## Monday, 25 August 2014

### What’s the big deal with consistency?

(Cross-posted at NewAPPS)

It is no news to anyone that the concept of consistency is a hotly debated topic in philosophy of logic and epistemology (as well as elsewhere). Indeed, a number of philosophers throughout history have defended the view that consistency, in particular in the form of the principle of non-contradiction (PNC), is the most fundamental principle governing human rationality – so much so that rational debate about PNC itself wouldn’t even be possible, as famously stated by David Lewis. It is also the presumed privileged status of consistency that seems to motivate the philosophical obsession with paradoxes across time; to be caught entertaining inconsistent beliefs/concepts is really bad, so blocking the emergence of paradoxes is top-priority. Moreover, in classical as well as other logical systems, inconsistency entails triviality, and that of course amounts to complete disaster.

Since the advent of dialetheism, and in particular under the powerful assaults of karateka Graham Priest, PNC has been under pressure. Priest is right to point out that there are very few arguments in favor of the principle of non-contradiction in the history of philosophy, and many of them are in fact rather unconvincing. According to him, this holds in particular of Aristotle’s elenctic argument in Metaphysics gamma. (I agree with him that the argument there does not go through, but we disagree on its exact structure. At any rate, it is worth noticing that, unlike David Lewis, Aristotle did think it was possible to debate with the opponent of PNC about PNC itself.) But despite the best efforts of dialetheists, the principle of non-contradiction and consistency are still widely viewed as cornerstones of the very concept of rationality.

However, in the spirit of my genealogical approach to philosophical issues, I believe that an important question to be asked is: What’s the big deal with consistency in the first place? What does it do for us? Why do we want consistency so badly to start with? When and why did we start thinking that consistency was a good norm to be had for rational discourse? And this of course takes me back to the Greeks, and in particular the Greeks before Aristotle.

Variations of PNC can be found stated in a few authors before Aristotle, Plato in particular, but also Gorgias (I owe these passages to Benoît Castelnerac; emphasis mine in both):

You have accused me in the indictment we have heard of two most contradictory things, wisdom and madness, things which cannot exist in the same man. When you claim that I am artful and clever and resourceful, you are accusing me of wisdom, while when you claim that I betrayed Greece, you accused me of madness. For it is madness to attempt actions which are impossible, disadvantageous and disgraceful, the results of which would be such as to harm one’s friends, benefit one’s enemies and render one’s own life contemptible and precarious. And yet how can one have confidence in a man who in the course of the same speech to the same audience makes the most contradictory assertions about the same subjects? (Gorgias, Defence of Palamedes)
You cannot be believed, Meletus, even, I think, by yourself. The man appears to me, men of Athens, highly insolent and uncontrolled. He seems to have made his deposition out of insolence, violence and youthful zeal. He is like one who composed a riddle and is trying it out: “Will the wise Socrates realize that I am jesting and contradicting myself, or shall I deceive him and others?” I think he contradicts himself in the affidavit, as if he said: “Socrates is guilty of not believing in gods but believing in gods”, and surely that is the part of a jester. Examine with me, gentlemen, how he appears to contradict himself, and you, Meletus, answer us. (Plato, Apology 26e- 27b)
What is particularly important for my purposes here is that these are dialectical contexts of debate; indeed, it seems that originally, PNC was to a great extent a dialectical principle. To lure the opponent into granting contradictory claims, and exposing him/her as such, is the very goal of dialectical disputations; granting contradictory claims would entail the opponent being discredited as a credible interlocutor. In this sense, consistency would be a derived norm for discourse: the ultimate goal of discourse is persuasion; now, to be able to persuade one must be credible; a person who makes inconsistent claims is not credible, and thus not persuasive.
As argued in a recent draft paper by my post-doc Matthew Duncombe, this general principle applies also to discursive thinking for Plato, not only for situations of debates with actual opponents. Indeed, Plato’s model of discursive thinking (dianoia) is of an internal dialogue with an imaginary opponent, as it were (as to be found in the Theaetetus and the Philebus). Here too, consistency will be related to persuasion: the agent herself will not be persuaded to hold beliefs which turn out to be contradictory, but realizing that they are contradictory may well come about only as a result of the process of discursive thinking (much as in the case of the actual refutations performed by Socrates on his opponents).
Now, as also argued by Matt in his paper, the status of consistency and PNC for Aristotle is very different: PNC is grounded ontologically, and then generalizes to doxastic as well as dialogical/discursive cases (although one of the main arguments offered by Aristotle in favor of PNC is essentially dialectical in nature, namely the so-called elenctic argument). But because Aristotle postulates the ontological version of PNC -- a thing a cannot both be F and not be F at the same time, in the same way -- it is difficult to see how a fruitful debate can be had between him and the modern dialethists, who maintain precisely that such a thing is after all possible in reality.
Instead, I find Plato’s motivation for adopting something like PNC much more plausible, and philosophically interesting in that it provides an answer to the genealogical questions I stated earlier on. What consistency does for us is to serve the ultimate goal of persuasion: an inconsistent discourse is prima facie implausible (or less plausible). And so, the idea that the importance of consistency is subsumed to another, more primitive dialogical norm (the norm of persuasion) somehow deflates the degree of importance typically attributed to consistency in the philosophical literature, as a norm an sich.
Besides dialetheists, other contemporary philosophical theories might benefit from the short ‘genealogy of consistency’ I’ve just outlined. I am now thinking in particular of work done in formal epistemology by e.g. Branden Fitelson, Kenny Easwaran (e.g. here), among others, contrasting the significance of consistency vs. accuracy. It seems to me that much of what is going on there is also a deflation of the significance of consistency as a norm for rational thought; their conclusion is thus quite similar to the one of the historically-inspired analysis I’ve presented here, namely: consistency is over-rated.

1. Catarina, it must be because inconsistent statements are false, and one wishes to avoid falsity: if a view V implies P and ~P, then V is false. Dialetheism doesn't deny this - it justy says a statement may be false but also true as well. To get this to make sense, one postulates a third truth value B (= "both true and false") and defines a 3-valued logic, LP, a kind of dual to K_3. Priest's logic, LP, denies that there are contradictions: i.e., statements which are false-come-what-may (in all interpretations). For Priest, every statement, even false ones, has a model in which it is true or true-and-false. So, in LP, there are no contradictions. Similarly, in K_3 there are no logical truths (statements which are true-come-what-may).

Inconsistency plays a central role in scientific progress, as Popper argued, because to falsify a theory T one shows that T contradicts some evidence E. I.e, T is inconsistent with the evidence, and therefore if the evidence is true, then T is false. On Priest's view, it seems to me that we could, in principle, maintain that Newtonian mechanics is true (and false), despite the fact that it is inconsistent with the evidence relating to fast moving objects, time dilation, etc.

Jeff

2. The dual thought to Jeff's is: many consistent theories are incomplete, and one wishes to avoid omitting truths. If a view V fails to imply P, and P is true, then V is inadequate.

Thanks, for the post, Catarina!

3. That's a good point - with dialetheias ("both-true-and-false") around, then truth-seeking \neq falsity-avoidance!

I probably forgot all the examples Priest uses of dialetheias: unrestricted comprehension, unrestricted T-scheme, some cases of vagueness, counterfactuals with impossible antecedents, quantifying-in with impossible indirect clauses (and I think some inconsistent racist legal sentences in the old Australian constitution). Of these, last two are the ones that seem hardest to deal with using non-paraconsistent semantics.

Jeff

4. Yes, the point is precisely that it's not a given that all inconsistent statements are false! If one takes this as a starting point, that will look a lot like begging the question.

5. And btw, I'm happy to grant that, all things being equal, a consistent discourse will prima facie be more persuasive than an inconsistent one.

6. Catarina, according to dialetheism, all inconsistent statements are false. What dialetheism adds is that inconsistent statements (at least some) may be both-true-and-false - what's called B or b in the 3-valued semantics for LP. LP is essentially Kleene's 3-valued logic logic K_3, but with U ("undetermined") replaced by B ("both"); and "designated" meaning "either T or B".

Jeff

1. I'm not sure, we should check with Graham and co. My impression is that in the case of (putative) inconsistent concepts, such as the concept of set,, to say that concept X has and does not have property A is simply true.
Also, the whole point of Branden and Kenny's paper I link to above is to show that accuracy (which is aiming at the truth) trumps consistency; so if they are right, an inconsistent set of beliefs can well dominate a consistent one.

2. Graham's idea is that some instances of naive comprehension are dialetheic - i.e., true-and-false! So, e.g., he says, of the Russell paradox, starting with comprehension, $\exists x \forall y (y \in x \leftrightarrow y \notin y)$, and via $\forall y(y \in R \leftrightarrow y \notin y)$, ending with $R \in R \leftrightarrow R \notin R$, that both of these, $R \in R$ and $R \notin R$, are true-and-false.

(Thanks for mentioning the paper by Branden and Kenny. So many footnotes ... plus I'm about to drive 400 miles to England's pastures green. Plus I owe you a report; I didn't forget.)

Jeff

3. Exactly, but the statement of this equivalence $R \in R \leftrightarrow R \notin R$, which is a contradictory claim, is simply true for Graham (I think), not true-and-false.

4. oops, sorry, typo,
----
Hi Catarina, in Priest's 3-valued logic LP, the biconditional $R \in R \leftrightarrow R \notin R$ is both-true-and-false ($B$). For if $|\phi| = B$, then $|\neg \phi| = B$, and then $|\phi \leftrightarrow \neg \phi| = B$ too. Jeff

7. So then, given the above, does the dialetheism vs. consistency debate come down to whether you think truth-completeness or falsity-avoidance is a more important virtue for (a) logic to have?

1. Well, this is a special case of the fairly large debate in epistemology on epistemic norms. It is very often said that the goals of knowledge are to maximize true beliefs and minimize false beliefs, but as noted by a number of people, these two are somewhat in tension with one another. What's worse, to overgenerate or undergenerate when it comes to beliefs?

8. Perhaps if you want to pursue 'deep geneology', the Homeric poems would be worth a thought, for example in the Odyssey, Telemachus demolishes O's first idea about what to do when back in Ithika (wander around the farmsteads drumming up support) by pointing out that it would lead to quick discovery by the suitors and consequent death, something that O doesn't want. So: if your plan will lead to results contradictory to your desires, discard it (which O does).

1. I have an impression that logic and epistemology are much more important in the Odyssey and the Iliad. In the former, Athena wanders around in disguise a lot, Odysseus repeatedly tells lies about who he is, Penelope pulls a clever identity check on him which he falls for, and there's the famous logic trick where Odysseus tells the Cyclops that his name is 'Nobody' (with the wrong pitch accent, but that didn't matter). While in the Iliad, things tend to be done more by brute force and social standing than anything else. So when Thersites tries to explain to the assembled army that their best interest lies in packing up and going home, right now, O does not refute him, but just beats the crap out of him. But it would take some real work to substantiate this.

9. Nice provocation, Catarina. I wonder if Matthew has already taken into account Łukasiewicz's well-known sharp analysis of Aristotle's remarks on the Principle of [Non-]Contradiction?

BTW, in view of what I read above, I believe it is worth emphasizing here yet again that dialethism is not identical with paraconsistency. The view that inconsistencies must be false" was in fact spoused by authorities like Tarski. However, it is by no means the view reflected in most usual semantics for paraconsistent logics, in which some inconsistencies are true (and not false). Of course, one might wonder, and with a good reason, what do such semantics even mean by true', if some inconsistencies are allowed by them to be true. But that's another matter. As Zach has put it, things might often be better understood, in paraconsistent logics, in terms of falsity-avoidance: a true contradiction' might not be just a pair of sentences p and ~p whose truth has been somehow simultaneously established, but might be thought instead as a pair of such sentences whose falsity has not been ascertained.

1. Joao, by "true" you mean "designated"? If one has a truth degree set $\{\top, \dots, \bot\}$ (usually some kind of algebra, e.g., a de Morgan algebra), then is the idea that there is an inconsistency $\phi$ such that $|\phi| = \top$? What is an example? Or do you mean there is an inconsistency $\phi$ such that $\phi$ is designated (in particular, not $\top$; i.e., $\phi$ is maybe less-than-true, or true-and-false, or something like that). But this is the standard view that inconsistencies are false in some way (albeit maybe false-and-true). The epistemic/normative point of avoiding inconsistency, as I said at the start, seems to be to avoid falsity (even if it's false-and-true).
Jeff

2. Yes, Jeff, "designated" is precisely what I mean. This is indeed the usual reading in the many-valued literature. Now, if one maps all designated values into "The True" and undesignated values into "The False", the same notion of entailment is determined (this observation is known as Suszko's reduction). It seems there is no particular objective reason why one must single out some designated value as "The Real Truth" and some single out some undesignated value as "The Real False", and call all the other values "half true", "less-than-true", or something else. In your example of LP, you want to enforce the reading "both true and false" for one of the designated values. Does that change the underlying notion of entailment, or is it just façon de parler? I assume dialethism is much more than this.

Having said that, I sympathize with everything else you said, and apologize in advance if we happen to be talking past each other.

3. Joao, thanks - yes, but it looks to me like, in general, this bivalent valuation $|.|_{s}$ given by Suszko's reduction, into $\{d,\overline{d}\}$, is not truth-functional?

E.g., for LP, there is no function f such that

$|\neg \phi|_{s} = f(|\phi|_{s})$

For suppose $|\phi| = \top$ and $|\theta| = B$. Then $|\phi|_{s} = d = |\theta|_{s}$. But $|\neg \phi| = \bot$, which is not designated, while $|\neg \theta| = B$. And so, $|\neg \phi|_{s} = \overline{d}$, while $|\neg \theta|_{s} = d$. Because $|\phi|_{s} = |\theta|_{s}$, but $|\neg \phi|_{s} \neq |\neg \theta|_{s}$, there is no truth function corresponding to $\neg$ under the Suszko valuation. I don't know this topic - is this right?

Jeff

4. Indeed, Jeff, such bivalent valuations must in general be non-truth-functional. But so is the (equivalent) bivalent relational semantics of LP, that induces one to talk about sentences being both true and false', right?

All logics respecting the Tarskian axioms on consequence are both many-valued and, according to Suszko's Thesis, also bivalent at their hearts'. Not all many-valued logics are truth-functional, though. A straightforward example of that phenomenon is obtained, for example, by simply adding a nullary connective to the logic of classic implication, and imposing no extra axioms. This logic was proposed long ago by Curry, and is nowadays known not to be characterized by any sort of truth-functional semantics. However, it does have an extremely simple characterization in terms of a two-valued nondeterministic semantics (as studied by Avron and others): just let the new connective be free to take either truth-value, in a given valuation. You can find a fuller discussion about this topic here.

In case you're interested in Suszko's Thesis, you can find a number of papers about it here. It is worth stressing that nothing of value is lost when passing from a semantics based on the so-called algebraic values' to an equivalent one based on the so-called logical values': the new bivalent semantics may indeed be constructed in such a way as to inherit pretty much all nice computational properties of the truth-functional semantics originating it. In fact, going bivalent has some practical advantages, first hinted by Dummett long ago. For instance, it is in principle much easier to compare two logics uniformly presented by way of a bivalent semantics than to do it with two logics presented by way of truth-functional semantics with a different number of truth-values, or with the same number of truth-values but different number of designated values.

5. Yes, thanks, Joao - I looked this up now, including the Arnon et al. work on "non-deterministic many-valued logic". The focus in this material are the consequence relations $\vdash$, $\vDash$. So, a logic has the form $(L,\vdash,\vDash)$, and, as you say, the underlying semantic functions, etc., can be varied keeping the overall "consequence facts" fixed. But still I'm not keen on reducing a truth predicate $\mathsf{True}(x)$ to a designatedness predicate, $\mathsf{Designated}(x)$. The main reason I'd give is the presence of the truth axiom for $\neg$:

$\mathsf{True}(\neg x) \leftrightarrow \neg \mathsf{True}(x)$

which, informally, says, "negation commutes with truth". But this won't hold for "designated". I.e.,

$\mathsf{Designated}(\neg x) \leftrightarrow \neg \mathsf{Designated}(x)$

won't hold - for there are formulas $\phi, \theta$, both of which are designated (e.g., one is $B$, one is $\top$), but $\neg \phi$ is designated while $\neg \theta$ is not designated ... So, this compositionality condition suggests not identifying truth with "designatedness".

(Admittedly, a fully compositional (classical) truth theory is inconsistent; I don't know what happens if one formulates fully compositional truth in some paraconsistent logic. Volker Halbach and Leon Horsten reformulated the Kripke-Feferman truth theory $KF$ in partial $K_3$ logic, and Volker tells me the result is so complicated that it's extremely hard to make sense of, either in terms of being practically usable, or in terms of its proof-theoretic analysis.)

Jeff

6. One extra thing maybe I'd add on the lattice of truth values/degrees: usually, for the set $V$ of truth degrees, we have some kind of bounded lattice $(V,\leq)$, with $\top \in V$ as greatest element and $\bot \in V$ as least element (usually complemented and distributive). For truth degrees $u,v \in V$, then the partial order,

$u \leq v$

expresses that "$v$ is, in some sense, at least as close to $\top$ as $u$ is, possibly closer". This is what motivates the idea that truth degrees strictly below $\top$ are somehow less-than-fully-true.

Jeff

7. Yes, the order-based notion of entailment is indeed a nice alternative to the matrix-based notion of entailment (and this is a nice paper by Font on the matter). In the case of logics such as First-Degree Entailment, however, the two perspectives characterize the same notion of consequence iff one partitions the four underlying truth-values by choosing precisely true' and both-true-and-false' as designated. For other partitions, taking as designated nothing but the true' or its dual, the matrix-based approach seems to serve better than the degree-based one.

10. I believe I got a better glimpse now about what really worries you, Jeff. I suspect a proper truth theory' had better simultaneously deal with Truth and Falsity predicates. I confess I am not very much into this, for I favor the view that logic is about consequence, and truth-talk is just one way of getting there. In particular, if two different semantics characterize the same consequence relation, I find it hard to prefer one over another if not in terms of how easy they might render the proofs of interesting meta-theoretical results. In any ase, one might well take the characterization of truth-for-a-logic' as a prime goal of investigation for Abstract Algebraic Logic. I wonder how much investigation has been done on truth-predicates from the latter perspective? (there was a talk precisely about this issue by Moraschini scheduled at the VSL this year, but unfortunately I have missed it)

On what concerns truth-functionality, do note that interesting examples of non-truth-functional logics need not have anything to do with a negation connective. Besides the example (by Curry) I gave before, where no primitive negation is present in the language, you might also recall the modal logics S1-S3 (by Lewis), where negation is fully classical. I wonder if there are particularly interesting challenges arising for truth theories from the study of such really non-truth-functional logics, and not only of logics whose circumstantial semantics happens to be presented in non-truth-functional terms?

I reckon you think the problem of consistency', which originated the present post by Catarina, is of particular importance for truth theories (irrespective of the presence of negation in the language)? How much do you think this study should be sensitive to a particular choice of semantics, or to a particular way of reading the underlying truth-values? Thanks for your thoughts on the matter!

11. Joao, I think Catarina's focus here and generally is primarily normative/epistemological, and logic(s) are to be connected to reasoning, interaction and/or persuasion. So, on that view of "logic-as-reasoning/debate", why should showing an interlocutor's position P inconsistent be persuasive? I think the answer is that P's inconsistency implies its not-being-strictly-true; its being, to some extent, false (even if not full-falsity). Then an epistemological norm of falsity avoidance explains why inconsistencies are to be rectified or avoided.

(As Zach added, this norm may omit truths ... e.g., dialetheic ones. And this point is much more general, because aiming at comprehensiveness will likely generate inconsistencies between competing theories. This is a Popperian point. A policy of trying to maintain consistency at all costs will then impede progress.)

For me, however, logic is the mathematics of strings, semantic values and consequence relations ... and so I don't even connect it to normativity or even reasoning at all!

Jeff