Give it a read. It is divided into 4 parts:

My Take on the Liar Paradox (Part I of IV)
My Take on the Liar Paradox (Part II of IV)
My Take on the Liar Paradox (Part III of IV)
My Take on the Liar Paradox (Part IV of IV)
All four articles combine to a total of about 8,000 words, so it will not take long for a dedicated reader to read through it.

Introduction

Abstract

In this essay I attempt to clarify what it means to say that an argument begs the question. One may think that it is a fairly straightforward matter but my analysis reveals that it isn’t so.

Shorthands

“BTQ” means begging the question, or begs the question whichever is grammatically correct on the context.

The phrase “begs the question” in english

The phrase “begs the question” has at least two meanings in english. The first and perhaps most common meaning is that of raising an important question. As it is written on FallacyFiles.org:

The phrase “begs the question” has come to be used to mean “raises the question” or “suggests the question”, as in “that begs the question” followed by the question supposedly begged. The following headlines are examples:

• • • • Warm Weather Begs the Question:
        To Water or Not to Water Yard Plants

      • Latest Internet Fracas Begs the Question:
        Who’s Driving the Internet Bus?

      • Hot Holiday Begs Big Question:
        Can the Party Continue?

This is a confusing usage which is apparently based upon a literal misreading of the phrase “begs the question”. It should be avoided, and must be distinguished from its use to refer to the fallacy.”¹

The second meaning of “beg the question” is in the informal logical fallacy of begging the question. It is this meaning that this essay attempts to clarify.

Proposed definitions

So what does it mean to say that an argument BTQ? There are surprisingly many different answers from good sources. Below I quote many of the different definitions given, some by authorities and some not.

FallacyFiles.org

In an article entitled “Begging the Question” FallacyFiles.org writes:

“The phrase “begging the question”, or “petitio principii” in Latin, refers to the “question” in a formal debate—that is, the issue being debated. In such a debate, one side may ask the other side to concede certain points in order to speed up the proceedings. To “beg” the question is to ask that the very point at issue be conceded, which is of course illegitimate. “²

And:

“Any form of argument in which the conclusion occurs as one of the premisses, or a chain of arguments in which the final conclusion is a premiss of one of the earlier arguments in the chain. More generally, an argument begs the question when it assumes any controversial point not conceded by the other side.”³

Notice how vague the one mentioned in the first paragraph is. To the defense of FallacyFiles.org, we may note that that paragraph is entitled “Etymology”, and is perhaps not meant to actually explain clearly what it means to BTQ but only to explain how the etymology relates to the meaning of the term.

The second paragraph is entitled “Exposition” and is clearly meant to explain the meaning of the term. However the paragraph features two independent definitions, a strict (which is a disjunction) and a general (or rather, broad) one.

butterfliesandwheels.com

In the article entitled “Bad Moves: Begging the question” butterfliesandwheels.com writes:

“Begging the question – assuming what needs to be argued for [...]”⁴

skepdic.com aka. The Skeptic’s Dictionary

In an article entitled “begging the question” it is written on skepdic.com:

“Begging the question is what one does in an argument when one assumes what one claims to be proving.”⁵

And a bit later:

“If one’s premises entail one’s conclusion, and one’s premises are questionable, one is said to beg the question.”⁶

Notice that these two definitions are not at all identical. Examples will show this later.

Wikipedia.org

In an article entitled “Begging the question” it is written on Wikipedia.org:

“The fallacy of petitio principii, or “begging the question”, is committed “when a proposition which requires proof is assumed without proof.”[3] More specifically, petitio principii refers to arguing for a conclusion that has already been assumed in the premise. The fallacy may be committed in various ways.

When the fallacy of begging the question is committed in a single step, it is sometimes called a hysteron proteron,[4] as in the statement “Opium induces sleep because it has a soporific quality”.[5] Such fallacies may not be immediately obvious in English because the English language has so many synonyms; one way to beg the question is to make a statement first in concrete terms, then in abstract ones, or vice-versa.[5] Another is to “bring forth a proposition expressed in words of Saxon origin, and give as a reason for it the very same proposition stated in words of Norman origin”,[6] as in this example: “To allow every man an unbounded freedom of speech must always be, on the whole advantageous to the State, for it is highly conducive to the interests of the community that each individual should enjoy a liberty perfectly unlimited of expressing his sentiments.”[7]

When the fallacy of begging the question is committed in more than one step, it is sometimes referred to as circulus in probando or reasoning in a circle[4] but incorrectly if we look at the definition Aristotle gave us in Prior Analytics.[1]

“Begging the question” can also refer to making an argument in which the premise “is different from the conclusion … but is controversial or questionable for the same reasons that typically might lead someone to question the conclusion.”[8]”⁷

And:

“In informal situations, the term begging the question is often used in place of circular argument. In the formal context however, begging the question holds a different meaning.[1] In its shortest form, circular reasoning is the basing of two conclusions by means of which there is demonstrated a reversed premise of the first argument. Begging the question does not require any such reversal.

Begging the question is similar to the Fallacy of many questions: a fallacy of technique that results from presenting evidence in support of a conclusion that is less likely to be accepted than merely asserting the conclusion. A specific form of this is reducing an assertion to an instance of a more general assertion which is no more known to be true than the more specific assertion:

* All intentional acts of killing human beings are morally wrong.

* The death penalty is an intentional act of killing a human being.

* Therefore the death penalty is wrong.

If the first premise is accepted as an axiom within some moral system or code, this reasoning is a cogent argument against the death penalty. If not, it is in fact a weaker argument than a mere assertion that the death penalty is wrong, since the first premise is stronger than the conclusion.”⁸

New York Times

In an article entitled “ON LANGUAGE; Take My Question Please!” it is written in New York Times:

“”This sentence fragment uses ‘begs the question,’ ” he writes, ”in the sense of a question that begs to be asked, usually because it is obvious to all. However, I am plagued by my logic course of some years ago, which taught me that begging the question is nothing of the kind. Rather, begging the question is a logically invalid form of argument that uses the point to be proven as part of the argument for its proof.”

Amen. Readers have been protesting this misuse of a term about a concept set down by Aristotle, a student of Plato Cacheris, in his book on logic written about 350 B.C. (Here comes mail on B.C.E.) His Greek term en archei aiteisthai was translated by the Romans as petitio principii, and rendered into English in 1581 as begging the question. In whatever language, it described the fallacy known as ”the assumption at the outset.”

In his 1988 book, ”Thinking Logically,” Prof. James Freeman explains: ”An argument begs the question when the conclusion, in the same or different words, or a statement presupposing the conclusion, is introduced as a premise. The case for the conclusion ultimately depends on accepting the conclusion itself.””⁹

Notice how it says that it is an invalid form of argument. But surely any argument that commits the strict fallacy of BTQ, that is, the conclusion is identical to a premise, is a valid argument. Why? Valid arguments are precisely those arguments where the premises logically imply the conclusion. Since any proposition implies itself [P⇒P], then any argument that BTQ in the strict sense is valid.

nizkor.org

In an article entitled “Fallacy: Begging the Question” it is written on nizkor.org:

“Begging the Question is a fallacy in which the premises include the claim that the conclusion is true or (directly or indirectly) assume that the conclusion is true. This sort of “reasoning” typically has the following form.

1. Premises in which the truth of the conclusion is claimed or the truth of the conclusion is assumed (either directly or indirectly).

2. Claim C (the conclusion) is true.

This sort of “reasoning” is fallacious because simply assuming that the conclusion is true (directly or indirectly) in the premises does not constitute evidence for that conclusion. Obviously, simply assuming a claim is true does not serve as evidence for that claim. This is especially clear in particularly blatant cases: “X is true. The evidence for this claim is that X is true.”

Some cases of question begging are fairly blatant, while others can be extremely subtle.”¹⁰

The Cambridge Dictionary of Philosophy

In an article entitled “Circular Reasoning” Robert Audi writes:

“circular reasoning, reasoning that, when traced backward from its conclusion, returns to that starting point, as one returns to a starting point when tracing a circle. The discussion of this topic by Richard Whatley (1787–1863) in his Logic (1826) sets a high standard of clarity and penetration. Logic textbooks often quote the following example from Whatley:

To allow every man an unbounded freedom of speech must always be, on the whole, advantageous to the State; for it is highly conducive to the interests of the Community, that each individual should enjoy a liberty perfectly unlimited, of expressing his sentiments.

This passage illustrates how circular reasoning is less obvious in a language, such as English, that, in Whatley’s words, is “abounding in synonymous expressions, which have no resemblance in sound, and no connection in etymology.” The premise and conclusion do not consist of just the same words in the same order, nor can logical or grammatical principles transform one into the other. Rather, they have the same propositional content: they say the same thing in different words. That is why appealing to one of them to provide reason for believing the other amounts to giving something as a reason for itself. Circular reasoning is often said to beg the question. ‘Begging the question’ and petitio principii are translations of a phrase in Aristotle connected with a game of formal disputation played in antiquity but not in recent times. The meanings of ‘question’ and ‘begging’ do not in any clear way determine the meaning of ‘question begging’. There is no simple argument form that all and only circular arguments have. It is not logic, in Whatley’s example above, that determines the identity of content between the premise and the conclusion. Some theorists propose rather more complicated formal or syntactic accounts of circularity. Others believe that any account of circular reasoning must refer to the beliefs of those who reason. Whether or not the following argument about articles in this dictionary is circular depends on why the first premise should be accepted:

(1) The article on inference contains no split infinitives.

(2) The other articles contain no split infinitives.

Therefore, (3) No article contains split infinitives.

Consider two cases. Case I: Although (2) supports (1) inductively, both (1) and (2) have solid outside support independent of any prior acceptance of (3). This reasoning is not circular. Case II: Someone who advances the argument accepts (1) or (2) or both, only because he believes (3). Such reasoning is circular, even though neither premise expresses just the same proposition as the conclusion. The question remains controversial whether, in explaining circularity, we should refer to the beliefs of individual reasoners or only to the surrounding circumstances. One purpose of reasoning is to increase the degree of reasonable confidence that one has in the truth of a conclusion. Presuming the truth of a conclusion in support of a premise thwarts this purpose, because the initial degree of reasonable confidence in the premise cannot then exceed the initial degree of reasonable confidence in the conclusion.”¹¹

What can we gather from this?

There is consensus about a strict definition of BTQ which is identical to circular logic. This is defined as: An argument is circular iff one of the premises is identical to the conclusion.

There is no consensus about a broad definition of BTQ. At best this is some intuitive notion. Further analysis could try to find a meaning appropriate for this broad sense. That task I will take up in a forthcoming essay.

Notes

In an earlier essay I mentioned that that meaninglessness is contagious with respect to sentences. One can pretty easily formulate the principle in normal english – if a sentence is meaningless, then so is any more complex sentence of which it is a part of. To get a proper, formal formulation of this we may simply think of the rules in logic systems used to form well-formed formulas (=wff’s) and then formulate some similar principles for the meaninglessness of sentences. Here’s what I have in mind:

Negation. For all sentences, iff it is not the case that a sentence is meaningful, then it is not the case that the negation of that sentence is meaningful.

(∀S)(¬M(S)↔¬M(¬S)

Conjunction part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the conjunction of that sentence with another sentence is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S∧Z)¹

Disjunction part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the disjunction of that sentence with another sentence is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S∨Z)

Implication/conditional part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the implication of the first sentence to the second is meaningful, and it is not the case that the implication of the second sentence to the first is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S→Z)∧¬M(Z→S))

Bi-implication/bi-conditional part. For all sentences, if it is not the case that a sentence is meaningful, then for all sentences, it is not the case that the bi-implication of the first sentence to the second is meaningful, and it is not the case that the bi-implication of the second sentence to the first is meaningful.

(∀S)(¬M(S)→(∀Z)¬M(S↔Z)∧¬M(Z↔S))

This should cover propositional logic. It is left to the reader can invent the relevant principles for modal logics and predicate logic.

Notes

By sentence theory I just mean a theory of truth carriers that implies that some sentences are true or some are false. Not necessarily a monist sentence theory (=theory that implies that sentences are the only kind of truth carriers) or a theory of sentences as primary truth carriers (=theory that implies that sentences are the primary truth carriers). For more about these terms, see my earlier writings on the subject.

Anyway, I read the newest post on my favorite logic blog (Blog&~Blog). It dealt with the sentences which I have given incredibly clever names (in footnotes):

For all sentences, if it is not the case that it is meaningful, then it is not the case that it is true.

NMNT.¹ (∀S)(¬M(S)→T(S))

For all sentences, if it is not the case that it is meaningful, then it is not the case that it is false.

NMNF.² (∀S)(¬M(S)→F(S))

With the obvious interpretation keys.

This seems like plausible sentences to many when faced with sentences such as the Chomsky:

C. Colorless green ideas sleep furiously.

Which Ben, btw, got wrong as he forgot the first word.

Let’s also agree that:

1. It is not the case that C is meaningful.

¬M(C)

However, this along with some other sentences is inconsistent (=implies a contradiction). First sentence bivalence:

SB.³ For all sentences, it is either true or it is false.

(∀S)(T(S)∨F(S))

The contradiction is easy to derive here:

2. ¬T(C) [from 1, NMNT, MP]

3. ¬F(C) [from 1, NMNF, MP]

4. T(C) [from 3, SB, DS]

5. T(C) ∧¬T(C) [from 2, 4, conj.]

Contradiction! So this doesn’t work. Here I told Ben (author of the blog) that I would drop SB.⁴ However that apparently doesn’t work either.

Say hi to the T-schema, or the semantic theory of truth:

TS1. For all sentences, iff it is true, then it is the case.

(∀S)(T(S)↔S)

TS2. For all sentences, iff it is false, then it is not the case.

(∀S)(F(S)↔¬S)

Now these are obvious to most people. Not something is that plausible to deny unless the alternatives are really bad. However from these one can get their contra-positional versions:

TS1-CP. For all sentences, iff it is not the case, then it is not the case that it is true.

(∀S)(¬S↔¬T(S))

TS2-CP. For all sentences, iff it is not the case that it is not the case, then it is not the case that it is false.

(∀S)(¬¬S↔¬F(S))

And from these, we can derive their converses (and we can do that because these are bi-conditionals that can be conversed without problems). Do the same for TS1 and TS2:

TS1-CP-C. For all sentences, iff it is not the case that it is true, then it is not the case.

(∀S)(¬T(S)↔¬S)

TS2-CP-C. For all sentences, iff it is not the case that it is false, then it is not the case that it is not the case

(∀S)(¬F(S)↔¬¬S)

TS1-C. For all sentences, iff it is the case, then it is true.

(∀S)(S↔T(S))

TS2-C. For all sentences, iff it is not the case, then it is false.

(∀S)(¬S↔F(S))

And these actually need to be simplified too before I can use them, but I’m too lazy to do that, so I’ll just add a simp. step. No big deal.

Now:

6. ¬C [from 2, TS1-CP-C, simp., MP]

7. F(C) [from 6, TS2-C, simp., MP]

8. F(C)∧¬F(C) [from 3, 7, conj.]

Contradiction. And I didn’t need to use double negation to get it though one could do that too with TS2-CP-C, and of course I didn’t use SB either. It seems to me that this is terrible and the best way out of the contradiction is to deny NMNT and NMNF, and believe instead that sentences like C cannot even meaningfully be said to be true or false, nor can they meaningfully be said to be not true or not false. Any complex sentence with a meaningless part is itself meaningless.⁵

There is a tendency for people to conflate denial of properties with the denial of the meaningful application of these properties to things. This seems to be the case here too. So instead of saying things like:

Meaningless sentences are not true.

Cars are not true.

We should say things like:

Meaningless sentences cannot meaningfully be said to be true.

Cars cannot meaningfully be said to be not true.

Maybe some people sometimes, confusingly, use the first versions as a shorthand for the second. If they do and really mean what the second ones mean, then they should use them.

In a web of beliefs approach one could set up an inconsistent set of sentences and see which one is the least plausible. I figure that my readers can do that in their heads without I needing to write it out in this case. Maybe the readers will agree with me that NMNT and NMNF are the least plausible ones in the set.

Notes

This is a common yet relatively unknown fallacy. The typical situation is this: Someone is defending some view or theory. That someone acknowledges the existence of a number of objections to the view/theory that he is defending. He then defeats these objections to his own satisfaction and concludes that there are no good objections. Presuming that the person is rational, this is where he ought to conclude that there are no good objections known to him. He should not conclude that there are none.

Interestingly, I found logician, Graham Priest, that commits this fallacy (oh well, even logicians commit fallacies but hopefully less or less frequently than other people). Graham Priest defends his dialetheism theory in his book In Contradiction. On pages 238-240 he defends a view about the transmission of obligations. He defends that view against some objections and then concludes:

“[...attempting to refute objections...] The principle of the transmission of obligation is, therefore, perfectly acceptable.” (p. 240)

Such a thing does not follow. It is possible and even probable that there are other good objections which render the view not perfectly acceptable.

Examples:

1. I cannot find my socks. / I couldn’t find my socks.

2. I cannot find a counter-example to the theory. / I couldn’t find a counter-example to the theory.

The word “cannot” is usually taken to express impossibility and be somewhat equivalent in meaning with the word “impossible”, thought they cannot be used in the same way because they are different word types.

However, if we rewrite the above sentences to use the other word, then the meaning changes. Consider:

1a. It is impossible to find my socks. / It was impossible for me to find my socks.

2a. It is impossible to find a counter-example to the theory. / It was impossible to find a counter-example to the theory.

Why does (1) and (1a) not mean the same? It is because, I think, that we use “cannot” to express a different kind of lack of possibility than we use with “impossible”. This possibility I call willful possibility. It is defined like this: It is willfully possible for agent S to A iff that S wills to bring about A materially implies that A would happen.

This is applicable to the above examples like this:

1b. S wanted to find his socks but did not. Thus, it was not willfully possible to for S to find his socks.

2b. S wanted to find a counter-example to the theory but did not. Thus, it was not willfully possible to for S to find a counter-example.

The meanings of (1) and (2) are also expressible in another way:

1c. I tried to find my socks but I failed.

2c. I tried to find a counter-example to the theory but I failed.

Though there are also some relevant considerations about time. Willful possibilities change from time to time. Suppose for instance that at some later time were S to look for his socks, then he would find them. Similarly with counter-examples to the theory. I did not find a way to easily incorporate this into the definition above.

One should not confuse willful possibility with logical, physical or some other kind of possibility. It would be odd to interpret (1) and (2) as logical etc. possibility, but it would not be odd to interpret (1a) and (2a) that way. Also the sentences in (1a) are odd and would probably not be used very often. However (2a) are not particularly odd since that if some theory is true, then there are no counter-examples to that theory. And since one cannot (willfully) find a counter-example that does not exist, then it is (willfully) impossible to find a counter-example to such a theory.

When interpreting sentences like (1) and (2) we should not be misled to interpret them as something along the lines of (1a) and (2a) but should (probably) interpret them as something like (1c) and (2c).

It seems to me that the formulas:

1. ◊P

2. (∃x)(Fx)

and:

3. □P

4. (∀x)(Fx)

are quite similar, if we translate the modal propositional ones into PWS. Here is the result:

1*. (∃w)(Pw)

3*. (∀w)(Pw)

Translated into a formal-ish english language, they should be read as:

1*. There exists a possible world, w, such that P is the case in w.

2. There exists an x such that Fx.

3*. For all possible worlds, w, P is the case in w.

4. For all x, Fx.

The similarity becomes stronger when an interpretation is added to the predicate logic formula though I leave that up to the reader to do.

Joyce does a rather strange interpretation in The Myth of Morality p. 121. He writes:

However, I doubt we even need concede that much. These “conditional reasons” are very shady customers. Take what seems to be a straightforward one mentioned above: one’s reason to save a drowning child if one exists. There are two readings:

(i) If there exists a drowning child, then S has a reason to save him/her.

(ii) S has a reason to save a drowning child if one exists.”

The absence of a comma after “child” in (ii) makes the difference. (ii) is saying that S has a reason all along: when there are no drowning children, when S is asleep, while S is witching TV, etc.”

Normally conditional sentences can be written in two ways in english (and other languages that I am familiar with), forwards and backwards. Forwards being what is similar to their logical structure and backwards what is not similar. Take the conditional sentence “If I don’t have a job, then I will not get money”. It is forwards for it is similar to its logical structure P→Q (with the obvious interpretation keys). The sentence “I will not get money if I don’t have a job” seems to express exactly the same conditional (i.e. proposition), but Joyce apparently thinks that it does not (if he accepted an analogy with his own example).

When I read the passage above I spent some time thinking about how to properly formalize his two interpretations. I came up with this:

I. (∃x)(Dx)→(∃y)(Ryx)

That there exists an x such that x is a drowning child materially implies that there exists an y such that y is a reason for S to save x.

II. (∃y)∧(∀x)(Dx)→(Ryx)

There exists an y and for all all x, that x is a drowning child materially implies that y is a reason for S to save x.

It seems to capture what he meant.

How Joyce made up these interpretations I don’t know. I note that (I) does not imply that (if there is no drowning child, then S does not have a reason to save one) [¬(∃x)(Dx)→¬(∃y)(Ryx)].

Joyce goes on to make distinction in a strange way:

“This observation is entirely generalizable, to the conclusion that there are not really any “conditional reasons.” Anything true of the form “S has a reason to Ø if C obtains” should be read as “If C obtains, then S has a reason to Ø,” not “S has a reason to Ø-if-C-obtains.”

Joyce would have benefited from using logic here for clarification instead of this. It’s not the case that Joyce wanted to completely avoid using logical symbols in his book anyway, for just two pages earlier we find some simplistic predicate logic formalizations of sentences. (Assuming that he would not have them there if he did not want logical symbols in the book.)

I once thought of a bridge scenario. It went like this: There is a bridge. Someone, a man, wants to find out whether it will break down in the future. So he keeps driving his car over it. It doesn’t break. So he rents a larger car which weighs more, but still the bridge does not break down. He continues this with larger and larger vehicles until the bridge finally breaks down.

An interesting part is that the closer we get to the break down of the bridge, the more justified is the man’s belief that it will not break. Another interesting thing is that we would not conclude like the man did in real life, for we know of bridges that have broken down in the past. That got me thinking but nothing concrete came out of it.

In a discussion of Hume Pyrrho wrote:

“Basically, a “higher level” induction is simply an induction about a broader or “higher” category of items. Thus, for example, one can think about cases in which one has touched flames, or one can think about cases in which one has touched something. The second of those is a broader category, and hence is a “higher level”. Normally, a higher level induction is considered to be better, as, for example, if you buy a new car, and on day 1, it does not break down, and on day 2, it does not break down, etc., such that one might be tempted to draw the conclusion that the car will never break down. However, if one applies a higher level induction regarding mechanical devices, one may then decide that there is a high likelihood of the car breaking down at some point, as mechanical devices have often been observed to do so.”

That seems to explain the scenario nicely.

Removed due to lack of quality.