'Therefore' in an illogical logic sense

Solution 1:

No, therefore should not be reserved for conclusions beyond a reasonable doubt. It is merely a transition similar to thus or as such:

Therefore, I ordered pepperoni.

Thus, I ordered pepperoni.

As such, I ordered pepperoni.

The extended conversation could have been:

I like meat on my pizza. Therefore, I ordered pepperoni.

You can test for an appropriate use of therefore by flipping the sentence order and using because:

I ordered pepperoni because I like meat on my pizza.

The contention that the therefore segment should be restricted to perfect logical use is forgetting that the word really only serves to draw a causal link between statements.

I like red. Therefore, I painted my wall red.

There is no logic here. I am just explaining why I painted the wall red.

Now, if someone is using therefore as a logical link and the logic sucks, you can say that the argument is bad. But the use of therefore isn't the problem.

This $10 item is 50% off; therefore I am saving $6.

This is wrong, but the use of therefore isn't incorrect because it is simply communicating the thought. The communication is accurate; that makes therefore the appropriate word. Replacing therefore with a different word changes the meaning of the sentence (and could correct the logic) but the intent of the speaker no longer matches the communication.

Edit: Since there seems to be some confusion about the actual definition of the word, here is what my dictionary says:

for that reason; consequently : he was injured and therefore unable to play.

Reason, in this context does not mean "logic". It just means "why".

Why couldn't he play? / He was injured.

He was injured and therefore unable to play.

He was unable to play because he was injured.

It is worth noting that there is a strict logical use for the term therefore that explicitly means something akin to "logically derived from the previous statements" but that would be applicable to formations of the following:

1. All men are mortal

2. Aristotle is a man

3. Therefore, Aristotle is mortal

This is commonly represented by three dots in a triangle (∴). But even in this case, the use of therefore is a signal of a specific meaning. If the conclusion is false, it was not an incorrect use of therefore but simply faulty logic. Removing or changing the word doesn't make the problem go away.

Solution 2:

Yes, you are right. 'Therefore' is used to link a statement which logically follows another one. It cannot be used in a standalone statement.

I would not say therefore should be used when something is being concluded beyond reasonable doubt. It can be used to connect two statements that

1) the user believes is logically correct

2) the user wants you to believe is logically correct.

For example, the following statement is grammatically correct, even though logically incorrect: "The sun moves from the east to the west. Therefore the sun is circling the earth".

Solution 3:

It depends what you mean by 'should'. Therefore is used in the context of an argument, and if you are following the proper rules of argument should only be used to link a premise to a conclusion. But if you are merely trying to win the argument (following the rules of rhetoric), it's a very useful device to bolster a weak argument: "You want more benefits and lower taxes; therefore you should vote for me".