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Why can't an argument have more than just one conclusion? If we assume some premises and we assume them to be true, then by some inference rules we are sometimes able to deduce more than just one true conclusion, is that right? Why do they argue then that an argument can have one and only one conclusion? They're proposing it for example in the following article: http://www.uky.edu/~rosdatte/phi120/lesson1a.htm

What do we call then such a system containing some specific premises, symbols, inference rules and ALL the conclusions that could be derived from given premises by given inference rules? In mathematics aren't we calling it a mathematical theory?

(I do not have a logic background.)

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    Arguments very often have more than one conclusion. Usually, you'll see C1, C2, C3, etc. (C stands for conclusion). This doesn't have a special name, it's just an "argument." A mathematical proof is an example of this, but usually the premises of a mathematical proof are more certain than premises of a philosophical proof (but not always). – Josh Aug 24 '19 at 18:25
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    What do they mean than by saying that a valid argument can have one and only one conclusion? For example in the following article? - uky.edu/~rosdatte/phi120/lesson1a.htm – TKN Aug 24 '19 at 19:07
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    I think they're trying to make the point that arguments should be separated in a way that's obvious to the reader of the argument. While there is no logical reason that multiple conclusions cannot follow from premises, it is often most helpful to present one conclusion at a time. – Josh Aug 24 '19 at 19:10
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    They do not argue it, they just state their convention. One can make a different convention where multiple conclusions are allowed, but then one can always turn them into a single one by taking the conjunction. So it makes no real difference. – Conifold Aug 25 '19 at 7:54
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    There is such a thing as muiltiple conclusion logic. It was developed by Timothy Smiley in the 1970s, but it never became popular. As Conifold says, having a singular conclusion is just a useful convention. – Bumble Aug 25 '19 at 10:36
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You are confusing two uses of the word argument.

In one sense, an argument is an extended discourse with limited aims such as education or persuasion.

In the second sense, argument is a synonym for the technical term inference which is the process by which a single proposition can be constructed from a collection of premises (sometimes unstated).

So, in the broader sense, an argument can have more than one conclusion (and usually does). In the narrow sense, it cannot by definition. Note that the broader use incorporates the narrower use generally.

How do we call than such a system containing some specific premises, symbols, inference rules and ALL the conclusions that could be derived from given premises by given inference rules? In mathematics aren't we calling it a mathematical theory?

And yes, once one begins to start reasoning from first principles axiomatically, the body of axioms or postulates undergo inferences which provide theorems, corollaries, and lemma, and collectively are referred to as a theory, which has been formalized mathematically as model theory.

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One of the earliest argument formats philosophy studied was the syllogism, in which two premises yield one conclusion. It could be argued all more complex arguments are made by chaining lots of syllogisms together, making the conclusion of one a premise of another. On this view, everything after the "atomic" premises is a conclusion obtained from them. (By atomic premises, I mean those that are never obtained as conclusions; there may be quite a few of these if we introduce more of them later.)

But how we write an argument is to an extent a matter of orthography. In principle, we can rearrange any argument to have one premise and one conclusion, as long as you admit the conjunction of finitely many statements "counts as" one statement. But we usually think of a long argument as having lots of premises and conclusions, most of them being a combination of the two as we go along. For example, if you read a mathematics textbook that leaves none of its claims' proofs unstated, you could treat the book as proving one conjunction of theorems from one conjunction of axioms, but you wouldn't. You would say, "here's a list of theorems, obtained from this list of axioms (not to mention this list of rules of inference)".

I'll mention one subtlety. Say you're studying a first-order theory with infinitely many axioms comprising a schema that can be summarised as one second-order statement, together with finitely many "standalone" axioms. (This is, for example, what Peano arithmetic & ZF set theory do in order to be first-order.) Then you can't collapse everything you're assuming into one first-order statement, nor everything you derive from it. So sometimes, the way we "count" statements gets into thorny technical aspects.

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    So is it just a matter of convention to argue that an argument can has one and only one conclusion? They argue that an argument can has one and only one conclusion in the following article for example: uky.edu/~rosdatte/phi120/lesson1a.htm . What do they mean by that? – TKN Aug 25 '19 at 7:09
  • @TKN Presumably the view is that, once you reach a conclusion, you start a new argument. The way we split discourse into "arguments" is typically to consider everything in the middle as neither a premise nor a conclusion, just an intermediate result, or else to say they're not "the middle" because you're looking at multiple arguments. But propositions cannot truly be counted in a well-defined way because they're closed under conjunction. – J.G. Aug 25 '19 at 7:12
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    But it is true that a particular set of premises can yield more than just one conclusion, right? – TKN Aug 25 '19 at 7:19
  • How do we call than a set of atomic premises together with a set of all conclusions deduced from them? If we add the inference rules used for this deduction, it reminds me of a mathematical theory.. – TKN Aug 25 '19 at 7:27
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An argument has only one conclusion because that is the accepted convention. As Bumble notes in a comment, multiple-conclusion logics exist. Wikipedia describes such logics as follows:

A multiple-conclusion logic is one in which logical consequence is a relation, ⊢, between two sets of sentences (or propositions). Γ ⊢ Δ is typically interpreted as meaning that whenever each element of Γ is true, some element of Δ is true; and whenever each element of Δ is false, some element of Γ is false.

They give these reasons why one might prefer a multiple-conclusion logic:

Some logicians favor a multiple-conclusion consequence relation over the more traditional single-conclusion relation on the grounds that the latter is asymmetric (in the informal, non-mathematical sense) and favors truth over falsity (or assertion over denial).

Perhaps a reason, besides convention, to prefer a single-conclusion logic is that it may be easier to check an argument with one conclusion.

Wikipedia also provides two examples of multiple-conclusion logics:

  1. Gerhard Gentzen's sequent calculus.
  2. D. J. Shoesmith and Timothy Smiley's Multiple-conclusion logic, Cambridge, 1978. For an overview see the review by Andreas Blass of this work in the BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 1, January 1980 available at Project Euclid.

Wikipedia contributors. (2010, December 30). Multiple-conclusion logic. In Wikipedia, The Free Encyclopedia. Retrieved 14:17, September 25, 2019, from https://en.wikipedia.org/w/index.php?title=Multiple-conclusion_logic&oldid=405064210

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