Why is it argued that an argument has one and only one conclusion?

Question

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.)

Answer 1

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.

Answer 2

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.

Answer 3

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.

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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