1 Introduction
This paper is a footnote to [RRM13]. [RRM13] is perhaps best described as a graph theoretical analysis of Yablo’s construction, see [Yab82]. We continue this work.
To make the present paper selfcontained, we repeat the definitions of [RRM13]. To keep it short, we do not repeat ideas and motivations of [RRM13]
. Thus, the reader should probably be familiar with or have a copy of
[RRM13] ready.All graphs etc. considered will be assumed to be cyclefree, unless said otherwise.
1.1 Overview

In Section 2 (page 2), we show that conjecture 15 in [RRM13] is wrong. This conjecture says that a directed graph is dangerous iff every homomorphic image of is dangerous. (The definitions are given in Definition 1.1 (page 1.1), (3) and (11).)
To show that the conjecture is wrong, we modify the Yablo construction, see Definition 1.3 (page 1.3), slightly in Example 2.1 (page 2.1), illustrated in Diagram 2.1 (page 2.1), show that it is still dangerous in Fact 2.2 (page 2.2), and collaps it to a homomorphic image in Example 2.2 (page 2.2). This homomorphic image is not dangerous, as shown in Fact 2.1 (page 2.1).

In Section 3 (page 3), we discuss implications of Theorem 24 in [RRM13]  see the paragraph immediately after the proof of the theorem in [RRM13]. This theorem states that an undirected graph has a dangerous orientation iff it contains a cycle. (See Definition 1.1 (page 1.1) (4) for orientation.)
We show that for any simply connected directed graph  i.e., in the underlying undirected graph from any two vertices there is at most one path from to see Definition 3.2 (page 3.2)  and for any denotation for we find an acceptable valuation for and
The proof consists of a mixed induction, successively assigning values for the and splitting up the graph into ever smaller independent subgraphs. The independence of the subgraphs relies essentially on the fact that (and thus also all subgraphs of is simply connected.

In Section 4 (page 4), we discuss various modifications and generalizations of the Yablo structure. Remark 4.1 (page 4.1) illustrates the argument in the Yablo structure, Example 4.1 (page 4.1) considers trivial modifications of the Yablo structure. In Fact 4.2 (page 4.2) we show that infinite branching is necessary for a graph being dangerous, and Example 4.2 (page 4.2) shows why infinitely many finitely branching points cannot replace infinite branching  there is an infinite “procrastination branch”.
1.2 Some definitions
Notation and definitions are taken mostly from [RRM13].
Definition 1.1

Given a (directed or not) graph will denote its set of vertices, its set of edges. In a directed graph, will denote an arrow from to which we also write if is not directed, just a line from to
We often use or etc. for vertices.

A graph is called transitive iff implies

Given two directed graphs and a homomorphism from to is a function such that, if then

Given a directed graph the underlying undirected graph is defined as follows: iff or i.e., we forget the orientation of the edges. Conversely, is called an orientation of

etc. will denote the set of propositional variables of some propositional language etc. the set of its formulas. and will be part of the formulas.

Given will be a valuation, defined on and extended to as usual  the values will be or so. will denote the valuation of etc. When the context is clear, we might omit the index

etc. will be a denotation assignment, or simply denotation, a function from to

A valuation is acceptable on relative to iff for all i.e. iff (When and are fixed, we just say that is acceptable.)

A system is called paradoxical iff there is no acceptable for

Given we define as follows: iff occurs in

A directed graph is dangerous iff there is a paradoxical system such that is isomorphic to
Definition 1.2
Let be a directed graph,

is a successor of iff
is a successor of

Call downward from iff there is a path from to i.e. is in the transitive closure of the succ operator.

Let be the subgraph of generated by is downward from i.e. is downward from and iff and
Definition 1.3
For easier reference, we define the Yablo structure, see e.g. [RRM13].
Let and
for a suitable language.)
Definition 1.4

Call a denotation or iff all have the form  as in the Yablo structure.

The dual notation expresses the analogous case with instead of i.e.

We will use and  for negation, and when we want to emphasize that a formula is not negated.
Remark 1.1
Note that we interpret in the strict sense of i.e., means that there is at least one which is true. In particular, if and then must contain a propositional variable, i.e. cannot be composed only of and so there is some arrow in the graph.
Thus, if in the corresponding graph is of the form is an acceptable valuation for then
Dually, for means that there is at least one which is false.
Thus, if in the corresponding graph is of the form is an acceptable valuation for then
2 A comment on conjecture 15 in [Rrm13]
We show in this section that conjecture 15 in [RRM13] is wrong.
Definition 2.1
Call (the integers) contiguous iff for all if and then too.
Fact 2.1
Let be a directed graph, for some contiguous and iff is the direct successor of
Then for any denotation :

may be (equivalent to) or
If or we abbreviate etc. (c for constant).
If is acceptable for then:

If then (if exists in

if then
if then

Thus:

If for some then for all for some

We have three possible cases:

for all

for no

there is some maximal s.t. so for all

In the first case, for all if is or then the valuation for starts anew, i.e. independent of and continues to etc. according to (2).

in the second case, there is just one acceptable valuation: we chose some and and propagate the value up and down according to (2)

in the third case, we work as in the first case up to and treat the as in the second case.

Basically, we work downwards from constants, and up and down beyond the maximal constant. Constants interrupt the upward movement.



Consequently, any on has an acceptable valuation and the graph is not dangerous.
Example 2.1
We define now a modified Yablo graph and a corresponding denotation which is paradoxical.

The vertices (and the set of language symbols):
We keep all of Fig.3 in [RRM13], and introduce new vertices for (When we write we tacitly assume that

The arrows:
We define (instead of writing we write  likewise for below):

(This is the main idea of the Yablo construction.)

for

Obviously, corresponds to and i.e.
Fact 2.2
and code the Yablo Paradox:
Proof
Let be an acceptable valuation relative to
Suppose then and for so for as in Fact 2.1 (page 2.1), (2). By there must be such that and or and and as in Fact 2.1 (page 2.1), (2) again, a contradiction.
If then as above for we find and and argue with as above for
Thus, with as above is paradoxical, and is dangerous.
Example 2.2
We first define :
We now define the homomorphism from to We collaps for fixed and all to more precisely, define by for all suitable
Note that only had arrows between “successor levels”, and we have now only arrows from to so is a homomorphism, moreover, our structure has the form described in Fact 2.1 (page 2.1), and is not dangerous, contradicting conjecture 15 in [RRM13].
Diagram 2.1
This is just the start of the graph, it continues downward through many levels.
The lines stand for downward pointing arrows. The lines originating from the correspond to the negative lines in the original Yablo graph, all others are simple positive lines, of the type
The left part of the drawing represents the graph YG’, the right hand part the collapsed graph, the homomorphic image YG”.
Compare to Fig.3 in [RRM13].
3 A comment on Theorem 24 of [Rrm13]
We comment in this section on the meaning of theorem 24 in [RRM13].
Definition 3.1
Fix a denotation
Let be the set of which occur in
Let be the set of relevant i.e. which influence for some E.g., in is relevant, is not.
Definition 3.2

Let be a directed graph. For let the subgraph of be the connected component of which contains and iff there is a path in from to together with the induced edges of i.e., if and then

is called a simply connected graph iff for all in there is at most one path in from to
(One may debate if a loop violates simple connectedness, as we have the paths and  we think so. Otherwise, we exclude loops.)

Two subgraphs of are disconnected iff there is no path from any to any in
Fact 3.1
Let be given,
If are two disconnected subgraphs of then they can be given truth values independently.
Proof
Trivial, as the subgraphs share no propositional variables.
Fact 3.2
Let be simply connected, and any denotation, Then has an acceptable valuation.
Proof
This procedure assigns an acceptable valuation to and in several steps.
More precisely, it is an inductive procedure, defining for more and more elements, and cutting up the graph into diconnected subgraphs. If necessary, we will use unions for the definition of and the common refinement for the subgraphs in the limit step.
The first step is a local step, it tries to simplify by looking locally at it, propagating [X] to with if possible, and erasing arrows from and to if possible. Erasing arrows decomposes the graph into disconnected subgraphs, as the graph is simply connected.
The second step initializes an arbitrary value (or, in step (4), uses a value determined in step (2)), propagates the value to for erases the arrow Initialising will have repercussions on the for so we chose a correct possibility for the (e.g., if setting requires to set too), and erase the arrows As is simply connected, the only connection between the different is via but this was respected and erased, and they are now independent.

Local step

For all

replace in by (or, equivalently, resulting in logically equivalent might now be empty),

erase the arrow
Note that will then be disconnected from as is simply connected.


Do recursively:
If then is equivalent to (or (it might also be etc.), so (or in any acceptable valuation, and is independent of

For replace in by (or might now be empty),

erase in
is then an isolated point in so its truth value is independent of the other truth values (and determined already).



Let be a nontrivial (i.e. not an isolated point) connected component of the original graph chose in If were already fixed as or then would have been isolated by step (1). So is undetermined so far. Moreover, if in then cannot be equivalent to a constant value either, otherwise, the arrow would have been eliminated already in step (1).
Chose arbitrarily a truth value for say

Consider any s.t. (if this exists)

Replace in with that truth value, here

Erase
As is simply connected, all such and are now mutually disconnected.


Consider simultanously all s.t. (They are not constants, as any must be a propositional variable.)

Chose values for all such corresponding to here).
E.g., if and the value for was then we have to chose also for and

Erase all such
is now an isolated point, and as is simply connected, all are mutually disconnected, and disconnected from all with considered in (2.1).

The main argument here is that we may define and for all and independently, if we respect the dependencies resulting through


Repeat step (1) recursively on all mutually disconnected fragments resulting from step (2).

Repeat step (2) for all in (2.2), but instead of the free choice for in (2), the choice for the has already been made in step (2.2.1), and work with this choice.
4 Various remarks on the Yablo structure
We comment in this section on various modifications and generalizations of the Yablo structure. We think that the transitivity of the graph, and the form of the are the essential properties of “Yablolike” structures.
We make this official:
Definition 4.1
A structure is called Yablolike iff is transitive, and of the form.
Remark 4.1
This remark is for illustration and intuition.
In the Yablo structure, after some which is true, all have to be false. After some which is false, there has to be some which is true.

We can summarize this as the following two rules:

After only  may follow, abbreviating:
after some with all with have to be

After , there has to be some

This has only finite solutions: a sequence of , ending with last element
Lastbutone does not work, as then the last is , but we need an after this one.

If we have an infinite sequence, there has to be a somewhere, followed by  only, contradiction.

if we start with then the first  imposes a somewhere, contradiction

if we start with , then there has to be somewhere, say at element so has to be , so some has to be contradiction.



An alternative view is the following:
(or constructs defensive walls, (or ) attacks them.
The elements of the walls () themselves are attacks on later parts of the walls, the attacks attack earlier constructions of the walls.
Example 4.1
We discuss here some very simple examples, all modifications of the Yablo structure.
Up to now, we considered graphs isomorphic to (parts of) the natural numbers with arrows pointing to bigger numbers. We consider now other cases.

Consider the negative numbers (with 0), arrows pointing again to bigger numbers. Putting at 0, and  to all other elements is an acceptable valuation.

Consider a tree with arrows pointing to the root. The tree may be infinite. Again at the root,  at all other elements is an acceptable valuation.

Consider an infinite tree, the root with successors and from each originating a chain of length as in Fig. 10 of [RRM13], putting at the end of the branches, and  everywhere else is an acceptable valuation.

This trivial example shows that an initial segment of a Yablo construction can again be a Yablo construction.
Instead of considering all we consider extending the original construction in the obvious way.
Fact 4.2
Let be loop free and finitely forward branching, i.e. for any there are only finitely many such that in Then is not dangerous.
(d may be arbitrary, not necessarily of the form.)
Proof
Let be any assignment corresponding to Then is a finite, classical formula. Replace by the classical formula Then any finite number of is consistent.
Proof: Let be a finite set of such and the set of occurring in As is loop free, and finite, we may initialise the minimal (i.e. there is no such that in the part of corresponding to with any truth values, and propagate the truth values upward according to usual valuation rules. This shows that is consistent, i.e. we have constructed a (partial) acceptable valuation for
Extend by classical compactness, resulting in a total acceptable valuation for
(In general, in the logics considered here, compactness obviously does not hold: Consider Clearly, every finite subset is consistent, but the entire set is not.)
The following modification of the Yablo structure has only one acceptable valuation for
Example 4.2
Let as usual, and introduce new
Let with
If then by so, generally,
if then and
If then so if etc., so, generally,
if then or
Suppose now then for all and for all By there is and a contradiction, or for all again a contradiction.
But is possible, by setting and for all
Thus, replacing infinite branching by an infinite number of finite branching does not work for the Yablo construction, as we can always chose the “procrastinating” branch.
Diagram 4.1
The lines stand again for downward pointing arrows. Crossed lines indicate negations.
Fact 4.3
Let be transitive, and be of the type

If and then has no acceptable valuation.
Let acceptable be given, is for this
Case 1: So for all and there is such so (either by the prerequisite or by Remark 1.1 (page 1.1)) but a contradiction.
In abbreviation:
Case 2: So so and by prerequisite so there is such so by Remark 1.1 (page 1.1) so but a contradiction.

Conversely:
Let or
Thus, the valuation defined by iff and otherwise is an acceptable valuation. (Obviously, this definition is free from contradictions.)
References
 [1]
 [RRM13] L. Rabern, B. Rabern, M. Macauley, “Dangerous reference graphs and semantic paradoxes”, in: J. Philos. Logic (2013) 42:727765
 [Yab82] S. Yablo, “Grounding, dependence, and paradox”, Journal Philosophical Logic, Vol. 11, No. 1, pp. 117137, 1982