is not this shaggy, bearded creature here chewing its cud?" Dialectical proofs are compelling, but they require dexterity with abstract thought and thus they are not compelling for everyone. A person has to be reasonably smart and patient and attentive to be convinced by dialectic. Thus, while syllogistic reasoning is restricted to those areas of thought where absolutely irrefutable premises are available, dialectic is restricted to those areas of thought where expert opinions can be tested by intelligent people.
Like most educated people of his time, Aristotle felt that the law courts and the public assemblies were no place for intelligent argumentation (I am being only slightly facetious here). The people entrusted with making decisions were frequently incapable of following the kind of close reasoning that dialectic required, and often the experts were not listened to precisely because they were experts. Scientific reasoning was similarly useless because self-evident and irrefutable premises are often unavailable when considering the past (law courts) and the future (political assemblies). And sometimes what is thought to be irrefutably true is only a collective delusion or an unquestioned prejudice. Moreover, science is concerned with general principles, not specific cases, and of course one does not convict all murderers when trying only one, nor can every law foresee every possible abuse. Thus some form of reasoning that will work well for specific cases and be palatable to the easily bored must be employed. Hence, rhetoric.
Rhetorical argumentation, outlined in Rhetorica, can be either inductive (based
on generalization from specific examples) or it can be deductive. The deductive
kind employs enthymemes rather than syllogisms. An enthymeme is an abbreviated
syllogism premised on a common opinion. The conclusions of these arguments are
not necessarily true; they are not even necessarily probable. They are either
convincing or unconvincing, although they may be binding if they take place
in a court or assembly hall. Strictly speaking, rather than reaching a conclusion,
those involved in rhetorical argumentation reach a decision at a particular
place and time about a particular case, or in political matters, they reach
a decision about general policies that will inevitably be tried on a case by
case basis. Whereas syllogisms consist of a major premise, a minor premise,
and a conclusion, enthymemes consist of a claim and a proof. A claim is any
statement that you would have your audience believe, but which they do not yet
believe. A proof is any statement that your audience does believe which will
make the claim believable if a link between claim and proof can be established.
The effort to link claim and proof is often signified by the word because. Often
proofs take the form of maxims, succinct statements that "ring" true
and require no justification. Because proofs are statements that are believed
without the need for justification, they usually carry some unspoken premise;
hence the definition of enthymeme as a truncated syllogism. E.g. When saving
for your retirement, you should invest in the stock market because over the
long term the stock market has steadily increased in value.