The use of Aristotelian logic has been modified and improved by medieval logicians. So what introduction to logic textbooks like the one versions written by Irving Copi and also the versions written by Patrick Hurley are not the original version. What texts like the ones just mentioned give a summary of the rules and concepts accepted when you are to use Aristotelian logic correctly. So there is no longer a need to directly quote Aristotle verbatim because of the modifications to the subject. This subject has been NOW moved to a topic within Philosophy known as Epistemology. [I have edited this answer since I have received two down votes, as I am writing this portion, to include more details.]
In Epistemology more concepts are included in the subject area of what was deemed “deductive reasoning” —logic for a nickname— that Mathematics does not seemed to be concerned. For starters the concept of a “proposititon” is the building blocks of forming “premises” to an “argument”.
People outside of Philosophy tend to have different ideas what they “BELIEVE” propositions are. A common misconception of what a proposition is listed as “a statement that is either true or false”. A worse definition of a proposition can be found is listed as “a sentence that is either true or false”.
A proposition is simply an idea. Ideas exist only mentally and not in the physical realm. There are no such things a “literal proposition”. That is, you can not use any of your senses to detect a proposition: you can’t see one, hear one, touch one, etc. This context of proposition—among many— is a concept not found outside of Philosophy. Other subjects misuse the original ideas behind the terminology, which I will try to tackle along the way. As a result of the same words being used in DIFFERENT CONTEXTS some confusion arises.
A proposition must be expressed in some form of communication: namely a language. So a language can only EXPRESS an idea. In the English language we happen to use “sentences” to communicate some idea. This is where some people relate propositions to sentences and how they must be the same thing —which of course is incorrect. All sentences are not propositions. Furthermore, no propositions are sentences! Propositions are EXPRESSED in a language so we can discuss them —otherwise we would never be able to discuss them or let other people know our ideas. We can used different languages to express the exact same idea: these ideas may use different terminology or words. Because we may use different words to say the same thing to another human being DOES NOT mean we used a DIFFERENT proposition. Too many people outside of Philosophy tend to think because sentence A and sentence B use different words —even in English— that there are two distinct propositions. They err because they are thinking that propositions are sentences and we ought to know propositions are NOT sentences. In epistemology the idea expressed in communication is counted ONLY once regardless of how many varieties of sentences or varieties of different words are used. Propositions are more about what communication you are expressing to another human being. How you “SAY IT” does not MATTER. In psychological fields such as rhetoric, law, politics, work place environment, school, etc HOW one says something can make a difference. Philosophers don’t care HOW you say it. Whatever it is must be true or false regardless or it may be considered meaningless. You saying “You are fired!” expresses the same proposition as “your employment at this company is no longer need”, “you are terminated”, “you don”t need to return to work here anymore moving forward”, etc. we all know people who use profanity words and may misspell those words by leaving a space or substitute a character such as @ in place of an appropriate letter. You are STILL expressing profanity as far as propositions go. There is no ducking or hiding your real intent in the communication. This is why Aristotelian logic works. It is formatted in such a way to eliminate the emotions order reason or at least reduce emotions so reason can thrive. Deception is usually applied through a human beings EMOTIONS. So in this way the Aristotelian logic is a way to prevent deception! Mathematical logic is not there to detect or expose deliberate deception from other human beings. Aristotelian logic was not discovered to mimic how ordinary speak in a language. The idea was to limit the emotional and persuasive content from the propositions to better evaluate the propositions of an argument. The logic of RHETORIC is taught to persuade other human beings. There is a REASON attorney’s are not USING mathematical logic in a REAL courtroom to defend their clients. Would it work for some clients? There is a reason they refuse to put all arguments in symbolic form. The argument in symbols loses its persuasiveness to human beings. If I use the phrase “the majority of. . . “ in a sentence this may convince people more than if I just use “SOME” in it’s place. The word “some” just does not bring the same impact as “the majority of . . .” in a debate. How an attorney argues alone may get his client the result he desires be it deceptive or not. How an attorney speaks and argues can cause a jury to vote “not guilty” which is the goal. The attorney is not concerned with the truth of his premises about his client. The result is more important than the quality of the statements made.
In the English language there may be several ways to communicate some one is stupid. How you say it does not matter if the intent is THERE! Some people think because you did not directly state the word “a stupid” then they did nothing wrong. In psychology fields that flys —not in epistemology. Because you use different words in a sentence does not mean the sentences can’t be equivalent (or mean the same thing contextually). Too many people count differently worded sentences as distinct propositions. You are not to count the proposition “there is an apple on the table” as a sentence because the same idea can be expressed in Spanish, French, Russian, Italian, etc. All of those sentences use different words so each sentence would be counted as a distinct sentence, BUT there is only one proposition expressed. What you intend to convey is what is counted —not how you say it with words. Being direct is not objectively different to being indirect. If you reach your goal then how you did it makes no difference being direct or indirect. Sugar coating is the same as breaking the news gently and the same as being direct as far as propositions go.
A similar term confused to proposition and sentence is the term “statement”. Then people usually outside of Philosophy start thinking along the lines of well if all sentences are statements and all propositions are sentences, then that must mean all statements are propositions—which of course is incorrect as well. A statement does not have to be a sentence at all. A statement is a form of communication that expresses an idea whether meaningful or not. Well what does that even mean? Literally meaningful communication deals what we call REALITY. That is, if I say “a cat is on the mat”, then there better be a mat that I can see and (or sense verify) also a cat that is on that mat (that can be sense verified). So the statement “Unicorns are mammals” is not literally meaningful. There are no unicorns. So this means statements can be categorized as “meaningful” or “meaningless”. Statements don’t even have to be sentences: they can be gestures, signals, signs, or sentences. Me holding a loaded gun to your head expresses a statement to you doesn’t it?
Statements are distinct from sentences and distinct from propositions. Sentences are distinct from statements and distinct from propositions. Propositions are distinct from both sentences and statements. With that said, all meaningful statements do express a proposition. By meaningful statement we assert that reality must be included and we can ask “is this true or false” as far as reality is concerned. This is where people get the idea propositions are either true or false. If a sentence used and we can ask is the sentence true or false and answer that question,THEN the sentence expresses a meaningful statement AND expresses a proposition. Again I stress propositions are not sentences and sentences are never propositions. The very best one can do is say sentences (or sentence variables)physically represent a proposition. That is, now I have something to physically POINT to when discussing ideas.
Propositions are typically called premises in an argument. Conclusions can be drawn from premises that are related in such a way where the truth of the conclusion follows from the premises. Thus we call this an “argument”. Here argument doesn’t mean a disagreement between two (or more)people. A “syllogism” is typically defined as a argument form that includes two propositions and a necessary conclusion. Here is where people outside of Philosophy run into problems because they don’t understand related concepts. All syllogisms MUST HAVE two propositions (at a minimum) as premises. Syllogisms are not limited to only two premises and there are no syllogism with zero propositions. Because you can’t visually see the premise doesn’t mean it is not there! There are different TYPES of syllogisms.
Categorical syllogisms are known in Aristotelian logic to have a quantifier, a subject term, a copula and a predicate term. Every two propositions yield a conclusion (be it valid or invalid). A minimum of two premises is REQUIRED.
Polly syllogisms is what we call a chain argument. It has more than two premises. The conclusion proposition of one syllogism is a premise for the next syllogism. It simply links several syllogisms to appear as one.
Enthymemes are syllogisms that suppress or have a missing proposition. That is, clearly you don’t SEE the proposition written there. The missing proposition is typically a premise and is called a “tacit premise”. You the reader must supply the missing proposition. So what looks like an argument with one premise and just a conclusion printed on a page TO YOU is still a syllogism with TWO PREMISES and a conclusion in reality. All you need to do if find the missing proposition and like magic the syllogism would have a minimum of two premises. The question is are you too lazy to figure the missing proposition out or do you even care? An enthymeme can have a missing major premise, a missing minor premise or a missing conclusion. Two of the three propositions are typically listed. You can’t literally have an odd number of premises by definition. If you have an odd set of premises there will always be a hidden premise you can use to derive the conclusion.
Epicheirema are syllogisms that list a cause or reason within one or more of the propositions listed. That is, one of the premises literally mentions a cause or a reason within the same sentence. You have a complete thought followed by a reason in the same sentence. For instance, at least one of the sentences you see will appear in the form of “All a are b since c” or “All a are b because of c”. Once you have a premise with a cause or reason to support it all in on sentence, your second premise will come afterwards and the conclusion is still necessary. You can have the cause or reason in the major premise, the minor premise or the reason or cause is found in BOTH premises. That is a total of three distinct types of epicheirema alone.
A Sorites is a chain of syllogisms where the conclusion of the first syllogism is the major premise for the second syllogism and the conclusion of the second syllogism is the major premise of the third syllogism and so on. This syllogism links the subject of the first premise to the predicate of the final conclusion (also known as transitivity). However there can be SOME HIDDEN PROPOSITIONS. You would have to figure out the missing propositions here as well like you would in the enthymemes. Here the missing propositions are often sub-conclusions (not the main conclusion) derived from a set of two premises in a chain argument. So likely you may just see a listing of several premises with one visual conclusion. For every two,propositions you can derive a conclusion by that alone until you reach the final conclusion.
An Oblique syllogism is a NON-Categorical syllogism that includes or expresses a transitive relationship between propositions. It is non-categorical for the way it is written: may not have a quantifier listed, may not have the wording structure categorical syllogisms have, etc. Specifically the middle term of this type of argument is NOT the means of what holds the syllogism together. The validity of this syllogism is NOT based on the distribution of the middle term. There is usually a transitive relationship stated such as “son of . . .”, “father of . . . “, “another of . . . “, “sibling of . . . ", etc. There is also a set of arguments that are formally invalid but express a relationship where you can see how the relationship holds in the premises and yields a true conclusion as well. For instance, a is greater than b, b is greater than c, therefore, a is greater than c. You can likely make this into categorical form by identifying the transitive relationship and placing that verbiage into both premises.
You also have hypothetical syllogisms as well as disjunctive syllogisms which you are likely already familiar with so I will not detail those here.
Many of the discrepancies between Mathematics and Philosophy the use of some of the same words and how arguments should be formed. I will first address the latter: how arguments should be formed. Mathematical logic often defines logic as FORMAL or about VALIDITY. One often hears or may read that “Logic is about validity”. one may also hear or read “Validity is about form”. Some people may go so far to put one and two together to conclude “logic is about form” or perhaps “formal validity”. The problem arises when CONTENT of the premises can make a difference in validity.
Some arguments are dependent on the content of the subject matter. Consider, Human being is a species. All women are human beings. Therefore all women are a species. Immediately something is wrong: we have two true premises and a false conclusion. The argument is formally valid: it is an Aristotelian AAA formal argument known to be VALID. Why is the conclusion false? One may say, “well you committed the fallacy of four terms!” The problem with THAT is the word in question has more than one context which makes the error a CONTENT MATTER concern and not a pure formal concern! Many mathematicians have claimed that the logician is not concerned whether the premises are true or not. Well perhaps they should be! Epistemology considers both LOGICAL FORM and CONTENT of the propositions involved in deductive reasoning. I was taught logic is NOT a just about form or formal validity. Content of the premises clearly can make a difference. Look up what used to be called supposition theory. For this reason medieval logicians divided logic into two parts: FORMAL LOGIC and MATERIAL LOGIC. The latter is called by a new name which is epistemology. It considers both content and form as important —not just one aspect as form.
Other arguments outside math and philosophy have a universal format which we call, loosely, “formal”. If people can repeat the same pattern we can loosely say it is formal. It is formal because we recognize a repeating pattern. As discussed above there are instances of argumentation that is not valid by the mathematical standards but still hold true in reality. Secondly, there are mathematical valid arguments that do not hold true in reality (which is called Soundness). This raises the point to some students if I can conclude some things that are not true in reality what is the use of this so called LOGIC? Some examples of valid arguments are true in reality. Some examples are not. That is a 50 percent accuracy rating which may not be convincing to why this is important or worthwhile learning. People in general learn by experience in some way or another. The fact we can recognize a repetition of patterns MEANS there is some form to what we are describing. Perhaps people are confusing the term “INFORMAL” to express something outside of textbook context of FORMAL. If you can say a fallacy is informal How is it that you can recognize it is an error of reasoning? With no form you would fail to recognize it or describe it! Perhaps the argument doesn’t have Aristotelian logic form or a modern logic form and that what is meant. People are confusing contexts of the same words. What they really mean to say is that deductive arguments are considered formal while “INDUCTIVE ARGUMENT” is not as formal. So if you can distinguish inductive from deductive there is a repetition of patterns you recognize: thus it has a form. Furthermore, if there are INFORMAL FALLACIES aren’t they proof enough that subject CONTENT can affect the result of the conclusion in any argument?
Some terminology or words are identical in spelling and pronunciation but the context differs in Philosophy and Mathematics which should be acknowledged. The term contradiction is a common one. In epistemology a contradiction is strictly defined. A contradiction is a RELATIONSHIP between two propositions where both propositions cannot be true simultaneously in the same place, time, and meaning AND both propositions cannot be false simultaneously. Exactly one of the propositions must be true while the other is false. Contrast THAT definition with the mathematical logic definition of contradiction: expresses a statement whose truth value is always false. Notice this leaves out the RELATIONSHIP between propositions. A single proposition can be a contradiction ? Really? Is that by form or BY CONTENT? Surely a triangle having four sides is a single statement which we can agree is a contradiction. We know it to be a contradiction because WE KNOW a more than just the given statement! A being needs more than just a single statement to derive the statement is impossible. This used to be referred to as a synthetic proposition. That is, one needs more than just a definition to know what you know; usually requires some experience to KNOW a the information, some familiarity or expertise in a subject to KNOW anything. An analytic proposition used to defined as a proposition that is derived by meaning alone and not gained by experience of the senses as most knowledge is gained. We know by the meaning of the word alone triangle cannot have four sides. Thus we have a contradiction because one set of information is compared to another set of information (having MORE THAN ONE statement) that can’t both be true and both be false. We can break down propositions to distinct types: analytical and synthetic.
Truth is the next concept confused. There are distinct types of truth. There are universal truth which happen to always be true. Mathematics uses the term "Tautology” for a statement that always hold the value of true. In the real world that idea is referred to as an “objective truth”. An objective truth is always true with no exception in its correct domain of discourse. A statement that always holds true by itself is not an argument. In epistemology one says a tautology is a RELATIONSHIP between two or more propositions where the truth value is constant (it does not change or alternate). One can arrive at a tautology in at least two ways: by identity or by equivalence. Obviously if I say, “all men are human beings” in comparison to the statement “all men are human beings” is always going to be true if one of them is true. The exact same statement is made twice while there is only one proposition present. This has the logical form A & A. I am comparing A with a mirror image of itself. This is called a principle of identity. This principle is also found in the inferences of conversion, obversion and contraposition found in Aristotelian logic. We can prove two propositions are identical then they must have the same truth value. Some propositions are not alway true. Some statements depend on content matter and not form to know they are true. Consider, “Donald J. Trump is the President of the United States”. Well if the statement was uttered three years ago it would be true. We now hold the claim to be false! So some truths do not hold forever after all. The term for that type of truth is “contingent truth”. That is, a contingent truth is a proposition that is not forever true —it has false instances as well as true instances. The mathematician does not even mention there are distinct types of propositions nor does he mention the distinct types of truth we can find in our langue expressed in sentences. The way the mathematician uses the word tautology is awkward when you compare two statements that are forever true that are not even related: all women are human beings and all even numbers are divisible by 2 would be tautological? That sounds weird because the statements are unrelated. Furthermore all equivalence formulas would be considered a tautology by definition: so p —> Q is equivalent to not (p or q) and must be tautological. We see that an equivalence is a tautology type. How would one know what type of tautology you refer to if there is more than one type of tautology? A or A is simply an identity which is not every equivalence. That is some equivalence forms are not identical to its counterpart: the left hand side expresses something different in words than the right hand side of the formula.
Objective truth is the main goal in epistemology. [There are also Objective falsehoods, which is identical to the way mathematicians use the word contradiction.] We use deductive reasoning to argue from TRUE PREMISES from the start to derive conclusions that must NECESSARILY ALSO BE TRUE. We move from truth to truth. We cannot allow the process of moving from false propositions to truth because then we can derive a TRUE premise, TRUE premise FALSE conclusion argument pattern. You should not be forming arguments with propositions you have no idea about the truth or falsity. You must start with TRUE PROPOSITIONS to ensure the conclusion logically follows. I was never allowed to use false premises whatsoever. In every other academic field out there the writer needs to do his or her due diligence to ensure the writing is accurate. If that means do research before writing anything down to begin with, then so be it. In mathematics, rhetoric, psychology and other fields people have this attitude that they can make premises any kind of way: have it your way like at Burger King. First off many don’t care if the premises are false from the start: i.e., all pens are cows and all men are pens. thus, all men are cows. Clearly this raises the point of why study this if it is blatantly false in the real world and how will I know when this reasoning applies to the real world when clearly I see it fails at times? This is why you ought to ONLY abuse true premises to begin with. That example is a pattern I found with people in India learning what they call “Indian Logic”. Any kind of premises any kind of way and they just keep going about as if everything is fine. Mathematics does the same thing.
{I want to just note there is a specific type of logic called “Indian Logic” as a topic that differs from Aristotelian logic and differs from Mathematical logic (or any other kind of logic out there), which is why it has its own category and name); and NO, every race of people do not have their own logic; it is just coincidence that the name happens to be “Indian Logic” —it could of been called something else.}. In a mathematical proof there are NO FALSE propositions! To say a proposition is FALSE what you are really saying is “I ACKNOWLEDGE that this proposition is false” which by itself is a TRUE a proposition! That is, “2+2= 7 billion” is a false proposition which I will express as B. To say B is always false the mathematician would say NOT (B). I will use ~ to symbolize NOT. ~ is called a tilde in Philosophy. So we can say ~B is a true proposition can’t we? In other words, we read NOT B (~B) as it is not the case that B is true which if you think about it expresses a TRUE CLAIM—not one that is always false as the mathematician defines it as. The acknowledgment of ~B is a statement in itself which happen to be a real world truth that says b is false. Show me how does one really express a false proposition in mathematics while at the same time being false in the real world. You can’t do it!
Other terminology confused is “a contrapositive” or contraposition. In Aristotelian logic this is strictly defined for categorical propositions and syllogisms. One thing to note this inference is not always valid in Aristotelian logic but somehow in Mathematical logic it is. This seems to confuse people who don’t know there exists at least two contexts of the same word with the ambiguous word LOGIC. Today most people think of logic as math. So it is highly likely a younger person studying LOGIC — aka mathematical logic — they will be surprised to here some one say “the contrapositive is not always valid”. This result is due to the student not being aware of the original context of the same word. The correct terminology in philosophy is TRANSPOSITION. This rule of inference is what math teaches as the contrapositive. Transposition is always valid whereas the term contrapositive is not.
One also must notice how the phrase If . . . Then. . . Is used in ordinary language as well as used in other academic fields. By now most people know these uses are not identical. One cannot help but notice the context changes the truth value of the same term or word. One way to avoid this is to add more specific details as to make distinctions as much as possible so that people will not confuse contexts or err in reasoning. The If . . . Then . . . Does not always mean necessarily follows. For instance, “if I hit the mega million jackpot I will donate it all to you” does not mean I will do as I say even if I do win or hit the mega million jackpot. For many people in math or related fields learning so called logic they think that If . . .then must be a necessity. In reality this is not true. Stick to the domain and context of the terminology to prevent errors in translation.
[To all of the haters out there: if you down vote my answer here or disagree with anything stated then be rational and be able to JUSTIFY WHY you disagree. Please explain why you down voted the answer not just you being emotional. I am open to correction if I made errors and will edit as needed to correct the answer.]
