Overview

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

Overview

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

Contents

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

Example

What is a Graph?

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

Edge Covers

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

Paths & Cycles

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

Vertex Degree

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 $x$

Cycle Space

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 $H$.

What is an Infinite Cycle?

The double ray graph

Does the theorem work for infinite graphs?

1. Any vertex can be reached from any other vertex
2. Each vertex has degree two
3. No edge is repeated

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

Ends of an Infinite Graph

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

Ends & Connectedness

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

Infinite Cycles

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.

Infinite Cycles

The one-ladder graph

The infinite cycle contained in the one-ladder graph

Infinite Cycle Space

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.

Support Graphs

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

Support Graph Properties

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?

Theories of Justification

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?

Foundationalism

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

Foundationalism

Coherentism

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 $\mathcal{C}(G)$

Coherentism

Infinitism

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

Infinitism

Infinite Coherentism

Infinite Coherentism imposes a single condition on $G$:

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

Infinite Coherentism

The one-ladder graph

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

Conclusions

• 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

References

• 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.