Archive for December, 2009

Formalization of questions and answers is not a much discussed nor studied topic. Though there is a branch of logic dealing with it, erotetic logics. I admit not to have read much on the issue, in fact, almost nothing. This is because there is almost nothing on the internet about it, and the few books that deal with it are not to find in the danish library system.

The idea is to formalize questions and the answering of such. With formalization we have a framework for understanding when a question has been answered. It seems to me that we only have our intuitions to rely on until a relevant logic has been created, and intuitions are too often not trust worthy.

Formalization of questions

Consider the question:

Q1. Who is credited with discovering the element Uranium?

How should we formalize this? How about:

Q1F. (∃x)(Ux∧x=?)

Ux ≡ is credited with discovering the element Uranium

Though there is a hidden second condition for a correct answer to this question. Note the word “who”. That that word is used implies that the answer refers to a person. So we may add:

Q1F*. (∃x)(Ux∧Px∧x=?)

Ux ≡ is credited with discovering the element Uranium

Px ≡ is a person

Consider a more complex question such as:

Q2. Which scientist is credited with discovering the element Uranium and whose name is Martin and who is german?

Formalization:

Q2F. (∃x)(Ux∧Sx∧Mx∧Gx∧x=?)

Ux ≡ is credited with discovering the element Uranium

Sx ≡ is a scientist

Mx ≡ is named Martin

Gx ≡ is german

Note how we can now formally make sense of/explain complexity of questions. Complexity correlates with the number of predicates in the formalization. But since there are multiple possible formalizations one need to be careful with this correlation. It may be the case that it corresponds perfectly or nearly so with the deepest possible formalization of a question.

Answers to questions

When should we say that an answer has been answered truthfully? Iff substituting the question mark “?” with the answer gives a true formula.1

Consider the correct answer to the first question above:

A1. Martin Heinrich Klaproth

And its formalization:

A1F. m

m ≡ Martin Heinrich Klaproth

Now substituting “?” with “m”:

Q1W. (∃x)(Ux∧x=m)

(Q1W) is true, therefore, the answer is correct.

Similarly with the second question:

Q2W. (∃x)(Ux∧Sx∧Mx∧Gx∧x=m)

Again this is true.

Translating to Logic English

Logic English (LE) being the language where to formalizations are translated, as with propositional logic and predicate logic. This is a precise language that corresponds perfectly with logic. (Another thing that I am working on.)

Consider again the first question unanswered and answered:

Q1F. (∃x)(Ux∧x=?)

Q1W. (∃x)(Ux∧x=m)

I propose this translation to LE:

There exists one x such that x is credited with discovering the element Uranium and x is identical with who?

There exists one x such that x is credited with discovering the element Uranium and x is identical with Martin Heinrich Klaproth.

Questions as expressing propositions

It is clear from my chosen formalization and translation that I think questions express propositions. So they can express something that is true or false like other sentences. This should not be too surprising. Consider the loaded question fallacy. That happens exactly when one asks a question that expresses a false proposition. A proposition expressed by the question is false iff it is not the case that there exists an x such that [predicates]. The clause with the “x=?” is ignored when evaluating truth values.

So, by accepting that some questions express propositions we can make sense of the loaded question fallacy.

Multiple correct answers

It is possible formalize the multiple correct answers aspect of questions. Consider the question:

Q3. What is x identical with in the equation x2=4?

Formally:

Q3F. (∃x)(x2=4∧x=?)

There are two correct answers to this question:

A3a. 2

A3b. -2

Substituting “?” with the answers produces:

Q3Wa. (∃x)(x2=4∧x=2)

Q3Wb. (∃x)(x2=4∧x=-2)

Which are both true formula. So, there are more than one true answer. We could formalize this as (with help from set theory):

Q3F*. (∃x)(x2=4)∧((∀y)(y2=4⇒y∈A∧MA>1))∧(x=?)

MS ≡ the number of members of S

A ≡ (the set of) answers

We can also translate this to LE:

There exists an x such that x2=4, and for all y, that y2=4 logically implies that y is a member of A and the number of members of A is more than 1, and x is identical with what?

It seems to me to be best to always have the question clause “x=?” at the end of a formula. Otherwise a translation of the formula into LE would produce a sentence that has a question mark in the middle of it. That would be grammatically incorrect. I strive to make LE grammatically correct or close to. The goal being that LE is readable by people that can read english.

An idea is to get rid of the first part of (Q3F*) and only keep the part with the universal quantifier (∀x), but that would produce problems with the question clause at the end. Like this:

Q3F**. (∀y)(y2=4⇒y∈A∧MA>1)∧(x=?)

x” is not mentioned at all before the question clause at the end.

Notes

1More precisely, a formula that when interpreted with the supplied interpretation keys expresses a true proposition.

I’m currently reading The Myth of Morality by Richard Joyce. In the summery section of chapter three he presents a central argument thus:

1. If x morally ought to Ø, then x ought to Ø regardless of what his desires and interests are.

2. If x morally ought to Ø, then x has a reason for Øing.

3. Therefore, if x morally ought to Ø, then x can have a reason for Øing regardless of what his desires and interests are.

4. But there is no sense to be made of such reasons.

5. Therefore, x is never under a moral obligation. (p. 77)

This argument is not valid under a straightforward interpretation. However Joyce earlier clarified the structure of the argument. He stated that the form is:

1. If P, then Q

2. If P, then R

3. If P, then (Q and R)

4. Not (Q and R)

5. Not P. (p. 42)

However this does not correspond well to the words above. First, notice the wording in (3). It is “can have”, which expresses a possibility, not an actuality like (2) does; “has a reason”. (3) should be either reworded (probably the best solution) or the formalization changed to ◊R (“it is possible that R”).

Second, (4) does not correspond very well to the formalization at all.

Third, (5) contains the word “never” which is a temporal concept not found in any of the other premises. Indeed they don’t feature temporal words at all. Accordingly, the wording of (5) should be changed or the formalization changed (to something like GQ, using these formalization keys). If we ignore the word “never” in (5), then the argument is valid even though (3) is a about a possibility instead of an actuality. This is of course because (∀P), from P, ◊P follows. (P→◊P is a theorem of S5)

So, given the above, I think that Joyce is a bit careless. More careless than professional philosophers should be. Especially a professor!

Kennethamy

“A human being is a biological category, and can be defined by human DNA, it seems to me. The category, person, is not biological, and maybe your teacher is confusing “human beings” with “persons”. All human beings need not be persons. For example, small infants are not yet persons. “Person” is a legal and social category. And all persons need not be human beings. For example, Mr. Spock (of Star Treck) was not a human being (he was part [Vulcan]). But he was certainly a person.”

That seems to me to be just right.

Source.

One may talk of a reliable car. “Reliable” here clearly means a car that has a high success rate of doing what it is supposed to (e.g. getting one where one wants to go in a reasonable time). One may also talk of a reliable source of information, that is, an authority. Can we understand reliability in a similar way here? Some people think that two concepts/notions of reliability are necessary. It seems to me that we can do with just one. An authority about something is just a person that has a high success rate of telling true propositions (/have a % justified beliefs) about the matter.

As an example think of Searle’s chinese room example. Is the person in the room a reliable source?

Imagine a native English speaker who knows no Chinese locked in a room full of boxes of Chinese symbols (a data base) together with a book of instructions for manipulating the symbols (the program). Imagine that people outside the room send in other Chinese symbols which, unknown to the person in the room, are questions in Chinese (the input). And imagine that by following the instructions in the program the man in the room is able to pass out Chinese symbols which are correct answers to the questions (the output).” (Searle, 1999, ‘The Chinese Room’, in R.A. Wilson and F. Keil (eds.), The MIT Encyclopedia of the Cognitive Sciences, Cambridge, MA: MIT Press. Found here.)

It seems to me that the answer is “yes”.

The other explication that has been offered of a reliable source is that “the person knows what he is talking about”. It seems to me that this is a constant phrase (i.e. one that is always used in almost the exactly same way). Such phrases are more often than not best understood non-literally. I suggest that a non-literal understanding/interpretation is a good idea. It could be understood as something similar to the general explication of reliability that I offered in the beginning of this essay.

However, one could insist that the phrase is meant literally and that reliability of persons /experts/authorities implies that the person knows. However it seems to me that the chinese room is a counter-example to this. The person in the room is reliable and does not know.

I also note that a non-literal interpretation/understanding is consistent with a fictionalist account of the field of the matter (e.g. an error theory about ethics) because one could use the notion of justified belief

“Although I generally prefer negative theories – those which posit as few unempirical concepts* as possible – my own leanings in this particular case are toward Realism. My attraction to the theory is bolstered by one further consideration: I can see no way to account for the existence of certain items, e.g. pieces of music, plays, and novels, other than by conceiving of them as abstract entities. Here I am considerably influenced by the arguments of C.E.M. Joad (1891-1953).
Joad argued ([105], 267-70) that the play Hamlet, for example, could not reasonably be identified with any particular in the world: neither with an idea in Shakespeare’s mind, nor with any manuscript he wrote, nor with any printed edition of the text, nor with any particular production, nor with any audio or video recording of any particular production. For Hamlet could exist even if any one or several of these were not to exist. While Joad, himself, rightly expressed some diffidence about his own arguments, I think that they add considerable impetus to a theory which would posit abstract entities.

Although I am a Realist, I am a reluctant Realist. For, to be frank, there is something exceedingly peculiar about positing entities which exist (subsist) outside of space and time. I, personally, would prefer a theory which could dispense with such mysterious entities. But I find the problems inherent in the various anti-Realist theories even more troubling. Realism is simply the better, in my estimation, of the available theories. But, like many other Realists, I do not much care for Realism. Recently one of my colleagues professed his repudiation of Realism by saying that he found the positing of abstract entities “ unintelligible ”. I share his displeasure. But I find myself unable to adopt his own anti-Realist position because I cannot in turn believe that the anti-Realist theories provide any better answer or that they can be developed without themselves having to posit at least some abstract entities.”

Norman Swartz, Beyond Experience, pp. 270-272, available online for free.

ACB:

This is an interesting point. Is there a minimum level of understanding that someone must have in order to derive justification from an authority? For example, if you are completely ignorant of music theory, and a qualified musician tells you: “A tritone is an augmented fourth or a diminished fifth”, are you justified in believing it? You do not have a clue what it means, except that it is something to do with music. Imagine the following exchange:

Layman (L): What are you talking about?
Musician (M): I’m talking about intervals.
L: What are they?
M: The distances between two notes.
L: You mean, like when two players stand five feet apart…
M: No, you fool, I mean like when you play two different notes on the piano.
L: Oh, I see. So what is this ‘fourth’ and ‘fifth’ stuff? That’s more than two.
M: No, you have to count up from the bottom note…
[Some minutes later]
L: Ah, I’m beginning to understand you now. So an augmented fourth sounds the same as a diminished fifth.
M: Yes, that’s right.
L: But what the hell is a tritone? Three tones? How can that be the same?
[Some minutes later]
L: Ah, I understand. Now I believe your original statement.
M: But why didn’t you believe it in the first place? I’m an expert in music theory, and you know I wouldn’t lie to you.

At what point would L become justified in believing M’s original statement? At the beginning? At the end? Or at some point in between? Is the acquisition of justification an all-or-nothing affair, or can it be incremental? Can any clear rules be formulated about this?

Or am I looking at this the wrong way? Should the question be, not “when would L first have justification for believing the statement”, but simply “when could he first believe the statement”?

Any thoughts would be welcome.

Emil:

I’m wondering this myself. I haven’t found any persuasive argument though. I have nothing to add.

Source.

ACB:

1. S knows that every statement in his geography textbook is correct. (His well-qualified geography teacher has told him: “I have checked this book carefully, and everything in it is correct”. The teacher is right, and S believes him.)

2. One of the statements in the textbook is that Quito is the capital of Ecuador.

3. Therefore, S knows that Quito is the capital of Ecuador, even if he has never heard of Quito or Ecuador.

This doesn’t seem right to me. What do you think?

Emil:

Hmm. Formalization may help.

1. (∀P)(TP→K(P))
For all propositions, that a proposition is expressed in the textbook (TP) materially implies that S knows that P.
2. TA
(proposition) A is expressed in the textbook.
⊢, 3. K(A)
Thus, S knows that A.

This is valid. But I don’t think that is the argument expressed above. The argument expressed above relies on confusion/equivocation on premise 1. The difficulty is formalizing what the other thing that is meant by that phrase is. Hmm.

My (1) above is not a correct formulation of what if meant by the premise phrase in the natural language. What about:

1′. K((∀P)(TA→A))

S knows that for all propositions, that a proposition is expressed in the textbook (TP) materially implies that P.

That seems to capture what is meant. The equivocation has been explained to my satisfaction. (1′) and (2) does not logically imply the conclusion, so that argument is invalid.

Source.

Some people think that the identity notion is captured by the second proposal above. I think we need two notions of identity. I will not discuss that now.

Strict identity

For all things, for all things, for all predicates, that x and y are strictly identical logically implies that that x has predicate F is logically equivalent with that that y has predicate F.

(∀x)(∀y)(∀F)(x=Sy⇔(Fx⇔Fy)) [with obvious interpretation and =S meaning strict identity]

This is called Liebniz’s law.

Personal identity

Kennethamy:

“I think Liebniz’s law accommodates personal identity quite comfortably with the addition of a time quantifier: x and y are identical if any property possessed by x at time t is also possessed by y at time t. If you add a world quantifier it can also handle transworld identity rather well.“

For all things, for all things, for all predicates, for all times, that x and y are personally identical logically implies that that x has predicate F at time t is logically equivalent with that that y has predicate F at time t.

(∀x)(∀y)(∀F)(∀t)(x=Py⇔(Fxt⇔Fyt)) [with obvious interpretation and =P meaning personal identity]

This seems to work.

But I don’t understand the part about transworld identity. It seems to me that the above can handle transworld identity fine.

Merely a translation of the danish version here.

Translation keys

Domains

D:x = things

D:y = things

D:t = moments

One variable predicates

Ex = x exists

At = the world was created at time t

Two variable predicates

Cxy = x created y

Three variable predicates

Kxyt = x created y at time t

Particulars

g = God

a = The world

n English Symbols Explanation
1 God exists. Eg Assumption for reductio
2 That God exists, logically implies that God created the world. Eg→Cga Premise
3 God created the world Cga 1, 2, MP
4 For all things and for all things, that a thing created another thing logically implies that there exists a moment such that that moment is before another moment at the first thing created the second thing at that other moment (∀x)(∀y)(Cxy⇒[(∃t)(t<t1∧Kxyt1)]) Premise
5 There exists a moment such that that moment is before another moment and God created the world at that other moment. (∃t)(t<t1∧Kgat1) 3, 4, MP
6 For all things there exists a moment such that that moment is before another moment and that thing created the world at that other moment, logically implies that there exists a moment such that that moment is before another moment and the world was created at the other moment. [(∀x)(∃t)(t<t1∧Kxat1)]→[(∃t)(t<t1At1)] Premise
7 It is not the case that there exists a moment such that that moment is before another moment and the world was created at that other moment. ¬[(∃t)(t<t1At1)] Premise
8 There exists a moment such that that moment is before another moment and the world was created at the other moment, and it is not the case that there exists a moment such that that moment is before another moment and the world was created at that other moment. [(∃t)(t<t1At1)]∧¬[(∃t)(t<t1At1)] 4, 5, conj.
9 It is not the case that God exists. ¬Eg 1-8, RAA

Some explanations to the premises

(2) is true when we are dealing with traditional monotheism. Traditional monotheism in the sense that there exists a God and God created the world.

(4) is reasonable when one considers it. If something is created at a moment by something else, then the first thing did not exist immediately before it was created by the other thing. There is at least one moment before a thing was created by another thing where it did not exist. That is what “created by” means.

(6) merely removes the creator so that the moment may be isolated.

(7) since time is a part of space-time and that space-time did not exist before the world was created, then there wasn’t a moment before the world was created. The world is here understood as the physical world in some sense that makes it possible that there is a non-physical world wherein God exists.

Sometimes it is clear that “always” should be interpreted as various temporal logics suggest. Other times it should not be interpreted as anything that has to do with time.

Consider this fictive conversation:

“Generally women prefer men with high social above men without.”

“Not always.”

But the first utterance has nothing to do with time. When “not always” is uttered in such contexts it means the same as the phrase “not in all cases [¬(∀x)(Fx)/(∃x)(¬Fx)]”. Similarly with other temporal words such as “often” (in many cases) and “never” (¬(∃x)(Fx)/(∀x)(¬Fx)).