# Questions in basic logic

1. Apr 27, 2007

### Shing

1.
What is the exact and specific meaning of "a true statement"?

2.
I can't comprehend about conditionals ,when there are a false hypothesis and a false consequent still regarded as true .
I think it should be "a not false statement" rather than " a true statement".
And it is hard to believe that
A statement including a false hypothesis and a false consequent is regarded as a true statement.

Thanks!

Last edited: Apr 28, 2007
2. May 8, 2007

### guten

I can't give an exact and specific meaning of "a true statement." I have not figured that out yet. But I will try to share some of my understanding about deductive logic.

A proposition, simply defined, is a statement that is true or false. This definition means that there is no in between and there is certainly no true and false at the same time, there is no grey area.

Some propositions are true by definition. Some are true because they correspond to fact, "some condition of the world".

Consider the conditional statement:

A friend says:

"If there is rain tomorrow, then I will buy a car."

We have to wait until tomorrow to see if it has rained and if the person has bought a car, to determine the truthfulness or falseness of this proposed fact. But we can consider it now.

Use the next paragraph only between the labels BEGIN and END.

We will use the rule that false corresponds to "being untrue to their word." And since a proposition can only be true or false we will accept as true as anything that is not false, for example, if you say the person was anything but "being untrue to their word", like "being poopy to their word" it is true. So as long as you say they lied it is false. If you are a little confused but can't say they lied, you say it is true. There is no middle ground. Also since "being true to their word" is not "being untrue to their word" it is true.

BEGIN
1) Suppose you are downtown, it is raining, and your friend drives by in a new car. Is this false? (Has the friend been untrue to their word? You should say yes if you think your friend has lied, making the statement false. Anything else makes this statement true.)

2) Suppose you are downtown, it is raining, and your friend walks by. Is this false?

3) Suppose you are downtown, it did not rain, and your friend drives by in a new car. Is this false? (Hint: You say it has not rained, why do you have a car? Well, says your friend, I never mentioned what would happen if it did not rain.)

4) Suppose you are downtown, it did not rain, and your friend walks by. Is this false? (Hint: You say where is your new car? Your friend says it has not rained so I did not buy a new car. You say, that is right, that is what you said.)
END

Think of true as short hand for "not false". Think of false as short hand for "not true". Also a logical argument guaranties that if you have true premises you will have a true conclusion, this is why logic is a very important part of sound reasoning. It does not guaranty that if you have false premises you will have a false conclusion. This means that a logical argument can give true conclusions from false premises. Remember that logic is about the structure of an argument and not the content of the argument.

Try googling "deductive logic" and compare with what you get when you google "inductive logic".

I hope this helped.

3. May 9, 2007

### Hurkyl

Staff Emeritus
If you have a language and a truth function, then a "true statement" is a statement in your language that, when plugged into the truth function, results in the value "true".

I imagine that you have no trouble comprehending this theorem of integer arithmetic:
if x = 1, then x * x = 1.​
If it's a theorem, then it has to be true in any model. (i.e. for any truth function for which the axioms of integer arithmetic are true)

But x is a free variable -- I can find a model where x has any value I want. For example, if I choose a model where x = 2, then in the corresponding truth function, v(x = 1) = false and v(x * x = 1) = false. If I choose a model where x = -1, then v(x = 1) = false and v(x * x = 1) = true.

But v(if x = 1, then x * x = 1) = true no matter what model I choose...

Here's another way to think of it. We have the following rules of deduction:
A => B
A
-------
B

A => B
~B
-------
~A

and we do not have the following rules of deduction:
A => B
B
-------
A

A => B
~A
------
~B

Here's an exercise: use the fact the first two are rules of deduction and the second two are not rules of deduction to compute what the truth table for A => B must be. I'll get you started:

(assuming you've already proven (true => true) = true)
If it were the case that (false => true) = false, then if we are told:
(A => B) = true
B = true
then it is impossible for A to be false -- therefore, A must be true, and we would have the rule of deduction
A => B
B
------
A

However, we know that this is not a rule of deduction. Our hypothesis was wrong, and so (false => true) = true.

Knowing the negation of "X is false" is the same as knowing "X is true".