Boolean Logic inconsistent systems and their negations.

Question

I've been reading https://faculty.washington.edu/smcohen/120/Chapter5.pdf and I had a question on inconsistent systems. It says about inconsistent systems "We may not be able to show, using logic alone, which premise is false, but we can establish that at least one of them is false." does that mean the negation of that system must be true? Here is my reasoning.

A) B is false

B) A is true

These two statements have an implied AND between them.  So the system can be represented by the single boolean logic statement A AND B.  The negation would be NOT A OR NOT B and would be the following  I will call them nA and nB.

nA) nB is true

OR

nB) nA is false

If nA is true, then nB must be true. But nB cannot be true because it calls nA false, which makes nB false and in turn makes nA false because it asserted nB was true. so there is a contradiction in assuming nA is true therefore nA cannot be true.

If nB is true then nA is false. Looking at nA we see it calls nB true which cannot be correct. So nA cannot be True, therefor it is false. And that IS consistent with nB calling nA false. So nB being true does not create a contradiction in nB, thus we can call it true.

Since this is an OR statement nB being true is enough to call the system true.  And since the negation is true, the original statement A AND B must be false.

So in conclusion when we take the system of statements

A) B is false

B) A is true

we can say the system has the value of FALSE not inconsistent. In addition since the negation of B was True, we can call B False. A is therefore True by the same reasoning.

Was I correct? Is there no such thing as inconsistent? If you have inconsistent statements they must have an implied "AND" between them. So their negation is an OR between them. Since they said "we can establish that at least one of them is false." then one of the statements in the negation must be true. And with OR statements one being true means the statement is true. And the negation of True is False. So the original system was False.

Maybe it's my definition of False and True that are wrong. Please let me know where I went wrong because even in the document I cited they have inconsistent as a solution. So I am most likely wrong, I just want to know why I'm wrong.

This is not for a class. This is me opining on the rules of logic and if there is actually only a binary True False value to everything and not a True, False, and Neither/inconsistent.

Answer 1

It appears that you are thinking only in terms of one possible semantics: Boolean semantics. But there are many more out there that do not interpret formulas as either being true or false, but allow more truth values.

To start with, there is intuitionistic logic that changes semantics and proof rules to the effect of rejecting proofs by contradiction (or consequences like law of exclude middle etc.). In particular, a formula is not interpreted as being either true or false but, in Kripke semantics, it is said to be valid with respect to a world, a certain stage of knowledge. The semantics of the logical connectives then say when we change our world. This leads to an entirely different meaning of formulas than the Boolean semantics.

Once you have seen such a slight change of proof rules and semantics, you can move to other interesting logics that may also alter the logical connectives avialable: substructural logic, which includes the very import linear logic; Łukasiewicz logic and other quantitative or probabilistic logics; modal logic; and, what might be most interesting to you, paraconsistent logic.

There is a whole landscape of different logics with variations of syntax, proofs and semantics. One can take this even further and forget about explicit syntax, ending up with category theory.

I hope this helps you to get started with your exploration.