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What are the authoritative authors when it comes to more than true and false in logics?

I know that in Aristotle's logic there is no third option:

This corresponds to a modern notion of logical consequence: X results of necessity from Y and Z if it would be impossible for X to be false when Y and Z are true.


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You can look at the entry on Multivalue logic in the Stanford Encyclopaedia of Philosophy. – Guido Jan 6 '13 at 21:52
By "more than true and false" do you mean multi-valued logics, i.e. logics which allow for more than two truth-values? The passage you quote about logical consequence is only remotely connected to this topic. And it is an open question whether Aristotle did indeed think of a third truth-value ("undecided") regarding assertions about contingent future events; the SEP entry you quote addresses this as well. – DBK Jan 6 '13 at 21:52
also look at intuitionistic logic, dialithism & modal logic. – Mozibur Ullah Jan 7 '13 at 1:05
@Guido that's also what I was looking for, a summary. Thanks! – Camil Staps Jan 7 '13 at 7:41
To expand on @MoziburUllah, Dr Graham Priest has written extensively on dialetheism. He doesn't approach it as much with multivalue logics, but rather with paraconsistent logics. – wmjbyatt Jan 8 '13 at 4:51
up vote 4 down vote accepted

Aristotles logic is more complex than your brief question suggests. He's popularly known for the syllogism. He also enunciates the law of the excluded middle, but he then goes out to point out that this cannot apply to our knowledge of the future where possibility is the rule.

A major goal of his book On Interpretation is to discuss the thesis that, of a pair of proposition one of which asserts what the other denies, is that one and only one can be true, and the other false. However he discusses an example where this is not possible because of the nature of time, the future is not actual but a spectrum of possibility. Contemporary logic has a subdiscipline called modal logic that discusses this.

It has been suggested he adopted, or at least flirted with, a three-valued logic for future propositions, or that he countenanced truth-value gaps, or that his solution includes still more abstruse reasoning. There are more points of contact between contemporary logic and that of Aristotle than the pre-modern commentators suggested which focused more on the syllogism and the law of non-contradiction that he enunciated in his book metaphysics.

Dialethism is the position that there are true contradictions, of which one form is Paraconsistent Logics which denies the principle of explosion, that is given one true contradiction then every proposition is true is a major contemporary research area. Dignaga and Dharmakirti founded and developed the Buddhist school of logic which was dialethistic. This allowed them to assert that atoms were spherical but also point particles (ie without extensions)

Brouwer reacting against Hilberts formalism advocated intuitionism that denied the validity of the law of the excluded middle, this was formalised into Heyting algebra which codified the laws of intuitionistic logic in the same way that boolean algebra codified that of classical propositional logic. Contemporary work in this discipline is done in Topos Theory in which intuitionistic logic is hybridised with that of type theory, amazingly enough it also has a geometric interpretation (as a category of sheaves on a space).

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This is true and I knew that, but it's not what I asked for. But thanks for pointing that out to other members of the community! – Camil Staps Jan 7 '13 at 7:41
@Staps: How is it not answering the question? I wrote about the major figures in buddhist & mathematical logic; and contemporary programmes in non-classical logic and introduced my answer by pointing out why Aristotle was aware that there was more to true/false than true/false? – Mozibur Ullah Jan 7 '13 at 12:05
Eh, right, missed that - sorry, dont know what happened. – Camil Staps Jan 7 '13 at 12:10

The property that a logic has when the only truth-values are "True" and "False" and every well-formed formula must be one or the other is called "Bivalence". There are a number of ways to deny bivalence.

Two of the most popular options are to recognize the existence of truth-value "gaps" or truth-value "gluts" (or both).

Accepting gaps, as the name would suggest, is to accept that there are some statements which are neither true nor false--- they are indeterminate. You could treat indeterminacy as a third truth value, or as simply the lack of truth or falsity. As far as I know there isn't much of a philosophical reason to preference one approach or the other. The difference corresponds to a choice of semantic apparatus. If you take the assignment of truth-values to statements to be functional then you treat indeterminacy as a third truth-value. If the assignment of truth-values to statements is relational then you can avoid positing a third truth-value by simply having some statements relate to neither truth nor falsity.

The issues are much the same for truth-value gluts except that accepting gluts means that you accept some statements are both true and false. Dialetheism is the philosophy most commonly associated with the existence of truth-value gluts.

For further information of these issues and exploration of technical systems of both sorts I recommend Graham Priest's An Introduction to Non-Classical Logic. The book presents these systems in a tableuax style of proof, which combines sytactic and semantic proof. It has the disadvantage of being somewhat unnatural (it doesn't resemble normal reasoning as much as, say, a natural deduction system) but it is rather easy to get accustomed to.

EDIT: Since I can't comment yet, I'll note here that as the other answerer indicates, it is unclear whether or not Aristotle accepted that there were truth-value gaps (he almost certainly didn't accept gluts). One of the areas where it seemed that he accepted truth-value gaps is in his discussion of future contingents. You can find discussion of this issue in his De Interpretatione.

EDIT 2: I just noticed that you were specifically asking for authors, so here are some:

Graham Priest for Dialetheism/Truth-Value Gluts

L.E.J. Brouwer and other intuitionists for Truth-Value Gaps

Petr Hajek on Fuzzy Logic (i.e., infinitely many valued logic)

Saul Kripke on his theory of truth and the semantic paradoxes (gaps)

Hartry Field on the semantic paradoxes (gaps)

Those are the big figures I can think of off the top of my head.

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