Interesting paper: Logic and Reasoning: do the facts matter? (Johan van Benthem)

In my hard task to avoid actually doing my linguistics exam paper, ive been reading a lot of other stuff to keep my thoughts away from thinking about how i really ought to start writing my paper. In this case i am currently reading a book, Human Reasoning and Cognitive Science (Keith Stenning and Michiel van Lambalgen), and its pretty interesting. But in the book they authors mentioned another paper, and i like to loop up references in books. Its that paper that this post is about.

Logic and Reasoning do the facts matter free pdf download

Why is it interesting? first: its a mixture of som of my favorit fields, fields that can be difficult to synthesize. im talking about filosofy of logic, logic, linguistics, and psychology. they are all related to the fenomenon of human reasoning. heres the abstract:

Modern logic is undergoing a cognitive turn, side-stepping Frege’s ‘anti- psychologism’. Collaborations between logicians and colleagues in more empirical fields are growing, especially in research on reasoning and information update by intelligent agents. We place this border-crossing research in the context of long-standing contacts between logic and empirical facts, since pure normativity has never been a plausible stance. We also discuss what the fall of Frege’s Wall means for a new agenda of logic as a theory of rational agency, and what might then be a viable understanding of ‘psychologism’ as a friend rather than an enemy of logical theory.

its not super long at 15 pages, and definitly worth reading for anyone with an interest in the b4mentioned fields. in this post id like to model som of the scenarios mentioned in the paper.

To me, however, the most striking recent move toward greater realism is the wide range of information-transforming processes studied in modern logic, far beyond inference. As we know from practice, inference occurs intertwined with many other notions. In a recent ‘Kids’ Science Lecture’ on logic for children aged around 8, I gave the following variant of an example from Antiquity, to explain what modern logic is about:

You are in a restaurant with your parents, and you have ordered three dishes: Fish, Meat, and Vegetarian. Now a new waiter comes back from the kitchen with three dishes. What will happen?

The children say, quite correctly, that the waiter will ask a question,say: “Who has the Fish?”. Then, they say that he will ask “Who has the Meat?” Then, as you wait, the light starts shining in those little eyes, and a girl shouts: “Sir, now, he will not ask any more!” Indeed, two questions plus one inference are all that is needed. Now a classical logician would have nothing to say about the questions (they just ‘provide premises’), but go straight for the inference. In my view, this separation is unnatural, and logic owes us an account of both informational processes that work in tandem: the information flow in questions and answers, and the inferences that can be drawn at any stage. And that is just what modern so-called ‘dynamic- epistemic logics’ do! (See [32] and [30].) But actually, much more is involved in natural communication and argumentation. In order to get premises to get an inference going, we ask questions. To understand answers, we need to interpret what was said, and then incorporate that information. Thus, the logical system acquires a new task, in addition to providing valid inferences, viz. systematically keeping track of changing representations of information. And when we get information that contradicts our beliefs so far, we must revise those beliefs in some coherent fashion. And again, modern logic has a lot to say about all of this in the model theory of updates and belief changes.

i think it shud be possible to model this situation with help my from erotetic logic.

first off, somthing not explicitly mentioned but clearly true is that the goal for the waiter to find out who shud hav which dish. So, the waiter is asking himself these three questions:

Q1: ∃x(ordered(x,fish)∧x=?) – somone has ordered fish, and who is that?
Q2: ∃y(ordered(y,meat)∧y=?) – somone has ordered meat, and who is that?
Q3: ∃z(ordered(z,veg)∧z=?) – somone has ordered veg, and who is that?
(x, y, z ar in the domain of persons)

the waiter can make another, defeasible, assumption (premis), which is that x≠y≠z, that is, no person ordered two dishes.

also not stated explicitly is the fact that ther ar only 3 persons, the child who is asked to imagin the situation, and his 2 parents. these correspond to x, y, z, but the relations between them dont matter for this situation. and we dont know which is which, so we’ll introduce 3 particulars to refer to the three persons: a, b, c. lets say the a is the father, b the mother, c the child. also, a≠b≠c.

the waiter needs to find 3 correct answers to 3 questions. the order doesnt seem to matter – it might in practice, for practical reasons, like if the dishes ar partly on top of each other, in which case the topmost one needs to be served first. but since it doesnt in this situation, som arbitrary order of questions is used, in this case the order the fishes wer previusly mentioned in: fish, meat, veg. befor the waiter gets the correct answer to Q1, he can deduce that:

∃x(ordered(x,fish)∧(x=a∨x=b∨x=c))
∃y(ordered(y,meat)∧(y=a∨y=b∨y=c))
∃z(ordered(z,veg)∧(z=a∨z=b∨z=c))
(follows from varius previusly mentioned premisses and with classical FOL with identity)

then, say that the answer gets the answer “me” from a (the father), then given that a, b, and c ar telling the truth, and given som facts about how indexicals work, he can deduce that a=x. so the waiter has acquired the first piece of information needed. befor proceeding to asking mor questions, the waiter then updates his beliefs by deduction. he can now conclude that:

∃y(ordered(y,meat)∧(y=b∨y=c))
∃z(ordered(z,veg)∧(z=b∨z=c))
(follows from varius previusly mentioned premisses and with classical FOL with identity)

since the waiter cant seem to infer his way to what he needs to know, which is the correct answers to Q2 and Q3, he then proceeds to ask another question. when he gets the answer, say that b (the mother) says “me”, he concludes like befor that z=b, and then hands the mother the veg dish.

then like befor, befor proceeding with mor questions, he tries to infer his way to the correct answer to Q3, and this time it is possible, hence he concludes that:

∃y(ordered(y,meat)∧(y=c))
(follows from varius previusly mentioned premisses and with classical FOL with identity)

and then he needs not ask Q3 at all, but can just hand c (the child) the dish with meat.

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Moreover, in doing so, it must account for another typical cognitive phenomenon in actual behavior, the interactive multi-agent character of the basic logical tasks. Again, the children at the Kids’ Lecture had no difficulty when we played the following scenario:

Three volunteers were called to the front, and received one coloured card each: red, white, blue. They could not see the others’ cards. When asked, all said they did not know the cards of the others. Then one girl (with the white card) was allowed a question; and asked the boy with the blue card if he had the red one. I then asked, before the answer was given, if they now knew the others’ cards, and the boy with the blue card raised his hand, to show he did. After he had answered “No” to his card question, I asked again who knew the cards, and now that same boy and the girl both raised their hands …

The explanation is a simple exercise in updating, assuming that the question reflected a genuine uncertainty. But it does involve reasoning about what others do and do not know. And the children did understand why one of them, the girl with the red card, still could not figure out everyone’s cards, even though she knew that they now knew.15

this one is mor tricky, this it involves beliefs of different ppl, the first situation didnt.

the questions ar:

Q1: ∃x(possess(x,red)∧x=?)
Q2: ∃y(possess(y,white)∧y=?)
Q3: ∃z(possess(z,blue)∧z=?)

again, som implicit facts:

∃x(possess(x,red))
∃y(possess(y,white))
∃z(possess(z,blue))

and non-identicalness of the persons:

x≠y≠z, and a≠b≠c. a is the first girl, b is the boy, c is the second girl. ther ar no other persons. this allow the inference of the facts:

∃x(possess(x,red)∧(x=a∨x=b∨x=c))
∃y(possess(y,white)∧(y=a∨y=b∨y=c))
∃z(possess(z,blue)∧(z=a∨z=b∨z=c))

another implicit fact, namely that the children can see their own card and know which color it is:

∀x∀card(possess(x, card)→know(x, possess(x, card)) – for any person and for any colored card, if that person possesses the card, then that person knows that that person possesses the card.

the facts given in the description of who actually has which cards are:

possess(a,white)
possess(b,blue)
possess(c,red)

so, given these facts, each person can now deduce which variable is identical to one of the constants, and so:

know(a,possess(a,white))∧know(a,y=a)
know(b,possess(b,blue))∧know(b,z=b)
know(c,possess(c,red))∧know(c,x=c)

but non of the persons can seem to answer the other two questions, altho it is different questions they cant answer. for this reason, one person, a (first girl), is allowed to ask a question. she asks:

Q3: possess(b,red)? [towards b]

now, befor the answer is given, the researcher asks if anyone knows the answer to all the questions. b raises his hand. did he know? possibly. we need to add another assumption to see why. b (the boy) is assuming that a (the first girl) is asking a nondeceptiv question. she is trying to get som information out from b (the boy). this is not so if she asks about somthing she already knows. she might do that to deceive, but assuming that isnt the case, we can add:

∀x∀y∀card(ask(x,y,(possess(y,card)?)))→¬possess(x,card)

in EN: for any two persons, and any card, if the first person is asking the second person about whether the second person possesses the card, then the first person does not possess the card. from this assumption of non-deception, the boy can infer:

¬possess(a, red)

and so he coms to know that:

know(b,¬possess(a, red))∧know(b, x≠a)

can the boy figure out the questions now? yes: becus he also knows:

know(b,possess(b,blue))∧know(b,z=b)

from which he can infer that:

¬possess(b,red) – she asked about it, so she doesnt hav it herself
¬possess(b, blue) – he has the card himself, and only 1 person has the card

but recall that every person has a card, and he knows that b has neither the red or the blue, then he can infer that b has the white card. and then, since ther ar only 3 cards and 2 persons, and he knows the answers to the first two questions, ther is only one option left for the last person: she must hav the red card. hence, he can raise his hand.

the girl who asked the question, however, lacks the crucial information of which card the boy has befor he answers the question, so she cant infer anything mor, and hence doesnt raise her hand.

now, b (the boy) answers in the negativ. assuming non-deceptivness again (maxim of truth) but in another form, she can infer that:

¬possess(b, red)

and so also knows that:

¬possess(a, red)

hence, she can deduce that, the last person must hav the red card, hence:

know(a,(possess(c,red))

from that, she can infer that the boy, b, has the last remaining card, the blue one. hence she has all the answers to Q1-3, and can raise her hand.

the second girl, however, still lacks crucial information to deduce what the others hav. the information made public so far doenst help her at all, since she already knew all along that she had the red card. no other information has been made available to her, so she cant tell whether a or b has the blue card, or the white card. hence, she doenst raise her hand.

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all of this assumes that the children ar rather bright and do not fail to make relevant logical conclusions. probably, these 2 examples ar rather made up. but see also:

a similar but much harder problem. surely it is possible to make a computer that can figur this one out, i already formalized it befor. i didnt try to make an algorithm for it, but surely its possible. heres my formalizations.

www.stanford.edu/~laurik/fsmbook/examples/Einstein%27sPuzzle.html

types of reasoners, i assumed that they infered nothing wrong, and infered everything relevant. wikipedia has a list of common assumptions like this: en.wikipedia.org/wiki/Doxastic_logic#Types_of_reasoners

I was thinking and doing som reserch about one of my kurent projects: Making an English languaj that is syntakialy limited such that it makes posible automatik translation into lojikal formalism. I stumbled akros som prety interesting artikles listed below:

en.wikipedia.org/wiki/Controlled_language

One of them has a website wher one kan find an introduktion to the system. It is aktualy very good and worth reeding. It is a 80 paj powerpoint presentation turned into PDF.

en.wikipedia.org/wiki/Basic_English

The 1944 paper kritikal of Basic English: How Basic Is Basic English?

en.wikipedia.org/wiki/Simplified_English

en.wikipedia.org/wiki/Plain_English

en.wikipedia.org/wiki/Plain_language

The benefits of using plain languaj ar rather obivus and konkreet. Using non-plain languaj makes komunikation take longer and proseed les optimal. This is mostly just waste of time but somtimes it is a mater of life and deth.

My (it is shared) projekt has som on-going diskusion in my forum. However, the languaj i hav in mind is mor similar to formalism than ACE is (the one linked to erlyr). I think that it is too problematik to handle nested konditionals with quantifyrs like:

F1. (∀y)(∀xFxy→Gxy)→Fy

in sylogistik languaj, i.e., as in sentenses like:

S1. “All men are human.”

Rather, one needs sentenses that ar harder to understand and les like ordinary English but beter for formalization like:

S2. “For any X, if X is a man, then X is a human.”

In simple kases, such as the example sentenses with the form:

F2. ∀xMx→Hx

ther is no need for mor advansed sentense syntax, but in the kase of the formalization F1 ther is need for such sentenses.

The phrases “I cannot find …” and “I couldn’t find …” and willful possibility

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).

The Myth of Morality and interpretation

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.)

Interpretation of “always”

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)).