# Are temporal logics neccessarily paraconsistent?

If a logic is temporal presumably conflicting accounts of the 'world' can be given. This seems the same territory that is occupied by paraconsistent logics.

Hence the question: Are all temporal logics paraconsistent?

Disclaimer: I know very little about either forms of logics.

• -1, I'm both the down-voter and the answerer here (an odd position). This seems like something you could have cleared up with a quick Google search. – Dennis Jun 19 '13 at 0:53
• I think the question should be charitably interpreted such that the formula [Fp & F~p] would be paraconsistant. There are some papers on the NOT website, which explores extensions of Aristotle's square of opposition, which interprets negation in paraconsistant logic as ~□ in classical modal logic. – Kevin Holmes Aug 31 '13 at 20:58
• @KevinHolmes It was the "necessarily" that I responded to and thought could be cleared up quickly. JMarcos does bring up an interesting point in his answer which seems to expand upon your point here (that modal logics can be given paraconsistent interpretations). – Dennis Sep 1 '13 at 10:43

No, they aren't.

Temporal Logic

Paraconsistent Logic

The most popular temporal logics are modal logics. Certain applications might make marrying the two natural (see, e.g., this abstract), but nothing about temporal logic requires paraconsistency.

Any of the usual normal modal logics may be reformulated in a language extending positive classical logic by the addition of a non-classical negation ⌣ interpreted as follows, for any world w of a model M:

`M, w ⊩ ⌣ψ  iff  M, w' ⊮ ψ for some world w' of M accessible from w`

It is easy to check that such non-classical negation ⌣ is indeed ⌣-paraconsistent (if your model has more than one world, it might satisfy an atom p together with its negation ⌣p, without simultaneously satisfying all other formulas), and straightforward to use the above language to define classical negation (define a top element ⊤ as an arbitrary theorem, say α→α, and define ∼α as α→⌣⊤), and the usual modal operators box and diamond (define □α as ∼⌣α, and define ◇α as ⌣∼α). Conversely, it is obvious how to define the paraconsistent negation ⌣ using the classical negation and one of the usual modal operators (define ⌣α as either ∼□α or ◇∼α).

To be sure, this means that any of the usual normal modal logics may be entirely rewritten in a language containing just implication and an indigenous paraconsistent negation. One could argue, in this sense, that normal modal logics are paraconsistent. A similar assertion might be made about temporal logics, where the satisfaction of ⌣ψ at instant t means that there is some future (or past) instant in which ψ is not satisfied.

Notice, moreover, that the satisfaction condition for the above mentioned paraconsistent negation is dual to the standard satisfaction condition for intuitionistic negation.

The characteristic feature of paraconsistent logics is their rejection of the principle of explosion, which says that from a contradiction anything follows. Dennis linked to a survey article of various logics belonging to this family (see esp. §§3.6–7). Relevant for our purposes here is a fact mentioned in §1:

Fact 1. Many paraconsistent logics validate ¬(φ ∧ ¬φ) (aka the law of non-contradiction).

This means that paraconsistent logics don't necessarily tolerate conflicting accounts of the world, i.e., there are paraconsistent logics that validate no sentence of the form φ ∧ ¬φ.

Now, there is a general philosophical view known as dialetheism, according to which there are true contradictions, i.e., some formulas of the form φ ∧ ¬φ are satisfiable! It seems to me that the 'territory' you speak of is not paraconsistent logic, but rather this more general dialetheist view. That's not to say that the two aren't related: since dialetheists tolerate contradictions, in order for their theories not to be trivialized, they must reject the principle of explosion. This makes paraconsistent logics a natural platform for those who want to create dialetheist-friendly logics.

Temporal logics, as Dennis already mentioned, are usually just modal logics in disguise. Depending on your intuitions about the structure of time, you choose as your base from a number of different modal logics. None of the modal logics I've come across have been paraconsistent, but such a combination might, of course, be possible and even turn out to have useful applications. Although the existence of non-paraconsistent temporal logics (e.g. the Priorean tense logic) already answers your question negatively, I understand your question arose from the observation that in temporal logic, a formula φ can be true at time t and be false at time t'>t and then maybe again true at time t''>t', and so on. But fortunately there is no conflict here at all. For non-dialetheist temporal logics this is generally true:

Fact 2. For any moment t and any formula φ: either t |= φ or t |= ¬φ.

This just follows from the fact that the base modal logic is classical and thus accepts the law of non-contradiction. Of course, this wouldn't generally be true for paraconsistent temporal logics and would never be true for dialetheist temporal logics.