What are the differences between an argument and a syllogism?
Along with definitions and usages, I would like examples to understand the differences.
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These terms don't have universally agreed-upon definitions.
Syllogism is often associated with Aristotle, in particular with the (restricted) kinds of deductive inferences he described as "perfect" (teleios), but which others subsequently called "syllogisms", although Aristotle himself used the latter term in a boarder sense, according to SEP:
All Aristotle’s logic revolves around one notion: the deduction (sullogismos). [...]
A deduction is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so. (Prior Analytics I.2, 24b18–20)
Aristotle’s most famous achievement as logician is his theory of inference, traditionally called the syllogistic (though not by Aristotle).
A "perfect" Aristotelian syllogism always has two premises and one conclusion, following certain schemata that I won't fully detail here, but a classic example is
All people are mortal. Socrates is a person. Therefore, Socrates is mortal.
These "perfect" Aristotelian deductions are said to be composed of "categorical propositions" (e.g. by I. Copi) even though those are technically not propositions in the sense of (modern) propositional calculus but involve monadic (one-place) predicates like "is mortal" above. Roy Cook (in A Dictionary of Philosophical Logic) calls the (Aristotellian) deductions over these "categorical syllogisms".
In this context, an example of non-Aristotelian/non-categorial syllogism (in Cook's terminology) would be disjunction elimination (from A or B and not B, infer A), which Cook calls "disjunctive syllogism". More generally Cook defines in somewhat of a generalization from the Aristotelian/categorial ones that a
A syllogism is any argument with two premises.
Also, Cook defines
A polysyllogism (or sorites) is an argument consisting of a sequence of syllogisms, usually categorical syllogisms.
A Dictionary of Logic by Ferguson and Priest only defines
syllogistic inference is a form of inference investigated by Aristotle (384–322 bce) in Prior Analytics. There are two premisses of the form: Some [every, no] is [is not], and one conclusion of the same form. Aristotle theorized which inferences of this form are valid. [...]
but this work does not define syllogism more generally... although it also e.g. defines disjunctive syllogism (in the same way as Cook).
IEP likewise defines syllogism (at least on the Aristotle page) as:
We can define a syllogism, in relation to its logical form, as an argument made up of three categorical propositions, two premises (which set out the evidence), and a conclusion (that follows logically from the premises).
Gensler in A to Z of Logic covers both senses:
SYLLOGISM. The term "syllogism" can be applied broadly, to cover arguments of any sort, or narrowly, to cover just categorical syllogisms (see syllogistic logic).
SYLLOGISTIC LOGIC. A branch of logic that studies arguments using "all," "no," and "some." Syllogistic logic was created by Aristotle and was the first branch of logic ever developed. While syllogisms today are often subsumed under quantificational logic, they still are some- times studied by themselves, especially in introductory logic courses. [...]
Followed by the usual presentation of Aristotelian syllogisms.
So what's an argument? Cook defines it as:
An argument is a sequence of statements where all but one of the statements (the premises) are intended to provide evidence, or support, for the remaining statement (the conclusion). Sometimes, in technical contexts such as the sequent calculus, an argument can have more than one conclusion.
Cook also defines "deductive argument" and "inductive argument" (and some subtypes thereof, which I'll omit here).
A deductive argument is an argument where it is intended that it be impossible for the premises to be true and the conclusion false.
An inductive argument is an argument where it is intended that it be improbable (but possible) for the premises to be true and the conclusion false.
Aristotle also made this distinction albeit in his own terms (from SEP again):
Deductions are one of two species of argument recognized by Aristotle. The other species is induction (epagôgê). He has far less to say about this than deduction, doing little more than characterize it as “argument from the particular to the universal”.
Ferguson and Priest do not actually define the term "argument", but they do define:
deduction: An argument where each of its steps is deductively valid; that is, where if the premisses are true so must the conclusion be.
IEP actually has a very detailed article on "argument", but from a certain perspective of informal logic; it opens with
The word “argument” can be used to designate a dispute or a fight, or it can be used more technically. The focus of this article is on understanding an argument as a collection of truth-bearers (that is, the things that bear truth and falsity, or are true and false) some of which are offered as reasons for one of them, the conclusion. This article takes propositions rather than sentences or statements or utterances to be the primary truth bearers. The reasons offered within the argument are called “premises”, and the proposition that the premises are offered for is called the “conclusion”. This sense of “argument” diverges not only from the above sense of a dispute or fight but also from the formal logician’s sense according to which an argument is merely a list of statements, one of which is designated as the conclusion and the rest of which are designated as premises regardless of whether the premises are offered as reasons for believing the conclusion. Arguments, as understood in this article, are the subject of study in critical thinking and informal logic courses in which students usually learn, among other things, how to identify, reconstruct, and evaluate arguments given outside the classroom.
That IEP page makes certain assumptions that are used formally true only in some logic contexts such as relevance logic(s). Further, that article defines "argument" in a sense that's closer to how it's used in argumentation theory, i.e. that someone must put it forward as such:
Letting P1, P2, P3, …, and C range over propositions and R over reasoners, a structural characterization of argument takes the following form.
A collection of propositions, P1, …, Pn, C, is an argument if and only if there is a reasoner R who puts forward the Pi as reasons in support of C.
The structure of an argument is not a function of the syntactic and semantic features of the propositions that compose it. Rather, it is imposed on these propositions by the intentions of a reasoner to use some as support for one of them. [...]
Plausibly, if a reasoner R puts forward premises in support of a conclusion C, then (i)-(iii) obtain. (i) The premises represent R’s reasons for believing that the conclusion is true and R thinks that her belief in the truth of the premises is justified. (ii) R believes that the premises make C more probable than not. (iii) (a) R believes that the premises are independent of C ( that is, R thinks that her reasons for the premises do not include belief that C is true ), and (b) R believes that the premises are relevant to establishing that C is true.
As an example (that conforms to i-iii) they give someone saying:
John is not an only child; he said that Mary is his sister.
An older (and more extensive) version of the SEP page on informal logic went even further and talked e.g. about visual arguments. Not sure how Aristotle felt about those.
Gensler (A to Z of Logic) has quite a bit to say about argument, again covering a broad definition; I'm not gonna reproduce all that here, but the basic lines:
ARGUMENT. Set of statements consisting of premises and a conclusion. Normally we use the premises to give evidence for the conclusion; but sometimes we are just exploring what the premises lead to. Arguments put into words a possible act of reasoning. An argument is valid if it would be impossible for the premises to all be true while the conclusion was false; it is sound if it is valid and has only true premises.
Logicians like to express arguments clearly, with each premise be- ginning a new line and the conclusion prefixed by"∴" or "therefore." Arguments in real life are seldom so neat and clean. [...]
[...] A good argument in a broad sense is one that is logically correct and fulfills the purposes for which we use arguments. A good argument should be deductively valid (or inductively strong) and have only true premises; have this validity and truth be as evident as possible to the parties involved; be clearly stated; avoid circularity, ambiguity, and emotional language; and be relevant to the issue at hand.
Arguments can be useful even if they fall short of these ideals. We would like to use premises that are so obvious that everyone will immediately accept them; but in practice this is too high a standard. We sometimes appeal to premises that only some will accept-perhaps those of similar philosophical, religious, or political views. [...]
Logicians normally allow arguments with no premises; a premise- less argument is valid if and only if the conclusion is a logical truth. While logicians normally allow only premises that are true or false, defenders of imperative logic want to allow imperative premises, which tell what to do instead of making true or false assertions. And while logicians normally allow only premises and conclusions that are of finite length, infinitary logic allows ones that are infinitely long.
A syllogism is a logically valid argument.
Any logically valid argument is a syllogism. This follows from the definition given by Aristotle himself:
A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that they produce the consequence, and by this, that no further term is required from without in order to make the consequence necessary. -- Aristotle, Prior Analytics, Book I
A few examples of syllogisms...
A circular argument:
God exists; Therefore, God exists.
A syllogism some people are wrong to believe that it is not a syllogism:
P and if P, then Q; therefore Q.
Fake Old News:
If Oswald didn't kill Kennedy, somebody else did; Oswald didn't kill Kennedy; Therefore, somebody else did.
A syllogism is a simplified form of argument, mainly used for pedagogical purposes to demonstrate valid and invalid moves in argumentation. If I may make a chess analogy, syllogisms are like a description of how individual pieces can move — part of the rules of the game — while an argument is an actual game being played.
Additions (per comments)
A syllogism is a logical unit that shows how properties of categories are transferred across two propositions. To use the classical example:
All men are mortal Socrates is a man ∴ Socrates is mortal (where ∴ is a symbol meaning 'therefore')
We have the category 'men', a property 'mortality' that inheres in that category, and an object 'Socrates' that is a member of that category. Generalized, this looks like:
Category C has property P Object O is a member of category C ∴ Object O has property P
... or in English, that any member of a category shares any properties that are inherent to that category. A syllogism makes explicit and procedural something that we might normally consider to be mere common sense, but syllogisms are useful because they allow us to manipulate the symbols to discover what are and are not valid moves. For instance, if I switch things up a bit and write the following:
Socrates is mortal Socrates is a man ∴ All men are mortal
We can easily see that this logical unit does not work (technically, an illicit minor term fallacy). Why should everyone be mortal just because Socrates is? We've made an invalid move by asserting that the property of an individual must necessarily inhere in the category the individual belongs to. It doesn't matter that in fact all men are mortal (that the conclusion is 'true'); what matters is that the process by which we came to that 'true' conclusion was flawed.
So, syllogisms spell out the basic valid and invalid logical units. However, it's rare that any real-world debate can be reduced to a simple syllogism. I mean, occasionally you'll find someone who makes the kind of false induction presented in that last (illicit minor) syllogism. This is particularly salient in prejudice, where people have an unfortunate tendency to judge entire groups by the perceived or imagined behavior of their worst members. But generally speaking arguments normally rest on numerous premises — some spoken and some unspoken — that are combined in various complex ways to produce a number of far-end conclusions. We cannot normally even reduce an argument to a simple sequential chain of syllogistic units; arguments are rarely so linear.
For instance, if we consider the ever-present debates about 'free will' that can found on this site, we'll find various arguments resting on diverse presumptions: the presence or absence of metaphysical entities such as a 'soul', physical determinism vs cultural determinism vs psychological determinism vs various forms of conditioned or unconditioned indeterminism, differences in choice models between libertarian, spiritual, and phenomenological perspectives, etc. One group might argue that the divine spark allows us to make free choices within the determinism of God's plan; another might say that we have only the illusion of free will as part of a fully deterministic order, a third might say that we have a presumptive capacity for free will and that the underlying reality is indeterminate. All of these positions make complex argument in their own favor and against the other positions. We might pick out some particular syllogisms, e.g.:
All men have free will Sam Harris is a man ∴ Sam Harris has free will
(which, incidentally, Sam Harris might disagree with). But this syllogism merely opens the door for new questions:
The syllogism is valid as given (in a simplistic, 'yeah-yeah-sure' sort of way), but it doesn't answer any of the properly philosophical questions, which must be explored through argumentation.
Is this sense, syllogisms become a limitation on what claims one can make in an argument — no one (and certainly no philosopher) wants to get called out for making a boneheaded error transferring a property improperly — but syllogisms rarely define and never resolve philosophical arguments.