# Three-valued logic as foundation

Isn't it more natural to use Three-valued logic(false-true-unknown) as the foundation of mathematics? It is a better model for natural languages. And it also can model sentences like the lair paradox by giving it the unknown value. And in set theory we don't need the axiom of regularity which we never actually use. So my question is this: why mathematicians like bivalent logic more?

• It seems uncomfortable to accept a statement is "between" true and false, no? – Asier Calbet Aug 9 '14 at 10:43
• @Assaultous2 yesno. – Jam Aug 9 '14 at 10:45
• @Assaultous2 All men are mortal. true or false? we don't know yet. science is very promising in those areas. – Buddha Aug 9 '14 at 10:46
• So. Even if we dont know, the statement is either true or false, not somewhere in between. – Asier Calbet Aug 9 '14 at 10:47
• You might read up on intuitionistic mathematics ("intuitionism") and "their" view on the law of excluded middle. – Benjamin Dickman Aug 9 '14 at 10:50

There are several limitations of three valued logic. The first is that there are several competing varieties: Versions developed by Lukasiewicz, Kleene, and several others are described in the literature, and mathematicians have not agreed on which is best.

The second is that none of these systems as currently developed is adequate for the purposes of deduction. The most important rules of inference in logic have corresponding tautologies in two valued logic.

A third is that the various three valued systems and their successful interpretation are not well understood, even by their creators and originators.

One pastime of logicians is experimenting with systems in which one or other of these rules of inference is rejected. However, many of these tautologies actually fail in three valued logic, which limits their utility. For instance, $P \to P$, "If P then P", "If Socrates is mortal then Socrates is mortal" fails in one of Kleene's three valued logics, because if P has the third value, so does $P \to P$. This makes it hard to do any deduction at all with uncertain, doubtful, or equivocal statements.

The logic of Lukasiewicz does better, but modus ponens in its standard form $(P \& (P \to Q)) \to Q$ fails as a tautology (in the case where P has the third value and Q is false), and so does the transitive rule for the conditional, $((P \to Q) \& (Q \to R)) \to (P \to R)$. These make deduction difficult. It is not generally recognized that this is because Lukasiewicz logic allows conditional statements as well as simple ones to have the third value, and we should not be accepting deduction from doubtful inferences as logically valid in any case.

There is a modal extension of Lukasiewicz logic that helps. The remedy for the failure of modus ponens is to make it explicit that the conditionals used in deduction should be definitely true. The statement $(P \& \Box(P \to Q)) \to Q$ is tautological, and so is the transitive rule for definitely true conditionals. $(\Box(P \to Q) \& \Box(Q \to R)) \to \Box(P \to R)$.

Using the modal version of Lukasiewicz logic, it is possible to distinqush conditions of "Possibly true", "certainly true", "possibly false", "certainly false", "equivocal" and "categorical" for a given proposition. The conditions "certainly true", "certainly false", and "equivocal" correspond to the truth values; the others do not.

It can then be claimed that statement can be possibly true and possibly false if and only if it is equivocal. It is likewise true that a statement is definitely true or definitely false if and only if it is categorical. Classical two valued logic is restricted to this class of statement.

It is common error to attempt to reason about three valued logic by assuming the same rules that apply in classical logic. For instance, "unknown" could be interpreted in either an equivocal sense "possibly true and possible false", or a categorical one, i.e. "definitely true or definitely false, but not known which is the case". These two senses behave similarly but are technically the negation of each other, so assuming an equivocal sense and then applying the law of the excluded middle gives a contradiction. Likewise, assuming unknown means equivocal and then applying the negation rules of two valued logic ("not true" equals "false" and "Not false" equals "true") also results in quick contradiction and confusion.

Allowing paradoxical statements such as "this statement is false" to have the third value may provide "a" solution, but whether it is "The" solution is highly debatable.

Three valued logic does not solve the liar paradox. If 'this sentence is false' was 'unknown' then it would be false and therefore true. So no advantage over bivalent logic.

The liar paradox relies on the fact that negation cannot have a fixpoint, which is what we need to interpret the liar sentence. You can avoid the problem by removing negation from the logic altogether, so the liar sentence cannot be formulated, but you cannot solve it.

Constructive mathematics rejects the principle of the excluded middle and allows for more truth values. There are mathematicians who develop mathematics this way. However, topology compensates for all the deficiencies that classical (bivalent) mathematics has compared to constructive mathematics.

• "topology compensates for all the deficiencies that classical logic". well, I did not know that. can you give a reference to how? – Buddha Aug 9 '14 at 13:24
• @Buddha I recommend reading a little about topos theory. One can think of various logical systems, including three-valued logic, as geometric objects. Classical set theory lives, in some sense, on the one-point space, so one can think of intuitionistic mathematics as classical logic plus geometry. – Slade Aug 9 '14 at 14:32
• I am not so sure that the logic here about the liar paradox is correct. If sentence $P$ is unknown it seems rash to conclude that it is not false; the statement "P is false" has a truth value of unknown, not true. – Slade Aug 9 '14 at 14:38
• @Buddha Intuitionistic logic can't be modeled by a two-valued system or by any other finite model. Instead, one typically takes values to be subsets of $\Bbb R$. Letting the value of a formula $x$ be $V(x)$, one has $V(x\lor y) = V(x)\cup V(y)$ and $V(x\land y) = V(x)\cap V(y)$. The difference comes with the interpretation of $\to$: $V(x\to y) = ((\Bbb R - V(x))\cup V(y))^\circ$, where $S^\circ$ is the topological interior of $S$. Then one can show that one has $V(F) = \Bbb R$ precisely when $F$ is intuitionistically valid. Wikipedia explains this in more detail. – MJD Aug 9 '14 at 21:22
• Another 3-valued liar paradox, which might be easier to understand intuitively, is "This sentence is false or unknown." So it's true if and only if it's false or unknown, which seems to cause problems no matter which of the 3 values you try to give it. – Andreas Blass Aug 9 '14 at 21:47