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Logic Questions

I am having trouble with a few logic questions below.

If A implies B and A is true, can B be false?
If A implies B and A is false, is B false?
If A implies B and A is false, can B be false?
If A implies B and B is false, is A false?
Is one example enough to show a statement is true?
Is one example enough to show a statement is false?

Thanks for the help.

Reply 1

pmc:producer

Essentially, false represents an empty set... 'A' is the assumption, 'B' is the conclusion.

Reply 2

ghostwalker

Original post by paradoxequation

You may find a truthtable useful here,

But in answer to your questions, No, Not necessarily, Yes, Yes.

Is one example enough to show a statement is true?
Is one example enough to show a statement is false?

Thanks for the help.

What do you mean by a "statement" here?

Reply 3

AlwaysFitin'

Original post by paradoxequation

I am having trouble with a few logic questions below.

Is one example enough to show a statement is true?
Is one example enough to show a statement is false?

Thanks for the help.

You will confuse yourself if you think like that,

A proposition in logic is the set of sentences to which we can meaningly apply the terms true or false.

Another definition of a proposition is a statement that can be verified to be true or false.

We use symbolic logic to simplify statements, propositions are represents by letters much like sets or variables in mathematics.

EXAMPLE 1 --------------------- A -> B

A logical implication is a logical statement between two propositions, and the ENTIRE implication being true or false depends on the INDIVIDUAL TRUTH value of the individual propositions.

A logical implication is also read to be like this IF A... then B.. the "IF A" is called the premise, B is called the conclusion. Other books have called A the hypothesis and B the conclusion. You must realise that the laws of logic are what dictate whether an argument is valid or not, and not the individual semantics or wordings of the propositions.

A logical implication is always true whenever the conclusion, prediction, or B as in the example one, is true, regardless of the value of the premise, hypothesis.

A logical implication is also true when both premise/hypothesis and conclusion/prediction are FALSE.

A: If it is raining.
B: Then the road will be wet.

When A is true, then it is raining, and if the road is wet, then we say the implication holds, it has a truth value of TRUE.

When A is false, and it is not raining, but the road is still wet, then we still say that the implication holds true, B is true even though A is false. How is it possible? we do not discount other factors that make the road wet.

When A is true, it is raining, and B is false, the road is dry. Then we say the implication is false, it does not hold.

When A is false, and is false, that is, it has not rained and the road is dry, the implication is true.

You are probably wondering why when the premise is false and the conclusion turns out to be true is equivalent to when the premise and conclusion being false, you need to google logical implication and practice truth tables better. I can read up more and explain it, but I think I shall turn you to something more important.

The individual truth values of propositions can be true or false, a logical implication is simply a logical statement, not a scientific law. There does not have to be any actual "cause - effect" relationship between the two propositions in an implication, as far as I am aware!

In regards to your question, is one example enough to show whether a statement is true? in what context?

Are you familiar with the notion of sets? A set is a well defined collection of objects. Its actually a primitive notion and remains undefined in mathematics like the other basic terms, but intuitively we can say a set is a well defined collection of objects.

A proposition in mathematics often contains an object, a prescription: which is affirmative or negative, and the quantifier, universal or particular.

" ALL BOOKS IN THE UNI LIBRARY ARE RED"
The object is the books, the prescription is red.
Are red, the prescription is affirmative. It applies. "NOT RED" would mean the prescription is negative.
ALL is the quantifier, here it is universal. Universal because it refers to all the books in the uni library.

This logical statement is a universal affirmative statement. Its negation is a particular negative statement. (Google de morgans laws)

Is one example enough to show whether a statement is true?

It depends on the quantifier. For "ALL", no.
For "SOME" or "there exists", an existential quantifier, yes.

is one example enough to show whether a statement is false?
Depends on the quantifier, for ALL. Yes. It would mean that the statement is not true for at least, at least, one element, and thus, logically the statement is not true.

------------------------------------------------
A little bit of history of logic here, aristotle and george cantor were important figures in splitting up objects of our perception into sets.

Logic is about dividing such objects of our perception into categories, literally the mathematical sets we learn about today.

So, when I said ALL BOOKS IN THE UNI LIBRARY IS RED. that is actually a set relation as well, it means there are two sets:

1. The set of all books in the uni library, DENOTED by B.
2. The set of all red things, DENOTED by the letter R.

B is contained within R, or B is a subset of R.

So, If I say all books in the uni library are red, and there is at least one green one, then one example is enough to prove that this proposition is false.

BUT, if I say that "SOME" or "THER EXISTS" a book which is red in the library, then we would need to show that all books in the library are not RED to make this proposition false. It depends on the quantifier, which depends on the language used within the proposition.

"Divide and conquer" The thing about logical arguments is that for them to be valid the premises must derive a valid conclusion, regardless of the actual wording or content of the propositions. Enthimymes are hidden premises, which we are not aware of but exist. A premise is something we know to be, or strongly believe to be, true.

(edited 7 years ago)

Reply 4

AlwaysFitin'

A little bit of history of logic here, aristotle and george cantor were important figures in splitting up objects of our perception into sets.

Logic is about dividing such objects of our perception into categories, literally the mathematical sets we learn about today.

So, when I said ALL BOOKS IN THE UNI LIBRARY IS RED. that is actually a set relation as well, it means there are two sets:

1. The set of all books in the uni library, DENOTED by B.
2. The set of all red things, DENOTED by the letter R.

B is contained within R, or B is a subset of R.

So, If I say all books in the uni library are red, and there is at least one green one, then one example is enough to prove that this proposition is false.

BUT, if I say that "SOME" or "THER EXISTS" a book which is red in the library, then we would need to show that all books in the library are not RED to make this proposition false. It depends on the quantifier, which depends on the language used within the proposition.

"Divide and conquer"

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