It seems to me that an assumption is an untold premise in my argument. Is it right?
Perhaps the main difference between what might be called a premise and an assumption by different authors is their use in a proof with inference rules. Here is an example of this difference in a natural deduction proof using a Fitch-style of presentation.
Note that the first two lines above the horizontal line could be called either premises or original assumptions. One can use these sentences without deriving them to help derive the goal.
Note that on line 3 an assumption has been made, Q. That would be an additional assumption. It opens a subproof. It also has a horizontal line below it. When that subproof is closed through an inference rule this additional assumption can no longer be used. It is discharged.
Regardless of this use, one should consult the definitions of these terms in whatever logic textbook one is using. Here is how the forallx textbook describes such a subproof. Instead of P and Q, they use A and B: (page 107)
The general pattern at work here is the following. We first make an additional assumption, A; and from that additional assumption, we prove B.
For these authors, premise and assumption appear to be nearly synonymous: (page 98)
A formal proof is a sequence of sentences, some of which are marked as being initial assumptions (or premises).
Furthermore, (page 99)
Note also that we have drawn a line underneath the premise. Everything written above the line is an assumption. Everything written below the line will either be something which follows from the assumptions, or it will be some new assumption.
In this text premises and assumptions are similar. They are sentences one does not have to derive. To distinguish them is to distinguish their use in a proof. Some assumptions are called "initial assumptions" or "premises" which are the first lines in a Fitch-style proof above the horizontal line. Others are called "additional assumptions" which start subproofs.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf
First of all, a premise is a statement. As such, a premise is therefore explicit.
A premise is a statement which is assumed as true for the purpose of an argument, where the conclusion will be considered as following from the given premise (see note on assume).
As such a premise may be actually true or actually false. In many cases, it doesn't matter for the purpose of the argument whether the premise is true or false. We just need to be able to assume that it is true because we may be only interested in whether the argument is valid.
An assumption is usually understood as something else altogether. To confuse the matter somewhat, the noun "assumption" has the same origin as the verb "to assume" (13th, from Latin assūmptiō, the act of taking up, from Latin assūmere, which is ... to assume).
So, making assumption and assuming a premise may seem to mean the same thing. However, you assume a premise for the purpose of an argument and the premise is explicit.
And assumption is usually understood as a proposition taken for granted or accepted as true without proof.
It is routine for philosophers to talk about looking for hidden assumptions. Suppose your conclusion happens to be falsified one way or the other. Then the advice is to look for a hidden assumption that may need to be reconsidered.
The difference is rather slim, though.
We tend to talk of premises in the case of arguments considered in the context of discussing logical validity, somewhat like Aristotle discussed the validity of his example syllogisms.
In this context, premises are usually all explicit statements but in the case of enthymemes, one premise is usually left implicit. However, in this case, the implicit premise is normally absolutely obvious.
Suppose for example that some guys in a bar are talking about Barack Obama. After ten minutes, one of the guy says, "Yeah, well, politicians, they're all liars anyway", or something more colourful. Everyone listening to that will understand the libellous implication even though it isn't spelled out, and even though one premise, that Obama is indeed a politician, has been left out.
Nobody would talk of an assumption in such a case. Everybody present knows Obama is a politician. It's not an assumption, it's a fact. But it is nonetheless a premise and the premise is left out in such cases because we all understand what is meant without the need to spell it out. To spell it out would be somewhat counterproductive, like wasting everybody's time asserting trivia and thereby compromising the punch of the conclusion. Essentially, you let people draw the conclusion by themselves, which is an effective way of implicating them into your argument and into accepting the conclusion.
We tend to talk of assumptions in the context of real-life arguments, and often to suggest there is a hidden assumption that is simply false. The arguments we make in real life are never formally valid since that would be impractical. We have to make many hidden assumptions, sometimes consciously, sometimes unconsciously.
An assumption may be consciously hidden to avoid drawing everybody's attention on it if you think most people won't accept it. And we also all make arguments for ourselves when we reason and we usually can't possibly identify all our assumptions, hence the need sometimes to look for the one which may be false.
This also explains why philosophical arguments and debates are necessary, to draw out the assumptions that would remain otherwise hidden.
All philosophical views are based on any number of assumptions and usually it is impossibly to make them explicit. Which is why Leibniz worked on a calculus ratiocinator, a theoretical and universal method of logical calculation to solve the problem of endless and inconclusive philosophical debates.
Well, we are still having endless debates.
Note on assume
We may assume a premise because we believe it is actually true and we want to show that the conclusion is therefore true given the premise, or because we think the premise is in fact false, and we want to show that it is necessarily false.
We may also use an argument where we assume the premises as true even though we believe they are in fact false, and this to show that the conclusion implied by the premises contradicts a belief held by someone who also believes the premises true, thereby proving that this person holds two beliefs that contradict each other, and possibly that therefore this person is irrational.
That we assume premises is demonstrated by the fact that we start all arguments by asserting the premises, and this without any condition, unlike the conclusion of the argument, which is asserted only under the condition of the premises assumed as true.
The differences are subtle and the terms are often used interchangeably, but if one wants to make a distinction, it is roughly as follows:
Premises are those statements on which the conclusion of the argument depends, that is, the "if" part, provided the argument is of the form "If A1, A2, ..., hold, then B holds". Of course, we also have tautologies which don't depend on any premises.
Assumptions are hypotheses made in the process of a proof which may be discharged later on. Assumptions don't necessarily have to be true in order for the conclusion to be true -- it might be that they are only used to e.g. derive a contradiction in a proof by contradiction, and the conclusion can be independent of whether or not the assumption actually is true.
As a simple example, let's consider the following argument:
P → Q
¬P → (Q ∧ R)
P → Q and ¬P → (Q ∧ R) are the premises of the argument, and Q is the conclusion.
An informal proof of this argument could look as follows:
There are two cases to consider:
1) Assume P. With the premise P → Q, by modus ponens we have Q.
2) Assume ¬P. Modus ponens on the premise ¬P → (Q ∧ R) gives us Q ∧ R. From conjunction elimination it follows that also Q.
Since Q follows from all cases, the conclusion holds.
The proof presented here is a proof by cases: The law of the excluded middle tells us that either one of P or ¬P must hold, and if we can show that in any of those cases we can derive Q, then we know that Q holds no matter which out of P or ¬P is actually true. Throughout the proof, we made two assumptions: First we assumed that P holds, then we assumed that the negation holds.
Eventually, it does not matter which of these assumptions actually holds: The only statements on which the truth of Q depends are the premises P → Q and (Q ∧ R). If the premises are true, then the conclusion is true, whereas we don't care if P or ¬P is true, as the conclusion follows in any case. So the conclusion depends on the premises P → Q and P → (Q ∧ R), but not on the assumptions P and ¬P. The assumptions were only needed temporarily within the proof.
The same reasoning carried out in a formal proof (here: in Fitch style natural deduction) could look as follows:
In a Fitch proof, everything which is above a horizontal line is an assumption. Each assumption that is not already given (that is, each assumption that is not among the premises) opens a new subproof, which is marked by a vertical line going from the assumption to the final conclusion of the subproof. We have two subproofs, one from P to Q and from ¬P to Q. Everything which is below a horizontal line depends on the assumptions above that line -- so the occurrence Q on line 4 depends on the assumption P line 3, and the formula occurrences Q ∧ R and Q on lines 6 and 7 depend on the assumption ¬P on line 5. And everything depends on the assumptions P → Q and ¬P → (Q ∧ R), which are the premises. When applying the law of the excluded middle (= the argument "one of the cases P or ¬P must be true and Q follows in either case so Q must be true) in the final step, the two subproofs are closed and the assumptions P and ¬P discharged, meaning that the conclusion Q no longer depends on these assumptions, as we showed that it follows in either case. However, Q does still depend on P → Q and ¬P → (Q ∧ R). In a Fitch proof, assumptions which are made on the outermost vertical bar level (i.e. as assumptions in the main proof rather than in a subproof) and on which the conclusion will still be dependent are the premises.
Note, that assumptions do not need to be only temporary hypotheses which later on get discharged -- assumptions may remain open, so any premise is also, in a way, an assumption. Open assumptions are precisely those statements on which the conclusion depends. Here I'm using the terminology "open assumption" and "premise" more or less synonymously. Some people might want to make more fine-grained distinction between the two, but in practice, they will be used pretty much interchangeably.
As another example, let's consider the argument modus tollens:
P → Q
P → Q and ¬Q are the premises, and ¬P is the conclusion of the argument.
An informal proof could look as follows:
Assume the negation of the conclusion, i.e. assume P. With the premise P → Q, by modus ponens we can derive Q. But this is a direct contradiction to the premise ¬Q. Hence, the assumption P must have been wrong, and we have that ¬P.
This is a proof by contradiction: We assume the negation of the conclusion -- P -- derive a contradiction, and conclude that the assumption must have been wrong and ¬P does indeed hold. Note that in the proof by cases above I said we didn't care which out of the assumptions actually holds. Here, we even know that P can not hold, as we can show that it leads to a contradiction. Again, the truth of the conclusion does not depend on the assumption P: All we're saying is that if P → Q and ¬Q is true, then ¬P is true; the assumption P was only made temporarily within the proof to show that assuming anything else than ¬P will lead to a contradiction.
Formalizing this argument in Fitch will look as follows:
Again, the subproof is closed and the assumption P discharged at the point where we apply negation introduction to conclude that P couldn't be true as it leads to a contradiction. Again, everything above a horizontal bar is an assumption, which is the formulas P → Q, ¬Q and P in this case. Only P → Q, ¬Q remain open by the time we reach the conclusion ¬P, meaning that the truth of ¬P depends on the truth of P → Q and ¬Q, which are the premises.