Translation...Interpretation...Formal Validity...

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Reply Tue 17 Nov, 2009 11:08 pm
Hello everyone. I am working through the "Interpretations" section of my logic text. I am at the point where we are talking about the production of a valid argument, whose translation is valid, but how it might not necessarily be formally valid.

So for example from "HORACE IS MARRIED" I conclude that "HORACE IS MARRIED TO SOMEONE WHO IS MARRIED."

Now, so long as D= a set of persons; h=Horace; and M=[1] is married to [2]

The sentences, I have the premise: ExMhx; and the conclusion Ex[Mhx ^ EyMxy]..........

_____________________

We can show this argument is not formally valid by giving an interpretation under which the sentence ExMhx is true, and the sentence Ex[Mhx^EyMxy] is false.

_____________________

My job is to show by filling in the following blanks (D=...; h=...; M=....) how the premise can be true, but the conclusion false.


I tried the following: :

D= positive integers

h= even integers

M=[1] is less than [2]

_______________________

Premise ExMhx would become there are positive integers where even integers are less than other even integers...(a true claim)

However, the conclusion Ex[Mhx^EyMxy], would read there are some positive integers where even integers are less than other even integers AND where some positive even integers are greater than other even integers... (seemingly true, unfortunately the #2 is not gretaer than other positive integers, but the conclusion only says "some positive integers are greater...")

I think both claims are true but I am not at all confident in my interpretations. The existential quanitifcation is throwing me off. And help?
 
Emil
 
Reply Tue 17 Nov, 2009 11:41 pm
@Horace phil,
Horace;104221 wrote:
Hello everyone. I am working through the "Interpretations" section of my logic text. I am at the point where we are talking about the production of a valid argument, whose translation is valid, but how it might not necessarily be formally valid.

So for example from "HORACE IS MARRIED" I conclude that "HORACE IS MARRIED TO SOMEONE WHO IS MARRIED."

Now, so long as D= a set of persons; h=Horace; and M=[1] is married to [2]

The sentences, I have the premise: ExMhx; and the conclusion Ex[Mhx ^ EyMxy]..........


Symbols here.

Your formalization is not entirely correct. This is a correct formalization of the sentences that you wrote:
[INDENT]1. Horace is married.
(Mh)

2. Horace is married to someone who is married.
(∃x)(Nhx∧Mx)
[/INDENT] Where "Mx" means x is married, and "Nxy" means x is married to y.

We can translate my formalization back into english-ish:

[INDENT]1. (Mh)
Horace is married.

2. (∃x)(Nhx∧Mx)
There exists at least one person such that Horace is married to that person and that person is married.
[/INDENT]That seems to be correct.

An alternative formalization
[INDENT]1. Horace is married to someone.
(∃x)(Mhx)

2. Horace is married to someone and that someone is married to someone.
(∃x)(∃y)(Mhx∧Mxy)
[/INDENT](AFAIK it makes no difference whether the existential quantifier is located before Mhy here or located before the conjunction.)

My additions/changes are emphasized.

The first formalization is more similar to the original sentences. The second is more similar to the formalization that you gave of the first.

Horace;104221 wrote:
We can show this argument is not formally valid by giving an interpretation under which the sentence ExMhx is true, and the sentence Ex[Mhx^EyMxy] is false.


That's right.

Horace;104221 wrote:
My job is to show by filling in the following blanks (D=...; h=...; M=....) how the premise can be true, but the conclusion false.


I tried the following: :

D= positive integers

h= even integers

M=[1] is less than [2]

_______________________

Premise ExMhx would become there are positive integers where even integers are less than other even integers...(a true claim)

However, the conclusion Ex[Mhx^EyMxy], would read there are some positive integers where even integers are less than other even integers AND where some positive even integers are greater than other even integers... (seemingly true, unfortunately the #2 is not gretaer than other positive integers, but the conclusion only says "some positive integers are greater...")

I think both claims are true but I am not at all confident in my interpretations. The existential quanitifcation is throwing me off. And help?


Your interpretation is wrong. "h" is a particular but you're trying to use it as a class (even integers). That doesn't work. Try this:

D,xy ≡ things
h ≡ The blue stone
Mxy ≡ x is larger than y

[INDENT]1. (∃x)(Mhx)
There exists at least one thing such that the blue stone is larger than that thing.

2. (∃x)(∃y)(Mhx∧Mxy)
There exists at least one thing and there exists at least one thing such that the blue stone is larger than the first thing and that the first thing is larger than the second thing.
[/INDENT]Suppose that there are only two things: a blue and a green stone. Clearly there is no thing that the green stone is larger than. Thus the argument is invalid. Since all the premises are true and the conclusion is false.
 
Horace phil
 
Reply Wed 18 Nov, 2009 09:20 am
@Emil,
Thanks Emil.

Or... [1] murdered [2]...

Marriage is reciprocals, murder is not neceessarily. Right?
 
Emil
 
Reply Wed 18 Nov, 2009 09:41 am
@Horace phil,
Horace;104299 wrote:
Thanks Emil.

Or... [1] murdered [2]...

Marriage is reciprocals, murder is not neceessarily. Right?


Translation keys
When writing translation keys, I find it easier to write:[INDENT]Mxy means x is married to y
[/INDENT]Instead of what you do:[INDENT]M [1] is married to [2]
[/INDENT]Relations
An entire section of a logic textbook that I read is devoted to naming the different kind of relations. Unfortunately I don't have it with me here and I can't recall the names. There is a name for relations where if they hold one way (e.g. Mxy), then they also hold the other way (i.e. Myx). You can look it up in N. Swartz, R. Bradley, 1979.

ETA. I googled it a bit. Wiki has a useful resource on relations. I think I found the name of it. It is a symmetric relation:
[INDENT]"for all x and y in X it holds that if xRy then yRx. "Is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x."[/INDENT]
 
kennethamy
 
Reply Wed 18 Nov, 2009 09:57 am
@Horace phil,
Horace;104299 wrote:
Thanks Emil.

Or... [1] murdered [2]...

Marriage is reciprocals, murder is not neceessarily. Right?


I wonder how it could be?
 
Emil
 
Reply Wed 18 Nov, 2009 10:03 am
@kennethamy,
kennethamy;104315 wrote:
I wonder how it could be?


Haha. You are right about that!
 
 

 
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