The epistemic significance of
Hume’s Principle
Difei Xu
The department of philosophy
Renmin University of China
The outline
• Frege’s logicism
• Neo-Fregean Logicism
• The problems fronted by Hume’s principle as
an epistemic principle and Neo-Fregeanists’
ways to solve them
• The criterion for good abstract principles in
Neo-Fregean philosophy and the criterion of
new axioms in Gödel’s philosophy
Frege’s logicism
• The core thesis in Grundlagen is that the
arithmetic truths are analytic truths, which
can be justified by logic and definitions.
• He aimed to discover the logical foundation of
arithmetic without resorting to intuition in
epistemology.
• It is generally known that Frege’s logicism
failes because the axiomatic system in
Grundgesetze is not consistent.
Frege’s logicism
the epistemology without intuition
• Abstract principles
∀α∀β(#α = # β ↔ α ≈ β) ,
where # is an operation applying to some type
of expressions to form a singular term; 
expresses an equivalent relation.
Frege’s logicism
instances for abstract principles
• The direction of line a = the direction of line b
iff. line a is parallel with line b.
• The number of F = the number of G iff there is
a bijection between F and G.
• The extension of F = the extension of G iff F
and G are coextensive.
Frege’s logicism
the definition of number
• In § 60-67 of Grundlagen, Frege suggested that
Hume’s Prinicple could be an implicit definition of
numbers, but finally he rejected it as an implicit
definition because of Caesar Problem. In order to
solve the problem, Frege turned to an explicit
definition of numbers and defined “the number
of F” as an equivalent class consisting of concepts
with the same cardinality.
Frege’s logicism
disaster
• This explicit definition needs a theory of
extensions (classes), and in order to provide
this kind of theory, Frege introduced axiom V,
which added to the second-order logic system
leads to inconsistency. It is a vital attack to
Frege’s Logicism.
Neo-Fregeanists’ logicism
• The core thesis in Neo-Fregeanists’ logicism is
that Hume’s Principle as an implicit definition
gives a way to explain how we understand
numbers.
• Neo-Fregeanism is so called because it is so close
to Frege’s philosophy of arithmetic. NeoFregeanists try to give an epistemology of
arithmetic without resorting to intuition. The
foundation of the epistemology for arithmetic is
logic and definitions.
Neo-Fregeanists’ logicism
The differences
• Neo-Fregeanism is different from Frege’s
philosophy; especially Neo-Fregeanists take
Hume’s Principle as a definition while Frege
did not accept Hume’s principle as a definition
but as a derived theorem in his system.
Neo-Fregeanists’ logicism
Frege’s theorem
• The standard second-order logical system
added Hume’s principle can derive the axioms
of Dedekind’s second-order arithmetic.
Neo-Fregeanists’ logicism
difficulties
• This position is not lack of criticism even from
Frege’s philosophy.
Neo-Fregeanists’ logicism
the truly threaten
• the truly threaten is to give a reasonable
justification to HP.
Neo-Fregeanists’ logicism
The justification for HP
• Neo-Fregeanists cannot resort to Frege’s
theorem, since otherwise second – order
arithmetic is epistemically prior to HP.
The justification for HP
Boolos
• Boolos thought Neo-Fregeanists cannot
guarantee HP is analytic truth.
• As Boolos said, analytic truth should firstly be
true, but how can we guarantee that this
system is consistent?
The justification for HP
bad company problem
• Frege’s Axiom V is also in the form of abstract
principles, and defines identity of extensions
of predicates, but once it is added to secondorder logic system, it leads to a contradiction .
How to demarcate “good” abstract principles
from “bad” ones is now called “bad company”
problem.
The justification for HP
the condition of consistency is not sufficient
• Nuisance Principle
The nuisance associated with the concept F is
the same as the nuisance associated with the
concept G just in case the range of things which
are either F or G but not both.
• NP only be satisfied in the finite model.
The justification for HP
• Wright added that a good abstract principle
must be conservative, which means that the
principle added to the original theory will not
result in new conclusion about old ontology.
• NP leads to the conclusion that the domain is
finite, which failed to be conservative, so it is
not acceptable.
The justification for HP
• The condition of conservativeness is
equivalent to the statement: an abstract
principle is acceptable only if it does not
restrict the upper bound of the domain.
The justification for HP
challenge to the condition of conservativeness
• Alan Weir shows that there is a class of restrictive
axioms V, which are all conservative but they are
inconsistent with one another.
• condition of stability
• an abstract principle is stable iff if it is satisfiable in
the model with the cardinality of ,and ，then it
is satisfiable in the model with the cardinality of .
The justification for HP
• Maybe the story does not end.
Neo-Fregeanists and Gödel
• Confronted with the bad company problem, NeoFregeanists searched for the conditions to demarcate
good abstract principles from bad ones, and the
conditions are the higher principles or the general
ground for good abstract principles. These higher
principles should tell the difference between HP and
axiom V, NP and so on. Once these higher principles
are found, their position of Logicism is defendable.
Neo-Fregeanists and Gödel
• like logical principles, these higher principles
play a foundational role in the epistemology of
arithmetic.
• They search for “upper ward” justification for
HP but deny “down ward” evidence as the
justification for HP.
Neo-Fregeanists and Gödel
• Frege’s arithmetic is a theory to give an
explanation of finite numbers in the same way
that a physical theory gives an explanation to our
physical experiences.
• When we reflect justification of the theory, the
“down ward” evidence, like second-order
arithmetic and our physical experiences are
involved.
• Without downward evidence, it is hard to claim
the theory explains what we wish it to explain.
Neo-Fregeanists and Gödel
• A good axiomatic system of mathematics is
not arbitrary: it derives our known
mathematical statements, and the system
itself shows how the known mathematical
statements brought into a logical system.
Neo-Fregeanists and Gödel
• In this sense, Frege’s theorem is evidence
making us known HP explains finite numbers.
when we consider the problem whether
Frege’s system explains finite numbers, Frege’s
theorem must be evidence for supporting the
claim.
Neo-Fregeanists and Gödel
• Of course Neo-Fregeanists reject this question
and reject to find the answer to this question.
They won’t accept intuition as the epistemic
source of arithematic.
• From the history of the development of
foundation of mathematics, we do have some
basic beliefs and then to search for general
axioms to unify these beliefs.
Neo-Fregeanists and Gödel
• If an axiomatic system is consistent but does
not derive second-order arithmetic, in what
sense we can believe that this system explains
finite numbers? Without reference to our
knowledge of finite numbers, we cannot
answer this question.
Neo-Fregeanists and Gödel
Gödel’s conception of new axioms
• In “what is Cantor’s continuum problem”,
Gödel proposed two criteria for accepting new
axioms to solve the continuum problem. The
two criteria are: 1 intrinsic necessity and 2
fruitfulness in consequences.
Neo-Fregeanists and Gödel
• I find it is hard to explain Gödel’s first criterion
for the truth of the new axioms. Here what I
wish to emphasize is the second criterion for
the truth of axioms, which is feasible in
mathematical practice.
Neo-Fregeanists and Gödel
• ‘a decision about its truth is possible also in
another way, namely, inductively by studying
its “success”, that is, its fruitfulness in
consequences and in particular “verifiable”
consequences’
Neo-Fregeanists and Gödel
• From the second criterion, we could accept
the truth of HP, because of its verifiable
consequences. Unlike Frege, Gödel accepted
intuition as an epistemic source of
mathematical objects. And from the above,
we see Gödel take the down-ward knowledge
as the evidence for accepting the truth of an
axiom. So Gödel, not as a foundationist,
searched for new axioms.
Neo-Fregeanists and Gödel
• Neo-Fregeanists as foundationists, searched for
the higher conditions of accepting HP, and
rejected Frege’s theorem as its justification. The
aim of the conditions is to demarcate good
abstract principles from bad ones. The conditions
for good abstract principles, logic, and accepted
abstract principles form the epistemic
foundations of arithmetic. As I said before, when
we say a theory is reasonable, we should take
some knowledge as a reference.
Neo-Fregeanists and Gödel
• Neo-Fregeanists do not say whether there is a
limit of abstract principles as a definition of
mathematical objects.
• And I am not so sure about whether this limit
exists , i.e. there is a kind of mathematical
objects that cannot be defined by abstract
principles or there is some mathematical truth
that cannot be proved by logic and abstract
principles.
Neo-Fregeanists and Gödel
• Even though there exists such a limit, it won’t
threaten Neo-Fregeanism.
• Let us assume that there is a statement in set
theory which may accept as truth but cannot
be proved by logic and any abstract principle.
Neo-Fregeanists then say, “That’s fine.
Arithmetic and set theory have different
epistemic foundation.”
Neo-Fregeanists and Gödel
• At least, if Neo-Fregeanists abandon the
endeavor to seek for the accepting conditions
but just justify HP by its consequences, like
assumptions in physical theory, then their
philosophy may lose its color of foundationism.
• At the same time, it costs too much for NeoFregeanism to take this step.
Thank you!