These tables seem to capture the relationships involved and thus allow us to form and evaluate complex sentences built up of statements that may be neither true nor false.
Truth Value Gluts
There is also another, quite different, reason for being attracted to three valued logics. Consider the familiar troublesome sentence: 'This sentence is false'. If it is true, then by virtue of its meaning, it must be false. If it is false, then since it claims this very fact, it must be true. In either case, it would appear to be both true and false. We might then be tempted to have a new truth value to represent being both true and false. We could denote it by 'b' for 'both'.
One objection to this is that it is a mistake to assume that the liar sentence must be true or false — perhaps it is neither. This would lead us once again to truth value gaps. Alternatively, we might think that there is some other form of mistake here which is not immediately analysable, but which will be solved at some point in the future. This is quite plausible and thus philosophers need not be committed to truth value gaps or gluts.
Perhaps surprisingly, the truth tables for three valued logic using b turn out to be exactly the same as those using n. However, the difference between the two logical systems comes in the way that proofs are deemed acceptable. Classically, we think that the hallmark of logical validity is truth preservation: for an argument to be valid, it must be impossible for its premises to be true without the conclusion also being true. There are two obvious ways to generalise this notion to three valued logics and these correspond to whether or not we think of the third value as being true.
Thus, if our third value is n, the appropriate constraint on truth preservation is that valid arguments cannot start with premises valued 1 and deliver a conclusion valued 0 or n — valid arguments cannot take us from something with truth to something without. On the other hand, if the third value is b then it possesses truth so, the truth preservation constraint dictates that valid arguments cannot start with premises valued 1 or b and deliver a conclusion valued 0.
In the first case, n acts much like 0 and in the second case, b acts much like 1. We say that in the first case 1 is designated while n and 0 are not. In the second case 1 and b are both designated while 0 is not. Being designated corresponds to having the truth that we want to preserve in logical implication. Thus we can say that valid arguments can never take us from designated premises to non-designated conclusions.
Another, particularly important difference stems from the role of contradictions in these logics. In classical logic, it is well known that contradictory premises can entail any conclusion — a law known as ex falso quodlibet. This seems rather bizarre, but is typically brushed over on the grounds that we should never adopt contradictory premises in the first place. However, life is not always that easy, and sometimes contradictory premises sneak in unawares or are foisted upon us. For example, unbeknownst to its creators, the first formulation of calculus (which used infinitesimal quantities) was inconsistent. So too are the national laws that govern us. Even small sections of such laws typically contain many inconsistencies. For example, it can be shown in the British Immigration act that certain people both can and cannot become citizens.
Non-classical logics can deny the rule of ex falso quodlibet and those that do so are known as paraconsistent logics. An important example of such a logic is the three valued logic with b. This allows us to reason formally about such systems as the original infinitesimal calculus or the British Immigration act without being reduced to gibberish. Contradictory conclusions can be drawn, but they are limited to the the area in which the contradictions arose and the paraconsistent reasoning can therefore be more robust. Thus, even if one does not believe that statements can be both true and false, it may still be tempting to use a logic involving b for pragmatic reasons.
Given the reasons both for truth value gaps and gluts, one might wonder why we shouldn't allow both. Indeed there is a system of four valued logic known as FDE which allows sentences to be true, false, both or neither. We can then define the standard logical operations using the following diagram.
The negation of a value is the value found by reflection in the line between n and b. Thus 1 and 0 swap places, while n and b are unchanged by negation. The disjunction of two values is their least upper bound and the conjunction of two values is their greatest lower bound. Note that the partial ordering used here ranks values according to increasing truth and decreasing falsity. Thus disjunction can be said to maximise truth and minimise falsity, while conjunction minimises truth and maximises falsity. These definitions can be seen to agree with our intuition and can also be represented as truth tables:
2) Philip Welch (2001). "On Gupta-Belnap revision theories of truth, Kripkean fixed-points, and the next stable set". Bulletin of Symbolic Logic 7(2001), 345-360.
2) Volker Halbach and Leon Horsten (2005), "The deflationist's axioms for truth", in Beall, J.C and Armour-Garb (2005), Deflationism and Paradox, Clarendon Press, 203-217.
Between the end of the 19th century and the beginning of the 20thcentury, the foundations of logic and mathematics were affected by thediscovery of a number of difficulties—the so-calledparadoxes—involving fundamental notions and basic methods ofdefinition and inference, which were usually accepted asunproblematic. Since then paradoxes have acquired a new role incontemporary logic: indeed, they have led to theorems (usuallynegative results, such as unprovability and undecidability) and theyare not simply confined to the realm of a sterile dialectic. Severalbasic notions of logic, as it is presently taught, have reached theirpresent shape at the end of a process which has been often triggeredby various attempts to solve paradoxes. This is especially true forthe notions of set and collection in general, forthe basic syntactical and semantical concepts of standardclassical logic (logical languages of a given order, the notionof satisfiability, definability). After the first forty years, theby-products of the paradoxes included axiomatizations of set theory, asystematic development of type theory, the foundations of semantics, atheory of formal systems (at least in nuce), besides theintroduction of the dichotomy predicative/impredicative whichwas important for conceptual reasons, but also for the future of prooftheoretical methods.
The latter concerns logic and the paradoxes.
We argue that what one can and must say about the logic of truth and the Liar paradox is influenced, or even in some cases determined, by what one says about the metaphysical nature of truth.
Truth Vagueness And Paradox An Essay On The Logic Of Truth
This Logos is just as much God as is the Father, since power or substance cannot exist without form. But form also cannot exist without substance and power to extricate it from infinity and render it actual; so that the Father and the Son are not two separable existents, but two incommensurable and equally original features of existence itself. . . .
Truth, Vagueness, And Paradox: An Essay On The Logic Of Truth
This simple dissolution of superstition yields three of my realms of being: matter . . . ; essence . . . ; and spirit . . . . There remains the realm of truth . . . .