a question about some logic

[MattcAnderson]

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

[Nick]

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!

[Batalov]

To illustrate Nick's point about (at least) two broad views on logic, it is helpful to consider the eixstence of two types of courses, exemplified by these (on Coursera):
- https://www.coursera.org/learn/logic-introduction
and
- https://www.coursera.org/learn/understanding-arguments (used to be a longer course which is now apparently enhanced and split into a series)

For me, the second one was an eye-opener. It is the practical side of what everyday people use for the 'human logic'. I am not a specialist, but it very weakly resembles Bayesian statistics - in the human world, frankly, there is no 100% truth or 100% false, instead there are degrees of belief, and consequently the size of bets that people are making every day of their life. For example, if a person goes to a car salesman with little knowledge of both the subject (the car) and the training in recognizing faulty "arguments" (that is, word constructions that are carefully built in advance to bias the prior beliefs of a buyer into the new, and perhaps very incorrect set of posterior beliefs), then that buyer is doomed to make very bad bets (decisions, purchases).

Recognizing exactly through what mental process most people make decisions (which is, for me, a very different process from mine) was very helpful to much better understand (but not condone) the ways modern politics works. How does one lead millions of people into making very impractical decisions that directly affect their lives for worse? -- No longer a mystery to me (and let's say for the sake of the argument is that I am talking about politics in Russia and/or in France). Car salesmen are everywhere (and of course they always were)!

[Dr Sardonicus]

From the Attached File a question in logic.txt

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."

[MattcAnderson]

Hi Mersenne forum,

Thank you for the good replies so far. I'm english speeking colledge graduate. My father was colledge professor. RIP . Mother good.

Read, games for the super intelligent.

Here is an example problem. It should be spoken.

John,, while James,, ,, had had HAD had had HAD HAD
Had Had Had Had a better effect on the teacher.

grinz

Regards,
Matt

[Harrywill]

Originally Posted by MattcAnderson

Hi Mersenne forum,

Thank you for the good replies so far. I'm english speeking colledge graduate. My father was colledge professor. RIP . Mother good.

Read, games for the super intelligent.

Here is an example problem. It should be spoken.

John,, while James,, ,, had had HAD had had HAD HAD
Had Had Had Had a better effect on the teacher.

grinz

Regards,
Matt

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.

[Degutis]

Originally Posted by Batalov

To illustrate Nick's point about (at least) two broad views on logic, it is helpful to consider the eixstence of two types of courses, exemplified by these (on Coursera):
- https://www.coursera.org/learn/logic-introduction
and
- https://www.coursera.org/learn/understanding-arguments (used to be a longer course which is now apparently enhanced and split into a series)

For me, the second one was an eye-opener. It is the practical side of what everyday people use for the 'human logic'. I am not a specialist, but it very weakly resembles Bayesian statistics - in the human world, frankly, there is no 100% truth or 100% false, instead there are degrees of belief, and consequently the size of bets that people are making every day of their life. For example, if a person goes to a car salesman with little knowledge of both the subject (the car) and the training in recognizing faulty "arguments" (that is, word constructions that are carefully built in advance to bias the prior beliefs of a buyer into the new, and perhaps very incorrect set of posterior beliefs), then that buyer is doomed to make very bad bets (decisions, purchases).

Recognizing exactly through what mental process most people make decisions (which is, for me, a very different process from mine) was very helpful to much better understand (but not condone) the ways modern politics works. How does one lead millions of people into making very impractical decisions that directly affect their lives for worse? -- No longer a mystery to me (and let's say for the sake of the argument is that I am talking about politics in Russia and/or in France). Car salesmen are everywhere (and of course they always were)!

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.

[Neutron3529]

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

[jwaltos]

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?

[chalsall]

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?