12 Nov 2016 Version
The logic outlined here is essentially that of Jan Lukasiewicz, with one added definition. The notation is based on that of C.I. Lewis and the logic is formally analogous to his system S5, although not identical to it. Lewis accepted the Law of the excluded middle as a general law, while Lukasiewicz rejected it. This system uses a different defnition of strict implication than Lewis used. This logic is truth functional, while the Lewis logics are not.
There are three truth values, symbolized T, U, and F (for true, uncertain, and false). The meaning of these truth values is disctated and constrained by the rules of the logic and does not necessarily corresond exactly to intuitive notions.
T is used for statements that are definitely, certainly or necessarily true. It is not possible, in this logic, to make a distinction between statements that are necessarily true and those that are contingently true. This would require a logic of at least four values, but this is beyond the scope of this treatment.
U is used for statements that are uncertain, undecided or doubtful. Given the ambiguities associated with these terms, none of them corresponds precisely. Another term, equivocal, can be used to describe statements with this truth value. This could be used for statements that are contingent, or contingently true or false, provides one forfeits the ability to distinguish between contingently true and contingently false. Use of this truth value is strictly incompatible with the classical Law of the excluded middle.
F is used for statements that are definitely, certainly or necessarily false, or impossible. As with true statements, it is not possible using only three truth values to consistently make a distinction between necessily false and contingently false.
In spite of the limitations of this logic, the ability to consistently and practically extend classical logic to three truth values is a significant step forward.
There are three principal unary operators, with the following truth tables:
This is a standard negation. It will be noted that if a statement is considered equivocal, it has the same truth value as its own negation. If "there will be a sea battle tomorrow" has the value of U, then so does its negation "There will not be a sea battle". This means that an equivocal statement and its negation are truth-functionally equivalent, even though they are semantically distinct. Although this may seen counterintuitive, there are technical advantages to adopting this treatment.
This corresponds to a generalized "not impossible". It does not have the same meaning as "possible" or "Concevable" in the philosophical sense. Although a statement "<>P" could intuitively be uncertain, signifying that it is not known whether P is possible or not, in this logic it is treated as a two-valued statement that must be either true or false.
This corresponds to "Necessarily" or "certainly" true. The various properties of statements do not correspond exactly to "necessary" in the traditional sense of alethic modality. Rather, an epistemic modality "It is certain" that P is true is indicated.
There are other statements that can be expressed by combinations of these.
The classical law of double negation holds true in this logic
This is a "strong" negation, signifying that P is impossible. It is only true when P has a value of false.
This is a "weak" negation, signifying that P is not necessarily true.
P and <>P are equivalent to P.
<>P and <><>P are equivalent to <>P. Modal operators to the left of the one closest to the expression are superfluous.
~<>~P is equivalent to P, and ~~P is equivalent to <>P.
There are only sixteen possible binary operators (2 ^ (2x2)) in classical two-valued logic, and it has been established that they can all be defined using only one. Three valued logic is far more complex, with 3 ^ (3x3) possiblities. While the interdefinability of the binary operators in classical two valued logic has more or less effaced the distinction between mathematical operators and mathematical relations, it is useful in three valued logic to maintain that distinction, and not treat the relations as combinations of operations.
There are two principal binary combination operators:
This is the inclusive or: Its value is the most true of the component variables or expresssions
& (and) This has the value of the least true of the component variables or expressions
There are four relational operators.
The first is implication or the conditional "if--then". This is the definition used by Lukasiewicz It should be noted that this is not the same as the material conditional of classical two-valued logic. (~P \/ Q) and cannot be reduced to it. It should also be noted that this allows conditionals (P->Q) to take values of U, which has important implications for the practical utility of straight Lukasiewicz logic.
The second is the strict Lukasiewicz conditional, definable as (P -> Q). This is definition mentioned earlier, and its use makes a vast improvement in the practical utility of Lukasiewicz logic. This is not the same as Lewis strict conditional, (given by his fishhook symbol -3), which can be defined as ~<>(P & ~Q). As a matter of fact, (P -3 Q) => (P =>Q) but not conversely. The statement (P =>Q) can be understood as a claim that Q is no less true than P. It shares this property with the classical two valued material conditional.
It may deserve emphasis that the equivalence of the two-valued material conditional (P -> Q) to the formula ~P \/ Q is an artifact of two-valued logic and is seriously misleading when the attempt to generalize the rules of logic to three or more values is made. The essence of deductive reasoning, that valid rules of deduction do not introduce error that is not present in the original assumptions, is best preserved using this version of the conditional..
The third is the Lukasiewicz biconditional. ((P -> Q) & (Q ->P)). This is actually not very useful. It employs the idea that logical equivalence is a biconditional, "P if and only if Q" but it is not sufficient to express truth functional equivalence "P has the same truth value as Q".
The fourth is logical equivalence, or the strict Lukasiewicz bioconditional, defined as either (P <->Q) or ((P =>Q & Q =>P)) This, like the classical two-valued biconditional, expresses the logical equivalence of the biconditional, and truth functional equivalence.
Althought the system is technically quite simple, it has a number of important consequences and implications.
As in classical two valued logic, formulas that are tautologies, true in all cases regardless of the values of the logical variables, may be considered laws of logic.
Another is that the laws of bivalence and the excluded middle are not laws of this logic. They are conditional statements which apply to some propositions, but not others. (P \/ ~P) and ~(P & ~P) and equivalent statements do not necessarily hold. Except for these, however, the commutative, associative, and distributive, the various laws of negation, and De Morgan's laws that are familiar from Boolean algebra do apply. Although this does not correspond to a Boolean algebra, it does have the properties of a de Morgan algebra.
The modified statment <> (P &~P) is a claim that P has the middle truth value. (P & ~P) is a contradiction. It is not obligatory to accept contradictories as true. It may be required to suspend judgement and accept them as equivocal. Likwise, <>~(P & ~P) is a permissive noncommittal statement that is true regardless of the truth value of P. ~ (P & ~P) or equivalently ~<>(P & ~P) is a statement that the middle value is exluded, in which case classical two valued logic applies.
A number of attempts have been made to show that Lukesiewicz logic is contradictory, by assuming that a statement has the third middle value, and then claiming that it allows <>(P &~P) and is thefore wrong. This is erroneous. In three valued logic, P & ~P are not necessarily contradictory. The contradiction is in allowing the middle on the one hand , and then trying to exclude it again on the other. If "There will be a sea battle tomorrow" and "There will not be a sea battle tomorrow" are both allowed to have a value of U, they are truth-functionally equivalent, and there is no point in trying to claim that "there will and won't be a sea battle tomorrow" is contradictory and impossible. For generality and consistency, this statement must also be left with a truth value of U until it is determined whether there will or will not be a sea battle. Intuition is a perilous guide when it comes to mixing principles of two-valued and three-valued logic.
~<>P <=> ~P
"Not possible" is equivalent to "Certainly not"
~P <=> <>~P
"Not necessarily" is equivalent to "Possibly not".
P => P
If a statment is certainly, or necessarily true, then it is true. Not conversely.
P => <>P
If a statement is actually or necessarily trrue, then it is possible. Not conversely.
The logical principles of modus ponens and transitivity must be modified.
P & (P->Q) -> Q (if P and "if P then Q", then we may conclude Q)
((P -> Q) & (Q -> R)) -> (P ->R) if "if P then Q" and "If Q then R", then "if P then R"
These principles were observed not to hold for Lukasiewicz logic, which was long thought to be an almost insuperable obstacle. It is true that these do not hold. However, they should not. It may be recalled that the Lukasiewicz conditional allows conditional statements to be doubtful. Allowing modus ponens and transitivity with doubtful conditionals can indeed result in doubtful, even false conclusions. The remedy, one it has been seen, is quite simple and obvious in hindsight: Forbid doubtful conditionals and require them to be strict.
(P & (P=>Q)) =>Q) and
((P =>Q) & (Q=>R)) => (P=>R) are both valid.