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After a lengthy discussion with WillO here, we can't seem to find a common ground and I am interested in whether there really could be an argument without a single premise.
Another question whose answers would answer this question as well would be: What is the most accepted definition of an argument in (philosophcal) logic?
Please include references to logicians or serious quoteable textbooks/online sources in your answers.
Yes.
(disclaimer: I am a mathematician, and may be unaware of any connotation that philosophical logic might imply that diverges from how I understand logic)
For example, the argument
P or not Phas zero premises and arrives at a tautological conclusion (as must all valid arguments with zero premises). And the validity can be expressed syntactically, as its sole step is justified as the introduction of a known tautology.
Now, in hindsight, I am not surprised there might be people who adopt a convention where an argument is defined in a way that requires a nonempty set of hypotheses. This convention is not* useful. Fortunately, it's a level of detail that can be ignored for most discourse, and for the few times it matters, I would expect someone who does adopt this convention to be able to make the appropriate mental translation from what I say about what I mean by "argument" to an equivalent statement about what he means by "argument".
Although I might try to speak in a more neutral way if I knew the other person adopted the convention. (but if the person was making a point about refusing to make the translation and I cared about conveying my meaning anyways, I would do so in a way that pointedly demonstrates I'm rejecting the spirit of their convention while adhering to the letter)
But this is sort of how it goes with all sorts of conventions. If someone believes that domains of discourse must not be empty, I expect them to have some way of understanding a discussion involving domains of discourse that might actually be empty. If someone believes that the word "number" should only apply to positive things, I still expect them to be able to understand discussion about negative and complex numbers.
*: I'm sure such a convention could be useful for certain narrow purposes; e.g. when temporarily dealing with a very restricted notion of "argument" that is easier to reason about, before building a more generally useful notion of "argument" on top of it.
P or not P assume something like "reality is consistent"? Then again, I guess I have to assume that it's consistent to state that, otherwise "is" doesn't mean anything.
Jan 12, 2015 at 14:21
P or not P is false, then that implies that the classic rules of logic are insufficient to be useful with regards to P. This can still be dealt with by a Formal System, but it means we'd need to be using a formal system so different from the norm that one would be obliged to specify the formal system ahead of time (or be considered "wrong").
Jan 12, 2015 at 15:52
First of all, this is not a question to which there is a right or wrong answer; it's a question about conventions. That said, the standard convention, so far as I'm aware, is that an argument begins with a set of premises and there are no restrictions on that set (except, in a formal setting, for the requirement that it be recursive --- i.e. that we ought to be able to recognize the difference between a premise and a non-premise).
You seem to be strangely passionate about an issue of pure convention, but for all your passion you don't seem to be able to produce a single standard reference that requires the set of premises to be non-empty.
In fact, if you're doing propositional (as opposed to predicate) logic, it can be convenient to define a tautology as a conclusion that follows from no premises. (Of course you have to be careful about what "follows" means if you're going to do this --- and again, this is a matter of convention, not absolute truth.)
If you do feel inspired to pick up a book, I feel sure that any standard logic textbook will do. They tend to specify that an argument starts with a set of premises, and I have never seen any restrictions placed on the cardinality of that set. Disallowing sets with exactly zero premises strikes me as exactly as unnatural as disallowing sets with any odd number of premises. What would be the point?
A quite authoritative source can be Aristotle...
See Aristotle's logic :
All Aristotle's logic revolves around one notion: the deduction (sullogismos). A thorough explanation of what a deduction is, and what they are composed of, will necessarily lead us through the whole of his theory. What, then, is a deduction? Aristotle says:
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)
Each of the “things supposed” is a premise (protasis) of the argument, and what “results of necessity” is the conclusion (sumperasma).
The core of this definition is the notion of “resulting of necessity” (ex anankês sumbainein). This corresponds to a modern notion of logical consequence: X results of necessity from Y and Z if it would be impossible for X to be false when Y and Z are true. We could therefore take this to be a general definition of “valid argument”.
The word "argument" is our modern translation for logos : speech; thus, it is something "complex", involving a "chain" of statements.
See Peter Smith, An Introduction to Formal Logic (2003), page 1 :
The business of logic is the systematic evaluation of arguments for internal cogency. And the kind of internal cogency that will especially concern us is deductive validity.
[...]
1.1 What is an argument?
By "argument" we mean, roughly, a chain of reasoning in support of a certain conclusion. So we must distinguish arguments from mere disagreements and disputes. The children who shout at each other "You did", "I didn't", [...] are certainly disagreeing: but they are not arguing in our sense, i.e. they are not yet giving any reasons in support of one claim or the other.
[...]
In brief, it is one thing to consider whether an argument starts from true premisses; it is another thing entirely to consider whether it moves on by reliable inferential steps.
See also Ernest LePore & Sam Cumming, Meaning and Argument : An Introduction to Logic Through Language (2nd ed - 2009), page 5 :
an argument is any set of statements one of which is the conclusion and the others are the premises. The relationship between the conclusion and the premises is such that the conclusion purportedly follows from the premises.
In spite of the "intuitive" notion of argument, the "technical" definition of logical consequence allows for 0-premises arguments : a logically valid sentence (or formula) is a logical consequence of a the empty set of premises.
No, there cannot.
Couple of modern interpretations of various logical systems allow for infinitely many valid zero-premise arguments (tautologies and wffs), but only deductive and in formal languages exclusively.
Example 1: Restall and Asmus - History of the Consequence Relation,
Example 2: Gensler - Introduction to Logic
Example 3: Wikipedia - Formal Logic.
On the other hand, one could argue - and that would be my claim - that none of those superficially standalone sentences are in fact unsupported, since in those cases vocabulary and rules of a given language serve directly as premises. For instance:
(P V ~P)
A zero-premise argument can be formulated exclusively in a formal language for it to work, since no natural language on its own provides a clear set of logical axioms. Unless we consider a sentence like "Nouns do not conjugate." as a zero-premise argument expressed in a natural language, but it seems to me more of a naturalised instance of a formal language of grammar.
Consequently, the law of noncontradiction, for instance, is either a sentence of aristotelian logic expressed in naturally due to lack of sufficient means to formalise it, or sentence of a natural language expressed with an underlying assumption that one accepts the two-valued logic; the underlying assumption most of us seem to share pretty much automatically. However, there is a number of many-valued logical systems like Łukasiewicz's, Kleene's, or Zadeh's under which such statement is not at all obvious.
So, to reformulate my claim - I think that what we call zero-premise arguments are in fact sentences of a logic or a formal language which use its axioms or rules and vocabulary as premises. We tend not see them during our proceedings, hence the entire fuss.
Your side of the argument has some interesting side effects which lead me to consider the requirement that an argument have premises be faulty.
I would agree with WillO that the most accepted form of argument would be a Formal Proof. This form of argument is the most accepted because its language is the most precise, leading to a surprising resilience against vacuous arguments. Each Formal Proof starts with a finite set of axioms, which I believe can safely be compared to your word "premises" without loss of generality.
WillO's claim is that it is valid for that finite set of axioms to be the empty set, and nothing immediately comes to mind to countermand that. Forbidding the empty set also has an interesting implication: A proof with zero axioms may be deemed "invalid," but a proof with one axiom: "P or not P" magically becomes valid, even if P never appears in the proof.
Forbidding this rather absurd looking tactic is difficult. How do you claim "an axiom is assumed true," but forbid axioms that are provably true? What about proofs which depend on an unproven theory (such as the Riemann Zeta theorem)? Do all such proofs become invalid if the Riemann Zeta theorem is proven, simply because one axiom is now provably true?
I think it would be difficult if not impossible to meaningfully reject a Formal Proof with an empty set of axioms without causing a large number of undesirable artifacts. Given the popularity of that form of argument, I would say that should be case closed for the debate unless someone can find quotes otherwise.
In your comment argument, Lukas, you mentioned "Pick up any logic textbook, it will tell you that there has to be at least 1-2 premises, depending on the author" This quote caught my attention because, in context, it looked like a poor argument. WillO had cited a single source (which actually does say it can be empty: "finite sequence" is a very specific mathematical term that includes empty sequences), and you argued any textbook should contradict him, without providing one yourself, leaving him to dig through dozens of textbooks in search of a sentence to prove your point that he doesn't believe in.
However, it is true that most arguments have premisses, and often they are of small number. If it was not a rule that you have premises, might there still be some reason behind your position? I think there might be.
The number of things which can be successfully argued without premisses is vanishingly small. It's certainly not reasonable to make meaningful debate about something important like "freedom," "meaning," or "good" with an empty-set of premisses. Those words are too hard to discuss without dragging human ideals into the debate, and those do not have mathematical or logical meanings which could allow them to be manipulated in a Formal Proof.
This leaves arguments without premisses as suspect. An argument without premisses cannot prove anything which was not already in the formal system used as part of the proof. This creates a sticky situation where either:
The latter is a bother because the issues in comparing formal systems are innumerable. It requires vastly more study and attention to detail to identify inconsistencies in formal systems than it takes to identify inconsistencies in arguments made with the formal systems. Mathematicians may debate for centuries over whether or not a particular class of proof is valid, and they literally make a living at it.
Thus, if you claim something profound from an argument with no premises, it is common for people to reject the claim outright, knowing that there is a good chance they lack the mathematical prowess to attack the formal system issues. It isn't to say that an argument without premises is invalid, but it does serve as a strong indicator to a listener that the speaker has not fully thought out the argument. There are likely premises sneaking in part-way through the argument, and someone will have to tease them out (this is known as "missing premisses"). The presence of premises is not proof that an argument is good, but the lack of premisses calls it into question.
So for the final round trip, I ask when does it make sense to make an argument with no premisses. For this, I must hand-wave away one thing which could be considered a premise: the formal system itself. No argument can be made without a system to make the argument in. If that counts as a premise, then your answer is correct. However, I see no evidence that this is a standard definition of "premise.*" So I will make the assumption that "premise" and "axioms for a formal proof" are identical while trying to make a meaningful argument with no premise.
Consider a proof style called "proof by contradiction." The goal of this style is to argue the opposite of your claim and demonstrate that doing so leads to an unacceptable proposition proven true (such as proving a contradiction). Consider that you have made an argument of your own, with premises, using a proof system which is different than mine. Consider that I agree with your premises, but not your conclusion. Clearly there is a disagreement within the proof system, but even if I am mathematician who can spend a decade dissecting your system, you wouldn't understand the results unless you too were a mathematician.
A proof by contradiction could discredit your proof system without either of us having to truly understand why the proof system is invalid. To make such a proof, I will make an argument in your proof system without premises. I will then use the proof system to generate an unsatisfactory proof. For example, if I can use your proof system to prove "It is acceptable to kill red-headed women on Tuesdays, but only if they are wearing earrings," without a single premise, that would leave your proof system highly suspect (unless that's your religion's sort of thinking!).
By making such a proof, I force your hand. You must either accept that, given any set of premises you choose, I can twist the argument into killing red-heads, or you are forced to include a special premise which forbids it in every single proof you make. The constant presence of this bothersome premise discredits the strength of your proof system, and eventually you will choose a different one.
For a less violent more mathematical example, consider "the axiom of choice (AoC)." AoC is an axiom is set theory that states "If I have a set containing non-empty "child" sets, I can create a new set by taking one element from each of the child sets." This is intuitive: if I have 10 bags with items in them, I can easily make an 11th bag by taking one item from each bag. This is so intuitive, that we often assume it as part of our formal system, as is done in the set theory called Zermelo–Fraenkel set theory with the axiom of choice (ZFC)
However, there are some very nuanced details that show up as we look at infinite sets of infinite sets. It takes years of college math to appreciate why these issues arrive, but it takes far less study to appreciate the consequences. There is a proof in ZFC which can take a sphere, cut it into 5 parts, rotate them, and re-assemble them into two spheres, each with the same volume as the original. No cheating, no empty cores, no spongy material. The math just works.
This bothers mathematicians so much that they often do not accept ZFC proofs, just because they've seen something that unnerves them and seems intuitively false. They limit themselves to ZF proof (Zermelo–Fraenkel without AoC). This proof that doubles two spheres has no premises in any proof system which has ZFC baked into it.
A classic resolution to this is to use proof systems which have ZF baked into their syntax, but not ZFC. One then adds the axiom of choice as a premise. Thus, most arguments for ZFC are made with at least one premise, but the original trouble-maker that cause people to reject ZFC based proof systems was an argument with no premises... thus showing such arguments are useful.
So, in summary: An argument with no premises is reasonable and meaningful. One purpose for such arguments is to discredit proof systems which appear to be generating undesirable relationships between premises and conclusions.
* Given that I just called you on an argument like this, claiming overwhelming literature support without actually citing any, let me know if you disagree with this claim and I'll actually go dig up a few examples or concede that part of the argument. I'm hoping you simply agree with me, because this is a linguistic topic, so there will be a lot of muddiness in the literature. I want to avoid that digging if at all possible. Clarity is good.
Yes, some logic texts allow that an argument can have no premises, though as others have noted, it is a matter of convention whether we extend the definition of “argument” to include single sentences or not.
Note that though logic teachers sometimes say that “logic is about arguments,” the word “argument” is dispensable in framing what the study of logic is about. Ted Sider in Logic for Philosophy (2010) defines logic as the study of “truth preservation by virtue of form.” Sider then explains the concepts of truth preservation and logical consequence without defining “argument.” If he even mentions the word “argument” in his presentation of logic, I don't see it. So, it's not the case that we need to understand what an argument is in order to do logic, or that the study of logic even depends on defining it one way or another. However, presentations of logic that do make the term central to their presentation need to define it, and they do it in very different ways.
One example of a contemporary logic textbook which includes single sentences under its definition is Nicholas J. J. Smith's Logic: The Laws of Truth (2012):
In our usage, an argument is a sequence of propositions. We call the last proposition in the argument the conclusion…. The other propositions are premises …. There may be any finite number of premises (even zero). (page 11)
Other philosophers' definitions include or presuppose that an argument includes premises. An example is Jennifer Fisher (2008), On the Philosophy of Logic. She defines “argument” as
a set of sentences in which some sentence (sentences) is (are) supposed to give some kind of support to another sentence. (page 214)
Like many, many other features of logic, we have a lot of flexibility about how we define the terms and symbols we employ — and even which terms and symbols we choose to make central to our presentation. The idea of “an argument” is certainly subject to such flexibility.
Neither deduction nor induction can do away with premises.
"Deduction tells you what follows from your premises, but does not tell you whether your premises are true."*
Scientific method assumes the principle of induction by simple enumeration, but the principle itself cannot be proved inductively. Bertrand Russell suggested that "the postulates required to validate scientific method may be reduced to five."** They are:
a. The postulate of quasi-permanence. b. The postulate of separable causal lines. c. The postulate of spatio-temporal continuity in causal lines. d. The postulate of the common causal origin of similar structures ranged about a centre, or, more simply, the structural postulate. e. The postulate of analogy.
However, it is possible to start an argument without common premises. The goal of such arguments is to find out if there are common premises. If there is no common premise, politely walk away like our Lord Bertrand Russell:
While in Princeton, I came to know Einstein fairly well. I used to go to his house once a week to discuss with him and Gödel and Pauli. These discussions were in some ways disappointing, for, although all three of them were Jews and exiles and, in intention, cosmopolitans, I found that they all had a German bias towards metaphysics, and in spite of our utmost endeavours we never arrived at common premises from which to argue. Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal 'not' was lad up in heaven where virtuous logicians might hope to meet it in hereafter.***
**Russell, Bertrand. The Art of Philosophizing. New York: Philosophical Library: 1968
**Russell, Bertrand. Human Knowledge, Its Scope and Limits. New York: Simon and Schuster, 1948
***Russell, Bertrand. Autobiography. Longdon and New York: George Allen & Unwin, 1967.
Yes, there is at least one zero-premise argument, in English:
Therefore, there is at least one zero-premise argument.
I heard tell of this argument a few years ago in discussion with some friends/philosopher colleagues. I believe it was attributed to Robert Stalnaker, but I can't find the reference.
The nice thing about this proof by example is that it provides a wholly intuitive, non-theoretical reason to suppose that such arguments exist.
Yes, here is one:
The surprising fact, C, is observed. But if A were true, C would be a matter of course. Hence, there is reason to suspect that A is true. (CP 5.189)
As for terms, propositions and arguments, the meaning of these words tends to drift, so should be thought of together.
For example from “On a New List of Categories”, Peirce: 1) Symbols which directly determine only their grounds or imputed qualities, and are thus but sums of marks or terms; 2) Symbols which also independently determine their objects by means of other term or terms, and thus, expressing their own objective validity, become capable of truth or falsehood, that is, are propositions; and, 3) Symbols which also independently determine their interpretants, and thus the minds to which they appeal, by premissing a proposition or propositions which such a mind is to admit. These are arguments.
Another place for which a definition for “Argument” is given is in “A Neglected Argument for the Reality of God” but even there, you end up viewing Argument contra Argumentation.
“An "Argument" is any process of thought reasonably tending to produce a definite belief.
An "Argumentation" is an Argument proceeding upon definitely formulated premisses.”
Therefore, the difference in whether CP 5.189 is or is not an argument without definite premises depends on whether you accept C and A to be definitely formulated or not. That is, you ought to get ahead of the game by definitely stating criteria for what is meant by definitely formulated at the outset.
This will depend on the situation.
For example, you could use CP 5.189 as a pragmatic maxim to support general education. There, it would be an argument with premises because C and A.
It would be an argument without premises for a problem situation in which we are currently immersed because we have not yet definitely formulated the premisses that is uttered out loud in public. Clearly determining such terms is a step toward coming to agreement.
I suppose this distinction is what is meant by potential and the actual.
Suppose that I make the claim that because there are black swans, obama must be a swan. You'd then argue that's logically invalid, when perhaps I wasn't interested in validity at all, only asserting that there are black swans.
We're arguing, and I have made an assertion which is not a premise.
So I think that there can be an "argument" without premises, as long as the to and fro of the discussion isn't contextualising it in that way.
How could you define "argument without premises" so that this example is ruled out?
