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.
http://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: https://en.wikipedia.org/wiki/Doxastic_logic#Types_of_reasoners