Over the past decade, great strides have been made in analyzing the structure of epistemic justification mathematically and probabilistically

- Peijnenburg (2007) starts a research program studying the probabilistic properties of infinite regresses
- Berker (2015) provides a graph theoretic account of foundationalism, coherentism, and infinitism
- The Graphical Approach to Epistemic Justification

Extend the graph theoretic account using new results from graph theory on infinite cycles

Bounded Paths | Infinite Paths | |
---|---|---|

Acyclic | Foundationalism | Infinitism |

Edge Cover in Cycle Space | Coherentism | Infinite Coherentism |

- Graph Preliminaries
- Infinite Cycles
- Graphical Approach
- Support Graphs
- Foundational & Coherentism
- Infinitism & Infinite Coherentism

A graph is a set of vertices and edges: $G = (V, E)$

Every directed graph $G$ has an underlying undirected graph $G'$

$H = (V_H, E_H)$ is a subgraph of $G$ if $V_H \subset V, E_H \subset E$

Example: $( \{ a,b,c \} , \{ab, bc, ca\})$

An *edge cover* of $G$ is a subgraph $H$ where every vertex of $G$ is incident to an edge in $H$.

Example:

- Removing $ca$ gives an edge cover
- Removing $cd$ and $ed$ does not

A *path* a path is a sequence of edges connecting a distinct vertices

Example: $caed = (ca, ae, ed)$

The *length* of a path is the number of vertices in the path

A *cycle* is a path except that the first and last vertices are the same

Example: $abca = (ab, bc, ca)$

The *degree* of a vertex $x$ is the number of edges incident to $x$

- the
*in-degree*of a vertex $x$ as the number of incident edges incoming to $x$ - the
*out-degree*of a vertex $x$ as the number of incident edges outgoing from $x$

The *cycle space* of $G$, $\mathcal{C}(G)$, is the closure of the set of cycles for $G$ by edge disjoint union

**Theorem:** Let $H = (V, E)$ be a subgraph of a finite undirected graph $G$. Then $E \in \mathcal{C}(G)$ if and only if every vertex of $G$ has even degree in $H$.

The *double ray* graph

Does the theorem work for infinite graphs?

- Any vertex can be reached from any other vertex
- Each vertex has degree two
- No edge is repeated

“...common sense tells us that this can hardly be right: shouldn’t cycles be round?” - Diestel (2004)

The *double ray* graph

A *connected component* of a graph is a connected subgraph that is maximal

An *end* of a graph to be the set of rays that belong to the same connected component after any finite set of vertices are removed

Example: the double ray has two ends

The *one-ladder* graph

Think of ends as points at infinity

A graph $G$ is *topologically connected* if there exists a path between any two vertices that is either finite or traverses ends

A *ray* is a path from a vertex to an end

The *degree of an end* is the maximum number of edge disjoint rays contained in the end

The *one-ladder* graph

**Theorem:** Let $C$ be a subgraph of a locally finite graph $G$. Then $C$ is a cycle just in case $C$ is topologically connected and the degree of every vertex and end is two.

The *one-ladder* graph

The infinite cycle contained in the *one-ladder* graph

The *one-ladder* graph

**Theorem:** Let $G = (V, E)$ be a locally fininte graph. Then $E \in \mathcal{C}(G)$ if and only if every vertex and every end of $G$ has even degree.

Let vertices represent beliefs and directed edges represent relations of support between beliefs

Let $G = (V, E)$ be a support graph

Constructing a support graph (informally):

- Let a belief $p$ be given
- For beliefs $s_0, s_1, \ldots$ supporting $p$, add an edge from $s_i$ to $p$ for all $i$
- Repeat for each of $s_0, s_1, \ldots$ that have not appeared before and so on

Example: If $p$ and $q$ are beliefs and $p$ supports $q$ then the support graph for $p$ is a subgraph of the support graph for $q$

Importantly, there is much we do not know:

- Does construction terminate in finite or $\omega$ steps?
- Connected?
- Acyclic?

These theories make assumptions on properties for support graphs

- Are paths in $G$ bounded or unbounded?
- Are there no cycles in $G$ or can $G$ be deconstructed into cycles?

Justification and knowledge are ultimately derivative from a set of basic or foundational elements whose justification does not depend in turn on that of anything else. - BonJour (2010)

Foundationalism imposes two conditions on $G$:

- The length of the shortest path between $v, p$ for all $v \in V$ is bounded
- $G$ is acyclic

beliefs can only be justified by other beliefs...what justifies beliefs is the way they fit together: the fact that they cohere with each other - BonJour (2010)

Coherentism imposes two conditions on $G$:

- The length of the shortest path between $v, p$ for all $v \in V$ is bounded
- There is an edge cover of $G$ in $\mathcal{C}(G)$

Infinitism borrows both from foundationalism that justification cannot be circular and from coherentism that only beliefs can justify other beliefs

Infinitism imposes two conditions on $G$:

- For each vertex $v \in V$, there is an infinite path to $p$ containing $v$
- $G$ is acyclic

Infinite Coherentism imposes a single condition on $G$:

- There is an edge cover of $G$ in $\mathcal{C}(G)$

The *one-ladder* graph

Interpretation: two indefinite reason giving processes go to/from the same end when they are appropriately connected or coherent

- Support graphs are a useful tool for exploring the properties of epistemic justification
- By introducing infinite cycles, we can consider a fourth solution to the regress problem

- Peijnenburg, Jeanne, (2007). “Infinitism Regained.” Mind 116(463), 597-602.
- Berker, Selim (2015). “Coherentism via Graphs.” Philosophical Issues 25(1), 322-352.
- Diestel, Reinhard (2004). “On Infinite Cycles in Graphs: Or How to Make Graph Homology Interesting.” The American Mathematical Monthly 111(7), 559-571.
- BonJour, Lawrence (2010). Epistemology: Classic Problems and Contemporary Responses, 2nd Edition, Rowman & Littlefield.