Archive for the ‘Logic’ Category

From researchgate: www.researchgate.net/post/What_is_the_actual_difference_between_1st_order_and_higher_order_logic

What is the actual difference between 1st order and higher order logic?
Yes, I know. They say, the 2nd order logic is more expressive, but it is really hard to me to see why. If we have a domain X, why can’t we define the domain X’ = X u 2^X and for elements of x in X’ define predicates:
SET(x)
ELEMENT(x)
BELONGS_TO(x, y) – undefined (or false) when ELEMENT(y)
etc.
Now, we can express sentences about subsets of X in the 1st-order logic!
Similarly we can define FUNCTION(x), etc. and… we can express all 2nd-order sentences in the 1st order logic!
I’m obviously overlooking something, but what actually? Where have I made a mistake?

My answer:

In many cases one can reduce a higher order formalization to a first-order, but it will come at the price of complexity of the formalization.

For instance, formalize the follow argument in both first order and second order logic:
All things with personal properties are persons. Being kind is a personal property. Peter is kind. Therefore, Peter is a person.

One can do this with either first or second order, but it is easier in second-order.

First-order formalization:
1. (∀x)(PersonalProperty(x)→((∀y)(HasProperty(y,x)→Person(y)))
2. PersonalProperty(kind)
3. HasProperty(peter,kind)
⊢ 4. Person(peter)

Second-order formalization
1. (∀Φ)(PersonalProperty(Φ)→(∀x)(Φx→Person(x)))
2. PersonalProperty(IsKind)
3. IsKind(peter)
⊢ 4. Person(peter)

where Φ is a second-order variable. Basically, whenever one uses first order to formalize arguments like this, one has to use a predicate like “HasProperty(x,y)” so that one can treat variables as properties indirectly. This is unnecessary in second-order logics.

Someone needed this, so I made a quick collection.

 

Introductory and more philosophical

 

Introductory not so philosophical/more formal

 

More advanced and very formal

The assignment was:

Any aspect? :D I just wrote stuff about formal logic. So no more research was needed. Lucky.

SMU paper 1

From here. btw this thread was one of the many discussions that helped form my views about what wud later become the essay about begging the question, and the essay about how to define “deductive argument” and “inductive argument”.

Reconstructo

 Do you know much about Jung’s theory of archetypes? If so, what do you make of it?

Kennethamy

 I don’t make much of Jung. Except for the notions of introversion and extroversion. Not my cup of tea. As I said, we don’t create our own beliefs. We acquire them. Beliefs are not voluntary.

Emil

 They are to some extend but not as much as some people think (Pascal’s argument comes to mind).

Kennethamy

 Yes, it does. And that is an issue. His argument does not show anything about this issue. He just assumes that belief is voluntary He does talk about how someone might acquire beliefs. He advises, for instance, that people start going to Mass, and practicing Catholic ritual. And says they will acquire Catholic beliefs that way. It sounds implausible to me. It is a little like the old joke about a well-known skeptic, who puts a horseshoe on his door for good luck. A friend of his sees the horseshoe and says, “But I thought you did not believe in that kind of thing”. To which the skeptic replied, “I don’t, but I hear that it works even if you don’t believe it”.

 

Evolutionary Psychology and Feminism 2011

 

Abstract This article provides a historical context of

evolutionary psychology and feminism, and evaluates the

contributions to this special issue of Sex Roles within that

context. We briefly outline the basic tenets of evolutionary

psychology and articulate its meta-theory of the origins of

gender similarities and differences. The article then evaluates

the specific contributions: Sexual Strategies Theory and the

desire for sexual variety; evolved standards of beauty;

hypothesized adaptations to ovulation; the appeal of risk

taking in human mating; understanding the causes of sexual

victimization; and the role of studies of lesbian mate

preferences in evaluating the framework of evolutionary

psychology. Discussion focuses on the importance of social

and cultural context, human behavioral flexibility, and the

evidentiary status of specific evolutionary psychological

hypotheses. We conclude by examining the potential role of

evolutionary psychology in addressing social problems

identified by feminist agendas.

Keywords Evolutionary psychology . Feminism . Sexual

strategies . Gender differences

 

I came across this study while reading this article, which i think i will comment on later.

 

 

The fact that physical attractiveness is so highly valued

by men in mate selection, and contrary to conventional

social science wisdom is not arbitrarily socially constructed,

does not imply that the emphasis placed on it is not

destructive to women—a point about which many feminists

and evolutionary psychologists agree (e.g., Buss 1996;

Wolf 1991; Vandermassen 2005). Many feminist scholars,

evolutionary psychologists, and evolutionary feminists

concur that the value people place on female beauty is

likely a key cause of eating disorders, body image

problems, and potentially dangerous cosmetic surgery. As

Singh and Singh (2011) and others point out, it can lead to

the objectification of women as sex objects to the relative

neglect of other dimensions along which women vary, such

as talents, abilities, and personality characteristics. Finally,

in the modern environment, it seems clear that men’s

evolved standards of female beauty have contributed to a

kind of destructive run-away female-female competition in

the modern environment to embody the qualities men desire

(Buss, 2003; Schmitt and Buss 1996).

 

In our view, the key point is that feminist stances on the

destructiveness of the importance people place on female

attractiveness need not, and should not, rest on the faulty

assumption that standards of attractiveness are arbitrary

social constructions. Societal change, where change is

desired, is best accomplished by an accurate scientific

understanding of causes. The evolutionary psychological

foundations of attractiveness must be a starting point for

this analysis.

 

indeed, as is (nearly?) always the case: if one wants to change some state of affairs, then actually understanding WHY it is the way it is to begin with is of paramount importance.

 

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Adaptations to Ovulation

Ovulation attains special status within women’s reproduc-

tive biology because it provides the very brief window

(roughly 12–24 h) during women’s menstrual cycle during

which conception is possible. Conventional wisdom in the

field of human sexuality over the past century has been that

ovulation is cryptic or concealed, even from women

themselves (e.g., Symons 1979). Evolutionary psycholo-

gists over the past decade have begun to challenge this

conventional wisdom. The challenges have come in two

forms—hypothesized adaptations in men to detect ovula-

tion and hypothesized adaptations in women to adjust their

mating behavior around ovulation.

 

Ancestral men, in principle, could have benefited (in

reproductive currencies) if they could detect when women

ovulated. An ovulation-detection ability would afford men

the ability to selectively direct their sexual overtures toward

women when they are ovulating, as male chimpanzees do.

And already mated men might increase their mate-guarding

efforts when their partners are ovulating. Both strategies, in

principle, could have evolved in men. The key question is:

Did they?More than 20 years ago, Symons (1987) concluded

that such male adaptations to ovulation had not evolved:

“The most straightforward prediction I could have made,

based on simple reproductive logic and the study of

nonhuman animals, would have been that . . . men will be

able to detect when women are ovulating and will find

ovulating women most sexually attractive. Such adaptations

have been looked for in the human male and have never

been found . . .” (p. 133).

 

it seems to me that the authors need to learn more logic. the above case seems to be an example of an argument from ignorance, altho in a nonstraightforward way. heres how i interpret it:

 

1) Symons wrote that there is no evidence of such adaptations in humans.

2) thus, Symons thought that there is no evidence of such adaptations in humans.

3) thus, Symons thought that there are no such adaptations in humans.

 

(2) follows given normal conditions, that is, that he wasnt lying etc. it has a hidden premise stating that the conditions are normal, in a kind of default reasoning way.

(3) however attributes an argument from ignorance inference to Symons, which is not warranted. it may be that the adaptations are difficult to find and that science had per 1987 just missed them.

 

Symons might not have held the view the authors attribute to him.

 

-

 

[...] And no other framework suggests that adaptations to

ovulation might have evolved. Whatever the eventual

evidentiary status of the competing hypotheses, it is

reasonable to conclude that the search for adaptations to

ovulation has been a fertile one, yielding fascinating

empirical findings.

 

dat pun

 

-

 

The positive outcome for everyone is that evolutionary

psychological hypotheses, sex role/biosocial theory hy-

potheses, and gender-similarity hypotheses all share the

scientific virtue of making specific empirical predictions.

In this sense, we see this special issue of Sex Roles an

exceptionally positive sign that the discourse is beginning

to move beyond purely ideological stances and toward an

increasingly accurate scientific understanding of gender

psychology.

 

since evo psychs dont hav any ideological stance, this description is exceptionally nice to them. the only ones who need to move past any ideology are the marxist feminists.

 

-

 

 

Exam paper for Danish and Languages of the world

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.

-

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

plato.stanford.edu/entries/moore/#2

So, on the face of it, this thesis has here been inferred from Leibniz’ Law. Moore observes, however, that the step from (1) to (2) is invalid; it confuses the necessity of a connection with the necessity of the consequent. In ordinary language this distinction is not clearly marked, although it is easy to draw it with a suitable formal language.

Moore’s argument here is a sophisticated piece of informal modal logic; but whether it really gets to the heart of the motivation for Bradley’s Absolute idealism can be doubted. My own view is that Bradley’s dialectic rests on a different thesis about the inadequacy of thought as a representation of reality, and thus that one has to dig rather deeper into Bradley’s idealist metaphysics both to extract the grounds for his monism and to exhibit what is wrong with it.

 

Interesting. GE Moore is in good company, along with Leibniz.

Towardsabetterquantitativelogic (due to formatting)

docs.google.com/document/d/1vN7pFML8N_s8HUMVmpai1rXiP1dLzeB04A9OufssEkI/edit

(KK)If one knows that p, then one knows that one knows that p.


Definitions
A0is the proposition that 1+1=2.
A1is the proposition that Emil knows that 1+1=2.
A2is the proposition that Emil knows that Emil knows that 1+1=2.

Anis the proposition that Emil knows that Emil knows that … that 1+1=2.
Where “…” is filled by “that Emil knows” repeated the number of times in the subscript of A.

Argument
1. Assumption for RAA
(∀P∀x)Kx(P)→Kx(Kx(P)))
For any proposition, P, and any person, x, if x knows that P, then x knows that x knows that P.

2. Premise
Ke(A0)
Emil knows that A0.

3. Premise
(∃S1)(A0S1A1S1…∧AnS1)∧|S1|=∞∧S1=SA
There is a set, S1, such that A0belongs to S1, and A1belongs to S1, and … and Anbelongs to S1, and the cardinality of S1is infinite, and S1is identicla to SA.

4. Inference from (1), (2), and (3)
(∀P)P∈SAKe(P)
For any proposition, P, if P belongs to SA, then Emil knows that P.

5. Premise
¬(∀P)P∈SAKe(P)
It is not the case that, for any proposition, P, if P belongs to SA, then Emil knows that P.

6. Inference from (1-5), RAA
¬(∀P∀x)Kx(P)→Kx(Kx(P)))
It is not the case that, for any proposition, P, and any person, x, if x knows that P, then x knows that x knows that P.

Proving it
Proving that it is valid formally is sort of difficult as it requires a system with set theory, predicate logic with quantification over propositions. The above sketch should be enough for whoever doubts the formal validity.