@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.
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