Alethic undecidability doesn’t solve the Liar

Stephen Barker (2014) presents a novel approach to solving semantic paradoxes, including the Liar and its variants and Curry’s paradox. His approach is based around the concept of alethic undecidability. His approach, if successful, renders futile all attempts to assign semantic properties (truth, falsity, gap or glut) to the paradoxical sentences, whilst leaving classical logic fully intact. And, according to Barker, even the T-scheme remains valid, for validity is not undermined by undecidable instances. 
 
Barker’s approach is innovative and worthy of further consideration, particularly by those of us who aim to find a solution without logical revisionism. As it stands, however, the approach is unsuccessful, as I shall demonstrate below.


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( ) presents a novel approach to solving semantic paradoxes, including the Liar and its variants and Curry's paradox. His approach is based around the concept of alethic undecidability. His approach, if successful, renders futile all attempts to assign semantic properties (truth, falsity, gap or glut) to the paradoxical sentences, whilst leaving classical logic fully intact. And, according to Barker, even the T-scheme remains valid, for validity is not undermined by undecidable instances.
Barker's approach is innovative and worthy of further consideration, particularly by those of us who aim to nd a solution without logical revisionism. As it stands, however, the approach is unsuccessful, as I shall demonstrate below.

Barker's Account
Barker takes as his starting point a version of the truthmaker principle ( , ): (TM) If a sentence is true (or false), then it is true (or false) in virtue of non-alethic facts.
A non-alethic fact is something like a state of affairs not involving the properties truth or falsity: that students drink is one such fact, whereas that the proposition ⟨that students drink⟩ is true is not, since the latter involves the property being true.
Truth or falsity requires a sentence to connect, eventually, to non-alethic facts. Barker ( , ) offers two examples of this connection failing. Consider the following in nite sequence of sentences: ⋮ Or, consider the truth-teller sentence: (T) Sentence T is true.
In each case, no sentence has the right kind of connection to non-alethic facts. They are ground-unspeci able (Barker , ). Notice that there's nothing paradoxical about such cases: we can consistently assign them all the value true or all the value false. But Barker claims, based on (TM), that we should not, for truth or falsity require the kind of connection to non-alethic reality that's missing in each case.
He then argues as follows: Ground-unspeci able sentences are undecidable with respect to their grounding status. Given TM -which says that all alethic properties must be grounded -it follows that such sentences are undecidable with respect to their alethic properties. . . . Call this undecidability with respect to alethic properties alethicundecidability or A-undecidability for short. (Barker , ) As a consequence, for each example sentence above, it is undecidable whether it 'is either true or false, lacks truth and falsity, is both true and false, and so on' (Barker , ). Alethic undecidability thus implies unassertability: 'we cannot assert, in principle, whether they are true, false, either true or false, neither true nor false, both true and false, and so on' (Barker , ). Barker then applies this reasoning to the (strengthened) Liar sentence: (L) L is not true; and to the Curry sentence: (C) If C is true, then (where is any absurd sentence you like). The claim is that both L and C are ground-unspeci able, hence A-undecidable. Moreover, according to Barker ( , -), all attempts at 'revenge' fail under this analysis. So, for example: (Barker , ). But this result doesn't imply that R is true, since for any A-undecidable sentence, we're banned from saying whether it 'is either true or false, lacks truth and falsity, is both true and false, and so on' (Barker , ). That, at least, is Barker's story. In this paper, I'll demonstrate that Barker's approach is awed. I'll do so using only A-decidable reasoning from A-decidable premises. That guarantees our conclusions will likewise be A-decidable and hence assertable. That's because valid reasoning (from nite premises, at least) preserves A-decidability. For if the premises are A-decidable, then so is their conjunction; and if the reasoning is valid, then the conditional from the conjoined premises to the conclusion is valid, hence A-decidable. Since logical proof is one way to decide the truth of a statement, that conditional itself must be A-decidable; and a conditional with an A-decidable antecedent must have an A-decidable consequent. It follows that A-decidable reasoning from A-decidable premises results in A-decidable conclusions.
My case against Barker is in two steps. First, we can in certain cases establish (in an A-decidable way) facts about a sentence's dependencies ( § ). I then argue ( § ) that reasoning in this way allows us to reinstate the (strengthened) Liar paradox.

Dependency Graphs
We can establish, in an A-decidable way, certain facts about dependency between entities, including sentences like L and C. We can use these facts to build directed dependency graphs for the sentences in question.
The idea is simple: when entity e 1 depends on e 2 , we put an arrow from the e 1 node to the e 2 node in our graph. For full generality, we would need to capture different senses of dependency. (An entity may depend on another for its existence; but in the case of sentences, we often talk about S 1 depending on S 2 for its truth.) We could capture different dependency relations in our graphs with different colour arrows.
Full generality would also require us to capture the distinction between full and partial dependency. (A ∧ B) ∨ C's truth depends on the truth of A, B and C, but in different ways: C is a full ground, whereas A and B are individually mere partial grounds, but jointly a full ground, for (A ∧ B) ∨ C. We can track the difference using an AND-OR graph structure, familiar from the technical literature.
We needn't go into the details here, however, as the cases we're interested in -L, C and the R i s -involve simple -dependencies. Their graphs are as follows: We may interpret an edge A → B as saying that A's truth-value depends on B's truth-value; or alternatively, as saying that the fact that A depends on (or obtains in virtue of) the fact that B (obtaining). For our purposes, it doesn't matter.
Let's focus on the nite graph for L for the time being. It's clear that the process of associating a sentence like L with its graph is algorithmically decidable, given the facts (i) that the only referring term in L is 'L'; and (ii) that 'L' refers only to L. Fact (i) concerns syntax only and so is clearly A-decidable. Fact (ii) is about reference. It would be absurd to suggest that the reference-facts are A-undecidable in these cases. For that would imply that we can't assert that L is self-referential and hence that the set-up to the paradoxes cannot even be stated. Putting these points together, the association of L with its graph is A-decidable. (Similarly for C and its graph.) Given that these graphs are determinate, nite structures, we can clearly make determinate statements about them. We can say, for example, that both L's and C's graph consists of a single node and that neither graph has a leaf node. And from such facts, we can infer further facts about L's and C's dependencies: that L does not depend on C, for example. And crucially, we infer such facts from A-decidable premises using only A-decidable reasoning, hence our inferred conclusions are A-decidable.
One class of facts we can determine, using this kind of reasoning, concern whether given nodes correspond to alethic or to non-alethic facts. Since nodes are sentences, this amounts to a syntactic check for the predicates 'is true' and 'is false'. Clearly, that check is A-decidable. (And even if we thought the class of alethic predicates was open-ended, or otherwise A-undecidable, it's clear that 'is true' and 'is false' are alethic predicates. So we clearly have a suf cient condition for a node's representing an alethic fact.) This allows us to infer, A-decidably, that L's graph's sole node states an alethic fact and hence that no node in the graph states a non-alethic fact. Such inferences will be crucial in the argument below.
It isn't so clear whether this kind of reasoning applies to the in nite graph associated with R 1 . It isn't in general algorithmically decidable whether such graphs contain such-and-such nodes. (A tree automaton search for such nodes may never terminate.) But now consider how the regress case is presented. We give its rst few cases, R 1 , R 2 , R 3 explicitly; and then we say '. . .' or 'and so on'. By this, we mean to say that the case consists of in nitely many sentences, one for each n ∈ N, of the form: (R n ) Sentence R n+1 is true.
If that is the implicit content of the speci cation of the case, then we have a general premise from which to reason. We infer that each R n depends only on R n+1 and that there is no last R n . From this, it follows mathematically (and hence A-decidably) that R 1 's graph has certain features: in particular, that each node states an alethic condition and hence that there is no branch in the graph terminating in a non-alethic node. This kind of reasoning allows us to reason about in nite cases, including Yablo's paradox.