In 1917, the Polish Logician Jan Lukasiewicz proposed a three-valued logic in an attempt to express the kind of uncertainty associated with the future. Objections arose almost immediately with his proposed interpretation, his definition of implication, and the failure of important rules of inference in his logic. Apparently, he did not work on it very long, and proposed many valued systems along the same lines, as well as a 4-valued modal system that was even less well received. Although many 3-valued logics have been proposed since then, his is still the one that is most often used.
He attempted to formulate a modal logic (a logic of necessity and possibility) on the basis of his 3-valued logic. This effort was ultimately judged unsuccessful, although the reasons why it didn't work are seldom clearly presented. Modal logic follows lines laid out by C. I. Lewis. For various reasons, the systems he originated, and most of their successors, are incompatible with Lukaseiwicz logic.
Some time in the early 1990s, I discovered a way to extend Lukasiewicz logic so that it could serve as the basis for a modal logic. As I searched the literature for evidence that someone had either anticipated or refuted my ideas, I found that this system had far more power than I had ever expected.
From a mathematical point of view, the system presented here is exactly equivalent to Lukasiewicz 3-valued logic. Proofs of consistency and completeness have already been provided and there is nothing new here. I have only supplied a couple of definitions, which are, mathematically, only shorthand for already defined combinations.
However, from a practical point of view, this is a full-featured and consistent extension of classical logic into the realm of the uncertain. It is analogous to extending the arithmetic of whole numbers to fractions or real numbers to complex numbers. It is consistent and truth-preserving, and uses algebraic techniques similar to those of Boolean algebra. It incorporates a fully functional modal logic and gives a unified perspective and provides insights and comparison into intuitionism, fuzzy logic, paraconsistent logic, and other areas of non-classical logic. The implications are nothing short of revolutionary.
To the best of my knowledge after a reasonable search of the literature and standard references, this particular development of Lukasiewicz 3-valued logic is new. Although this work has been submitted for publication and presented to a number of academic logicians, it has been given at best a cursory and superficial glance, and then dismissed or ignored. Therefore, let the revolution begin without them.
Logical expressions in L3 are composed of variables, connectives, and grouping symbols. Variables represent sentences which express propositions and which have a truth value.
T is used for propositions that are considered definitely, certainly, necessarily, or unambiguously true.
U is used for propositions that are considered uncertain, unknown, undecided, doubtful, equivocal, or paradoxical. Further restrictions on the meaning of U are provided by the logical theorems. It's even possible to use this value to examine statements that are either true or false but you don't know which, provided you (shh, look around, are any spies for 2VL orthodoxy listening?) emphasize your ignorance, swallow contradictions, and ignore the inconvenient fact that they really are one or the other.
F is used for propositions that are considered definitely, certainly, necessarily, or unambiguously false.
These truth values are only used in evaluating expressions, not as logical constants that may appear within expressions. They are ordered so that T is "more true" than U, which is "more true: than F. This is useful in interpreting the binary connectives.
There are two kinds of connectives. Unary connectives are placed before variables and signify statements about the proposition. Binary connectives appear between two variables and signifiy some sort of logical operation or relation between the propositions. Each of these is associated with a truth table which defines how the truth value of the whole expression is to be determined from the truth value or values of its components.
(~ P) Negation, "Not P", or "It is not the case that P". The first and last entries in this table match classical two-valued logic. It's reasonable to suppose that the negation of a doubtful statement is also doubtful, although this has hidden implications.
| P | ~ P |
|---|---|
| T | F |
| U | U |
| F | T |
([]P) Certainty. This is roughly equivalent to "Necessarily", "Certainly", "Definitely", "Unambiguously", "Indubitably", "Provably" although each of these terms has connotations that don't exactly agree with the behavior dictated by the theorems.
| P | []P |
|---|---|
| T | T |
| U | F |
| F | F |
(<>) Possibility. As used here, this differs from the meaning more usually employed in philosophy or modal logic, to mean "Theoretically possible" or "non-self-contradictory". It has a rather broader sense of "At least possible", or "not impossible".
| P | <> P |
|---|---|
| T | T |
| U | T |
| F | F |
(?P) Equivocation . This might be read "P is equivocal". It is a claim that P has the truth value U. P is "not certain" is an ambiguous phrase which could be taken to mean either ~[]P or ?P. "Equivocal" is less ambiguous and better captures the "yes and no" character associated with the middle truth value.
| P | ? P |
|---|---|
| T | F |
| U | T |
| F | F |
(!P) Dichotomy "P is dichotomous", meaning certainly either T or F, with no middle ground allowed. In this logic, this is true of some statements but not of others.
| P | ! P |
|---|---|
| T | T |
| U | F |
| F | T |
It may be noted that other combinations are possible. There are no connectives provided that "degrade" T or F to U, although some other 3-valued logics do provide them. The connective [] asserts that statement has the truth value T. ? asserts that a statement has the truth value U. There is no single connective that asserts that a statement has the truth value F, but the combination ~<> accomplishes the same purpose.
In a complementary fashion, <>P does not state that P has a certain truth value. it says that its value may be T or U, but it isn't F. Likewise, !P declares that P's value may be T or F, but it isn't U. There is no single connective that states that a statement is U or F, but not T, but the combination ~[] accomplishes the same purpose.
( P v Q). The logical (inclusive) "or". The value of P v Q is the most true of the values of P and Q.
| P v Q | Q | |||
|---|---|---|---|---|
| T | U | F | ||
| P | T | T | T | T |
| U | T | U | U | |
| F | T | U | F | |
( P & Q) The logical "and". The value of P & Q is the least true of the values of P and Q
| P & Q | Q | |||
|---|---|---|---|---|
| T | U | F | ||
| P | T | T | U | F |
| U | U | U | F | |
| F | F | F | F | |
( P -> Q ) The conditional. This is probably the best equivalent to the English "if...then..." This is the conditional proposed by Lukasiewicz, and is not easily defined in terms of any others.
| P -> Q | Q | |||
|---|---|---|---|---|
| T | U | F | ||
| P | T | T | U | F |
| U | T | T | U | |
| F | T | T | T | |
( P => Q) The strict conditional, "Certainly, if P then Q". "If P then certainly Q" might be interpreted to mean this, [](P ->Q). or might also be used for (P -> []Q). This is generally the most useful conditional in 3VL.
| P => Q | Q | |||
|---|---|---|---|---|
| T | U | F | ||
| P | T | T | F | F |
| U | T | T | F | |
| F | T | T | T | |
Two other connectives of interest to logicians are not much used here. They are not adequate for L3 because they make no provision for the uncertain or doubtful. The role of the material conditional of classical two valued logic, (~P v Q) or ~(P & ~Q) is best played by the strict Lukasiewicz conditional.
The Lewis strict conditional (fishhook) ~<>(P & ~Q), commonly used in modal logic, is also generally not used, and the strict Lukasiewicz conditional is also used instead.
(P <-> Q) Biconditional (if and only if). This provides a logical equivalence.
| P <-> Q | Q | |||
|---|---|---|---|---|
| T | U | F | ||
| P | T | T | U | F |
| U | U | T | U | |
| F | F | U | T | |
( P == Q ) Equivalence. This is a truth-functional equivalence, stating that the two expressions have the same truth value. It is also the strict biconditional.
| P == Q | Q | |||
|---|---|---|---|---|
| T | U | F | ||
| P | T | T | F | F |
| U | F | T | F | |
| F | F | F | T | |
For those interested in such things, an axiomatic basis for L3 was given by Wajsberg. Along with the rules of modus ponens and substitution, the axioms are:
| 1) | Q -> (P -> Q) |
| 2) | (P -> Q) -> ((Q -> R) -> (P -> R)) |
| 3) | ((P -> ~P) -> P) -> P |
| 4) | (~Q -> ~P) -> (P -> Q) |
Negation and the non-strict conditional are defined by these axioms. Other connectives can be defined. The first four are standard. The next two have apparently not been previously defined and used in Lukasiewicz logic: an oversight which has prevented this logic from being properly appreciated for over 80 years. The last two have been previously defined but little used.
| []P | := | ~( P -> ~P) |
| <>P | := | ~P -> P |
| P v P | := | (P -> Q) -> Q |
| P & P | := | ~((Q -> P) -> ~Q) |
| P => Q | := | []( P -> Q) |
| P == Q | := | ((P => Q) & (Q => P)) |
| !P | := | []P v []~P |
| ?P | := | ~!P |
These are expressions that are tautologies or truth-functionally true, meaning that they are evaluated as T for every assignment of truth values to the variables. Many systems employ the rule of necessitation, in which if P is a theorem of the logic, []P can be introduced into a proof. This can also be used here, because according to the truth tables, []P is true for every tautology.
This is only a preliminary list: Other theorems will be added in future versions. Also, a bare list without interpretation or commentary isn't very interesting. More will be added later.
These apply to propositions in general.
Double negation
Idempotent law for v
Idempotent law for &
Commutative law for v
Commutative law for &
Associative law for v
Associative law for &
Distributive law for v
Distributive law for &
de Morgan's law for v
de Morgan's law for &
Contrapositive rule
Modus ponens
Transitive rule of the strict conditional
Strict biconditional equivalence
Equivalent means P is necessary and sufficient for Q.
Reflexive law of equivalence
Negation law of equivalence
Symmetric law of equivalence
Transitive law of equivalence
If definitely P then P
If P then Possibly P
Not necessarily is equivalent to possibly not
Not possible is equivalent to Certainly not
If the strict Lewis conditional holds, the strict Lukasiewicz conditional does also (but not conversely)
If the strict Lewis conditional holds, the ordinary material conditional does also, but not conversely.
If the strict Lukasiewicz conditional holds, the ordinary Lukasiewicz conditional does also, but not convesely.
If the ordinary material conditional holds, the ordinary Lukasiewicz conditional does also, but not convesely.
P is dichotomous if and only if ~P is also
P is equivocal if and only if ~P is also
If P is not dichotomous it is equivocal
If P is not equivocal it is dichotomous.
"Definitely P" is either true or false.
"Possibly P" is either true or false.
"P is equivocal" is either true or false.
"P is dichotomous" is either true or false.
P is either certain or not certain
P is either possible or impossible
P is either dichotomous or equivocal
"Either P or not P" is always at least possible.
"P and not P" is never necessarily true
"P or not P" is equivalent to Not "p and not p". The principle of bivalence is equivalent to the principle of non-contradiction.
"P and ~P" means that P is either impossible or equivocal.
An assertion that P is dichotomous is equivalent to asserting bivalence.
P and ~P are both possible if and only if P is equivocal.
A statement is equivocal if and only if it is equivalent to its own negation.
P is equivocal if and only if "p and ~p" is also equivocal.
Some of the more revolutionary consequences of this logic are mentioned here. More detailed verification and analysis of these claims remains to be done.
Classical two-valued logic has been successful enough that many people wonder why a 3-valued approach is even necessary. The principal reason is that we must routinely and daily deal with ambiguity and indecision. 2-valued logic is really only successful when it is possible to neatly divide statements into "True" and False". It has been far more successful in mathematics, where it is possible (however artificially) to eliminate indecision, than in wider contexts.. But even there, in many contexts, even mathematics where there is and has been a valiant effort to remove uncertainty, it is impossible to remove it entirely. This is even worse in every day living, where we must routinely and daily deal with ambiguity and indecision.
L3 includes classical 2VL as a subsystem. Any 2-valued theorem has at least one theorem of 3VL which reduces to it when uncertaintly is eliminated. Although direct proofs using either axiomatic or algebraic methods can be made, many of the indirect methods of classical logic (such as reductio ad absurdum) cannot be used in unmodified, unrestricted form. However, it should be able to develop modified versions of indirect methods and a complete system of natural deduction. This work is incomplete.
L3 logic is capable of reproducing the axioms of many other modal logics as theorems. It is not equivalent to them, because typically in modal logic, the axioms and theorems of classical 2VL are accepted entirely, while in L3, the excluded middle and related principles are not valid. Also, the strict implication used by Lewis and others is replaced by weaker but more effective strict Lukasiewicz conditional. Generally, it has theorems that other systems do not, some of which raise strong objections on the part of modal logicians, and the interpretation and semantics are markedly different..
John Halleck's web site has a description of the various recognized systems of modal logic and the axioms used to describe them. (There is also an expression evaluator for this logic. it does not include the strict conditional, butcan be supplied as [](P -> Q)
If the Lukaseiwicz strict conditional is used, and the combination ~<>P is used for negation, Heyting's axioms for intuitionistic logic can be reproduced as theorems, or tautologies, of L3.
This has connections with fuzzy logic, which represents truth values as points on the interval between 0 or F, and 1 or T. This logic absorbs all the intermediate points into the single value U.
There seem to be parallels to logic programming, in which the notation A+ is used in place of []A and A- instead of ~<>A. More detailed theoretical studies need to be done.
The logic has features of paraconsistent logic. The simple assertion (P & ~ P) is non-explosive. More detailed comparison remains to be done.