mersenneforum.org a question about some logic
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2017-04-28, 11:52   #1
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

2·3·7·17 Posts

Hi Mersenneforum,

For those of you who are familiar with logic, what is a syllogism?

I have been told that some syllogisms are invalid.

dictionary.com

I make this a double question. Also, what does a Ven diagram with more than
4 regions look like?

No response is a bad response.

Regards,
Matt
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 a question in logic.txt (527 Bytes, 325 views)

 2017-04-28, 16:30 #2 Nick     Dec 2012 The Netherlands 110011100112 Posts Broadly speaking, there are 2 ways to look at logic. You can focus on the history and philosophy of it, or instead concentrate on the practical side, as used in mathematics and computer science. A syllogism is a particular restricted pattern allowing you to deduce a new statement from 2 existing statements, and I would say it belongs to the first point of view. From the mathematical side, the idea known as "modus ponens" is more general and useful. Given a statement $$p$$ and a statement of the form $$p\Rightarrow q$$, it says you are allowed to deduce the statement $$q$$. For example, if we have two sets $$A$$ and $$B$$ and an element $$x$$ and we already know that $$x\in A$$ and that $$x\in A\Rightarrow x\in B$$ (i.e. $$A\subset B$$). then we may conclude that $$x\in B$$. I am not an expert on logic, so you may get a better answer from someone else!
2017-04-29, 16:27   #4
Dr Sardonicus

Feb 2017
Nowhere

25×139 Posts

From the Attached File a question in logic.txt

Quote:
 what is the difference between ... a valid logical inference and a true fact statement?
I'm not sure what "true fact statement" means. Especially in these times of "alternative facts"

However, I would point out that the rule of inference modus tollens AKA modus tollendo tollens AKA "denying the consequent" can be used to illustrate that a statement can be "logically true" even though it's nonsense.

The rule is, that the implication "A implies B" (where A and B are statements) is logically equivalent to "not-B implies not-A" (the "contrapositive" of A implies B). The statement A is the "premise" and B the "conclusion."

OK, now what happens if the statement A is false? Then not-A is TRUE, so the contrapositive is (logically) TRUE. The original implication, therefore, is also (logically) TRUE. That is, Any implication with a false premise is logically TRUE. I like to call such things "vacuously true," since the logical truth of the implication has no practical meaning.

The implication "A implies B" is formulated in terms of the logical operations AND, OR (inclusive OR) and NOT as "NOT-A OR B." The negation is "A AND NOT-B." The negation clearly is TRUE when A is TRUE and NOT-B is also TRUE; the original implication is thus FALSE when the premise A is TRUE and the conclusion B is FALSE.

An example of the utility of the contrapositive occurs with Fermat's "little theorem:"

Let p and a be positive integers, and gcd(a,p) = 1. If p is prime, then p divides a^(p-1) - 1.

The contrapositive of the implication is, "If p does not divide a^(p-1) - 1, then p is not prime." This means that compositeness can be detected without factoring.

There is also an amusing use of the contrapositive known as "Hempel's paradox."

Last fiddled with by Dr Sardonicus on 2017-04-29 at 16:29

2017-05-12, 06:42   #6
Harrywill

"Harry Willam"
May 2017
USA

22×5 Posts

Quote:
Two best friends, James and John, were asked on an English test to describe a women who suffered from a cold. John wrote: "The woman had a cold," which is the teacher marked as Wrong. James, however, wrote: "The woman had had a cold." Since James' answer was right, it had a better effect on the teacher.

2018-12-26, 09:34   #7
Degutis

Dec 2018

1 Posts

Quote:

I personally like the Logical Reasoning courses on KhanAcademy. There's also such a term as Symbolic Logic. That is why, I always treat logic as something ambiguous.

Last fiddled with by Degutis on 2018-12-26 at 10:15

2018-12-30, 07:04   #8
Neutron3529

Dec 2018
China

41 Posts

Quote:
 what is the difference between ... a valid logical inference and a true fact statement?
In my opinion, a true fact statement could not be "If 1+1=3, then the earth will blow up and every human beings will die." since it is too obsurd.

But such logical inference must be TRUE since 1+1 not equals to 3.

For the Ven diagram, there are 16 regions

 2019-02-15, 22:22 #9 jwaltos     Apr 2012 Brady 383 Posts I sometimes like to answer a question with another question..as in this case. This is like being on an island within an archipelago..you may not see any other islands so you must do your best to understand where you are and how to connect to those places you cannot see. In an archipelago ALL islands are defined as connected to one another. If you write something down within a known logical structure that is non-contradictory and consequential can you prove that what you write is true, factual, valid and that from what you have written at least one deductive/inductive inference may be made which must be true, factual....etc? Can you do the same for statements....say in religion or politics? Last fiddled with by jwaltos on 2019-02-15 at 23:12
2019-02-15, 22:36   #10
chalsall
If I May

"Chris Halsall"
Sep 2002

3×52×127 Posts

Quote:
 Originally Posted by jwaltos I sometimes like to answer a question with another question..as in this case.
I similarly like to run these experiments.

What, exactly, was the question?

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