# Bealer’s Intensional Logic - Overview

Clarence Protin

(October 2020)

###### Abstract

Many intuitively valid arguments involving intensionality cannot be captured
by first-order logic, even when extended by modal and epistemic operators.
Indeed, previous attempts at providing an adequate treatment of the phenomenon
of intensionality in logic and language, such as those of Frege, Church,
Russell, Carnap, Quine, Montague and others are fraught with numerous
philosophical and technical difficulties and shortcomings. We present Bealer’s
solution to this problem which hinges on an ontological commitment to theory
of Properties, Propositions and Relations (PRP). At the most basic level we
can distinguish two conceptions in the theory of PRPs. An objective one tied
to modality and necessary equivalence, and a mental (intentional) one tied to
concepts and the requirement of non-circularity in definitions. Building on
the work of Russell, Church and Quine, Bealer proposes two distinct
intensional logics T1 and T2 (presented in Hilbert form) corresponding to
these two conceptions, both based on the language of first-order logic
extended with an intensional abstraction operator. In T1 necessitation can be
directly defined and the axioms entail that we obtain standard S5 modal logic.
These logics have a series of striking features and desirable aspects which
set them apart from higher-order approaches. Bealer constructs a non-Tarskian
algebraic semantic framework, distinct from possible worlds semantics and its
problematic ontological commitments, yielding two classes of models for which
T1 and T2 are both sound and complete. Other features include being able to
deal with quantifying-in, and the various substitution puzzles, being free
from artificial type restrictions, having a Russellian semantics, satisfying
Davidson’s learnability requirement, etc. Bealer proposes his logic as the
basis of a larger philosophical project in the tradition of logicism (or
logical realism) concerning which we refer to his book Quality and Concept
(1982). This includes a neo-Fregean logicist foundation of arithmetic and set-
theory in which various (according to him) purely logical predication axioms (
and intensional analogues of ZF, NGB, or Kelley-Morse axioms) are adjoined to
T2, thereby explaining incompleteness as a property of pure logic rather than
of mathematics. Surprisingly, and rather ironically, Bealer’s logic also
fulfills Carnap’s thesis of extensionality due precisely to its ontological
commitment to the reality of PRPs. These notes are the introduction to a
project to give detailed proofs of all the mathematical logic results that
appear in Bealer’s _Quality and Concept_ and to clarify and simplify some of
the concepts and techniques.

The present note is the first part of a project involving a series of notes
about Bealer’s logic. In this part we give some motivation and a general
introduction to Bealer’s logic (based on a talk given at the University of
Lisbon)111https://ciencias.ulisboa.pt/pt/evento/12-10-2020/bealers-
intensional-logic-part-i. We have strived to simplify and clarify Bealer’s
logical systems to some extent. We hope, in fact, that a simpler alternative
presentation will be possible, perhaps in the style of Combinatory Logic. In
subsequent notes we will present our work-in-progress in furnishing detailed
proofs for all the results stated in Bealer’s _Quality and Concept_ which are
either unproven or given only very sketchy indications of a proof. We hope
thereby also to motivate researchers in logic, philosophy and related areas to
tackle the open questions and conjectures related to the more ambitious
speculative chapters of Bealer’s book. All comments and criticisms are
welcome.

##  1 Overview of Bealer’s Logic

###  1.1 References

  * •

G. Bealer, Quality and Concept, Oxford University Press (1982)

  * •

G. Bealer, Completeness in the theory of properties, relations, and
propositions, Journal of Symbolic Logic 48 (2): 415-426, (1983)

  * •

G. Bealer, Property theory: the type-free approach v. the Church approach,
Journal of Philosophical Logic 23: 139-171, (1994)

###  1.2 Motivation

Whatever $x$ believes is necessary. Whatever is necessary is true. Whatever
$x$ believes is true.

$\forall y.B(x,y)\rightarrow N(y)$ $\forall y.N(y)\rightarrow T(y)$ $\forall
y.B(x,y)\rightarrow T(y)$

Whatever $x$ believes is true. $x$ believes that $A$. It is true that $A$.

Treat that-clauses as singular terms, and represent for a formula $A$ the term
’the proposition that A’ by $[A]$:

$\forall y.B(x,y)\rightarrow T(y)$ $B(x,[A])$ $T([A])$

###  1.3 Motivation - Quantifying-in

Whatever $x$ believes is true. $x$ believes that $v$ believes something. It is
true that $v$ believes something.

$\forall y.B(x,y)\rightarrow T(y)$ $B(x,[\exists y.B(x,y)])$ $T([\exists
y.B(x,y)])$

$x$ believes that he believes something. There is someone $v$ such that $x$
believes that $v$ believes something.

$B(x,[\exists y.B(x,y)])$ $\exists v.B(x,[\exists y.B(v,y)])$

  * •

That-clauses should be treated as singular terms which can contain _externally
quantifiable_ variables.

###  1.4 Semantic Problems

  * •

Substitutivity failures for materially equivalent and co-referential formulas.

$x$ believes that everything runs. Everything runs if and only if everything
walks. $x$ believes that everything walks.

$B(x,[\forall y.R(y)])$ $\forall y.R(y)\leftrightarrow\forall y.W(y)$
$B(x,[\forall y.W(y)])$

$x$ wonders whether y is the author of _Waverley_ $y$ is the author of
_Waverley_ $x$ wonders whether $y$ = $y$

$W(x,[y=\iota z.A(z)])$ $y=\iota z.A(z)$ $W(x,[y=y])$

###  1.5 Motivation

Intensional entities can be different even if their _extension_ be the same.
What kind of entities do the $[A]$ correspond to ?

  * •

Linguistic entities (Carnap, Quine): names or structural descriptions of a
formula or inscription of a formula: violates the Langford-Church translation
test.

  * •

Nothing at all: the above metalinguistics names and descriptions can be
analysed away completely (Scheffler): violates Davidson’s learnability
requirement.

  * •

Extra-linguistic entities such as propositions (Gödel’s platonism, Bealer’s
logical realism).

###  1.6 Other Intensional Terms

Other nominative formations in language such as the gerundive and infinitive
are captured by _intensional abstraction_ over variables
$[A]_{x_{1},...x_{n}}$. Their semantic correlates are _propositions, relations
and properties_.

  * •

Bealer’s intensional logic is first-order logic with equality with a finite
number of predicates symbols extended with term-forming intensional
abstraction operators $[\quad]_{x_{1}..x_{n}}$ satisfying certain axioms (and
axiom-schemes).

  * •

Sound and complete semantics.

  * •

There are two kinds of intensional entities and thus two distinct axiomatic
systems of intensional logic, $T1$ and $T2$.

  * •

Using a single primitive binary predication predicate $\Delta$ and additional
”predication” axiom-schemes we obtain a neo-Fregean formulation of arithmetic.

  * •

Thesis of extensionality: we can embed Bealer’s logic into classical FOL and
use standard Tarskian semantics.

###  1.7 Comparison with other Approaches

The multi-modal approach originating with C.I.Lewis, Carnap, Hintikka, Kripke
and others:

  * •

Operators apply only to formulas and not to singular terms so many intuitively
valid arguments cannot be captured.

  * •

Eclectic and incomplete rather than based on a single intensional operator.

Higher-order approaches have the following disadvantages relative to Bealer’s
logic:

  * •

Does not have a sound and complete semantics.

  * •

Predicates can be linguistic subjects which leads us to consider two sorts of
meaning relation to avoid Frege’s ’a=a’/’a=b’ puzzle.

  * •

Needs type-theory to avoid paradoxes but this imposes implausible existence
restrictions on intensional entities.

  * •

Division of variables into sorts makes it impossible to deal with
”transcendental” predicates without violating Davidson’s learnability
requirement.

  * •

Requires possible-world semantics and commitment to non-actual possibilia.

###  1.8 The Two Conceptions of Intensionality

  * •

According to the first (Leibniz, one of Church’s formulation to Frege’s theory
of _sense_) two intensional entities are equal _if and if they are necessarily
equivalent_. Related to possible-worlds semantics and deals with necessity and
possibility. A definition need only be a necessary equivalence to be correct.

  * •

According to the second conception each intensional entity when defined
completely _must possess a unique non-circular definition_. Found in Leibniz’s
distinction between simple and complex properties and Russell’s doctrine of
logical atomism. Used to treat intentionality: belief, desire, perception,
etc.

Example of a deduction valid in the first conception but not in the second:

$x$ wonders whether there are trilaterals which are not triangles. Necessarily
all and only triangles are trilaterals. $x$ wonders whether there are
triangles which are not triangles.

###  1.9 The language $L^{\omega}$

###### Definition 1.1

The language $L^{\omega}$ consists of a countable ordered collection of
variables $x,y,z,...$ and $n$-ary predicate symbols $F,G,H,...$, with a
distinguished binary predicate $=$, logical connectives $\&,\sim,\exists$ and
the intensional abstraction operator $[\quad]_{x_{1}...x_{n}}$.

###### Definition 1.2

Formulas and terms of $L^{\omega}$ are defined by simultaneous induction:

  * •

Variables are terms.

  * •

If $t_{1},...,t_{n}$ are terms and $F$ is $n$-ary predicate, then
$F(t_{1},...,t_{n})$ is a formula.

  * •

If $A$ and $B$ are formulas, $v$ a variable, then $(A\&B),\sim A$ and
$\exists.A$ are formulas.

  * •

If $A$ is a formula and $v_{1},...,v_{m}$, $0\leq m$, are _distinct_
variables, then $[A]_{v_{1}...v_{m}}$ is a term.

  * •

The intensional operator generalizes the set-theoretic extension operator. Can
be seen as the $\lambda$ operator of ”version 2” second-order logic (Prawitz,
Nat.Ded., ch.V).

  * •

A term of the form $[A]_{v_{1},...,v_{n}}$ is called an _intensional
abstract_.

  * •

Intensional abstracts of the form $[F(v_{1},...,v_{n})]_{v_{1}...v_{n}}$ are
called _elementary_.

  * •

Intensional abstracts $[A]_{\alpha}$ where $A$ is not atomic are called
_complex_.

###  1.10 Bealer-Quine Decomposition

  * •

_we can decompose every formula in a unique way in terms of syntactic
operations on elementary abstracts_. These syntactic operations are then
interpreted as algebraic operations on the model whose underlying set consists
interprets elementary abstracts.

  * •

A term $[F(t_{1},...,t_{n})]$ is seen as the result of _substitutions_
(_predication_ operations) applied to the elementary abstract
$[F(x_{1},...,x_{n})]_{x_{1}...x_{n}}$. For instance, $[F(t_{1},...,t_{n})]$
would be seen as something like the predication
$Pred([F(t_{1},...,v)]_{v},t_{1})$.

  * •

The problems involved with defining such a substitution operation are the same
as there would be for the type-free $\lambda$-calculus: the problems that
bound variables are not named uniquely and of the same variable occuring in
different arguments.

  * •

Bealer-Quine decomposition is an algorithm that generates a unique
decomposition of any intensional abstract in terms of a set of partially-
defined syntactic operations. There is a certain arbitrarity in the way this
decomposition is defined; what is crucial is the uniqueness which allows
arguments by structural induction and the definition of denonation.

  * •

This process is similar to the elimination and untangling of bound variables
in Schönfinkel’s Combinatory Logic and specially to the approaches of Quine in
his 1960 paper _Variables explained away_ and 1981 paper _Predicate functors
revisited_.

  * •

We cannot see $[F(t_{1},...,t_{n})]_{\bar{x}}$ as the result of substitutions
unless _no variable in $\bar{x}$ occurs in more than one argument of $F$_.
Hence we must decompose our term first into terms of ”reflexion” operators
before decomposing it into a predication. For instance $[F(x,x)]_{x}$ is the
result of the reflexivization of $[F(x,y)]_{xy}$. This reflexivization
involves substituting occurrences of $x$ by $y$.

  * •

If a free variable $x$ in $t_{k}$ occurs in ${\bar{x}}$ then, due to
$\alpha$-equivalence, it does not make sense to define the predication
operation in terms of $t_{k}$ but we must consider $t^{\prime}_{k}$ which is
$t_{k}$ also abstracted over $x$.

  * •

For simplicity we do not desire a multitude of predication operators for
different arguments and atomic predicates of different arities. The same goes
for reflexivity operators in which there is the additional complexity of the
same variable occurring in any number of the arguments.

  * •

To use a smaller collection of predication operations and a single
reflexivization operator we must introduce permutation operations which simple
switch the order of the abstracted variables. Thus we can permute
$[F([G(x)],y)]_{xy}$ into $[F([G(x)],y),y)]_{yx}$ which is the same as
$[F([G(y)],x),x)]_{xy}$. We want as few permutations operations as possible.

  * •

We desire uniqueness of the syntactical decomposition.

Consider for instance $[F([G(x,y)],x)]_{x}$. Because of $\alpha$-equivalence
we cannot consider this to be the result of substituting $[G(x,y)]$ for $v$
into $[F(v,x)]_{xv}$.

###  1.11 Bealer-Quine Decomposition

  * •

A term $[A]_{\bar{x}}$ is _normalized_ if all the variables in $\bar{x}$ are
free in $A$ and they display the order in which these variables first occur
free.

  * •

If $A$ is atomic then we call $[A]_{\bar{x}}$ a _prime term_ if the variables
in $\bar{x}$ are free in $A$, independently of the order.

  * •

Given an atomic formula $F(t_{1},...,t_{n})$ if a variable occurs free in more
than one of the terms $t_{i}$ then it is called a _reflected variable_. A
prime term $[A]_{\bar{x}}$ is called a _prime reflection term_ if $\bar{x}$
contains a reflected variable in $A$.

  * •

Consider a prime term $[F(t_{1},...,[B]_{\bar{y}},...,t_{j})]_{\bar{x}}$ which
is not a prime reflection term. If there is a variable in $\bar{x}$ which is
free in $[B]_{\bar{y}}$ and all the previous arguments are variables in
$\bar{x}$, then the prime term is a _prime relativized predication term_ and
such a variable is called a _relativized variable_.

  * •

Consider a prime term $[F(t_{1},...,t_{k},...,t_{j})]_{\bar{x}}$ which is not
a prime reflection term. If there is term $t_{k}$ which is a variable not in
$\bar{x}$ or which contains no free variables which are in $\bar{x}$ and all
the previous arguments are variables in $\bar{x}$, then it is a _prime
absolute predication term_.

###  1.12 Bealer-Quine Decomposition

Given an intensional abstract $[A]_{\bar{x}}$ it falls into one and one only
of the following seven categories which depend only on $A$ and $\bar{x}$
independently of order (only on the underlying set).

  1. 1.

Complex and $\bar{x}$ contains variables not free in $A$.

  2. 2.

Complex and all the variables in $\bar{x}$ are free in $A$.

  3. 3.

Not complex and $\bar{x}$ contains variables not free in $A$.

  4. 4.

Prime reflection term.

  5. 5.

Prime relativized predication term.

  6. 6.

Prime absolute predication term.

  7. 7.

$[A]_{\bar{x}^{\prime}}$ is elementary for some permutation $\bar{x}^{\prime}$
of $\bar{x}$.

These will be the well-defined domains for our partial syntactic operations.

###  1.13 Bealer-Quine Decomposition

Consider a permutation $[m_{1},...,m_{n}]$ of the sequence $[1,2,...,n]$. We
decompose it in a specific way as sequence of permutations of the form
$conv=(1\quad 2\quad...\quad n)$ and $inv=(n-1\quad n)$.

Suppose $\pi^{\prime}=\pi_{n}...\pi_{2}\pi_{1}$ has already been defined. Then
let $s$ be the maximal subsegment of $\pi^{\prime}[m_{1},,,.m_{n}]$ equal to
an initial segment $b=[1,..,k]$ of $[1,2,...,n]$. If $b$ is the whole
$[1,...,n]$ we are done. Otherwise $k+1$ will correspond to a certain $x$ in
$\pi^{\prime}[m_{1},...,m_{n}]$. If $x$ is the last element then we choose
$inv$, otherwise we chose $conv$. This process terminates and the
decomposition is clearly unique. Given a permutation $\pi$ we denote its
decomposition by $b(\pi$).

  * •

For an intensional abstract $[A]_{\alpha}$ we construct inductively an
expression $\mathbf{B}([A]_{\alpha})$ consisting of unary operations
$C,I,E_{v},R,N,U$ where $v$ is a variable, and binary operations $A,P_{n}$ for
$n\geq 0$ and elementary abstracts on the atomic predicates of $A$. To a
permutation $\bar{x}^{\prime}$ of a sequence of (distinct) variables $\bar{x}$
we associate a permutation as above and we denote by
$b(\bar{x},\bar{x^{\prime}})$ the expression which results from substituting
$inv$ and $conv$ for $I$ and $C$ in its decomposition. For
$\bar{x}=v_{1},...,v_{n}$ let
$E_{\bar{x}}(s)=E_{v_{n}}(E_{v_{n-1}}(...E_{v_{1}}(s)...))$.

###  1.14 Bealer-Quine Decomposition

  * •

Suppose $[A]_{\bar{x}}$ falls into cases 1 or 3. And let $\bar{x}^{\prime}$ be
a permutation of $\bar{x}$ of the form $\bar{y}\bar{z}$ so that
$[A]_{\bar{y}}$ is normalized and $\bar{z}$ consists of the variables in
$\bar{x}$ that do not occur free in $A$ in order of their occurrence in
$\bar{x}$. Then
$\mathbf{B}([A]_{\bar{x}}):=b(\bar{x}^{\prime},\bar{x})(E_{\bar{z}}(\mathbf{B}([A]_{\bar{y}})))$.

  * •

Suppose $[A]_{\bar{x}}$ falls into case 2 and is not normalized. And let
$\bar{x}^{\prime}$ be a permutation of $\bar{x}$ so that
$[A]_{\bar{x}^{\prime}}$ is normalized. Then
$\mathbf{B}([A]_{\bar{x}}):=b(\bar{x}^{\prime},\bar{x})(\mathbf{B}([A]_{\bar{x}^{\prime}})))$.

  * •

Suppose it falls into case 2 and is normalized. Then we have

$\mathbf{B}([A\&B]_{\bar{x}}):=A(\mathbf{B}([A]_{\bar{x}}),\mathbf{B}([B]_{\bar{x}}))$

$\mathbf{B}([\sim A]_{\bar{x}}):=N(\mathbf{B}([A]_{\bar{x}}))$

$\mathbf{B}([\exists vA]_{\bar{x}}):=U(\mathbf{B}([A]_{\bar{x}v}))$

  * •

Consider case 4. Of the reflected variables in $\bar{x}$ consider the one $v$
with the right-most occurrence and let be $t_{k}$ the right-most argument of
$A$ in which $v$ has a free occurrence. Let $w$ be the alphabetically earliest
variable not occurring in $A$. Let $\bar{x}^{\prime}$ be the permutation of
$\bar{x}$ such that $[F(t_{1},...,t_{n})]_{\bar{x}^{\prime}}$ is normalized
and consider $\bar{y}v$ where $\bar{y}$ is $\bar{x}^{\prime}$ with $v$
removed. Then

$\mathbf{B}([F(t_{1},...,t_{n})]_{\bar{x}}):=b(\bar{y}v,\bar{x})(R(\mathbf{B}([F(t_{1},...,t_{k}[w/v],...t_{n})]_{\bar{y}vw})))$

  * •

Consider case 6 and let $t$ be the argument which is not a variable in
$\bar{x}$ or which contains no free variables which are in $\bar{x}$ and all
the previous arguments are variables in $\bar{x}$. Let $v$ be the
alphabetically earliest variable not occurring in $A$. Let $\bar{x}^{\prime}$
be the permutation of $\bar{x}$ such that
$[F(t_{1},...,t,...,t_{n})]_{\bar{x}^{\prime}}$ is normalized. Then

$\mathbf{B}([F(t_{1},...,t,...t_{n})]_{\bar{x}}):=b(\bar{x}^{\prime},\bar{x})(P_{0}(\mathbf{B}([F(t_{1},...,v,...t_{n})]_{\bar{x}^{\prime}v},\mathbf{B}(t))))$

  * •

Consider case 5 of a prime term
$[F(t_{1},...,[B]_{\bar{y}},...,t_{j})]_{\bar{x}}$, where there is a variable
in $\bar{x}$ which is free in $[B]_{\bar{y}}$ and all the previous arguments
are variables in $\bar{x}$. Let $\bar{z}$ be the sequence, as they first
occur, of all $m$ free variables in $[B]_{\bar{y}}$ which are in $\bar{x}$.
Let $v$ be the alphabetically earliest variable not occurring in $A$. Then if
$\bar{x}^{\prime}$ is the normalizing permutation of $\bar{x}$ it is of the
form $\bar{x}^{\prime\prime}\bar{z}\bar{x}^{\prime\prime\prime}$. Let
$\bar{w}=\bar{x}^{\prime\prime}\bar{x}^{\prime\prime\prime}\bar{z}$. Then

$\mathbf{B}([F(t_{1},...,[B]_{\bar{y}},...t_{n})]_{\bar{x}}):=b(\bar{w},\bar{x})(P_{m}(\mathbf{B}([F(t_{1},...,v,...t_{n})]_{\bar{x}^{\prime\prime}\bar{x}^{\prime\prime\prime}v},\mathbf{B}([B]_{\bar{y}\bar{z}}))))$

  * •

Finally for case 7 let $\bar{x}^{\prime}$ be such that
$[F(v_{1},...,v_{n})]_{\bar{x}^{\prime}}$ is elementary.

$\mathbf{B}([F(t_{1},...,t_{n})]_{\bar{x}}):=b(\bar{x}^{\prime},\bar{x})([F(t_{1},...,t_{n})]_{\bar{x}^{\prime}})$

###  1.15 The Logic of Necessity T1

The conception of intensionality behind T1 is that there is only one necessary
truth and that the intensional abstracts of necessarily equivalent formulas
are equal and interchangeable. We define $\square A\equiv_{df}[A]=[[A]=[A]]$.
A Hilbert system for $T1$ consists of the following axioms and rules over the
language $L^{\omega}$.

  * •

All Tautologies

  * •

(Ins) $\forall v.A(v)\rightarrow A(t)$, $t$ free for $v$ in A.

  * •

(QImp) $\forall v.(A\rightarrow B)\rightarrow(A\rightarrow\forall v.B)$, $v$
not free in $A$

  * •

(Id) $v=v$

  * •

(L) $v=w\rightarrow(A(v,v)\leftrightarrow A(v,w))$, $w$ free for $v$ in the
positions it replaces $v$.

  * •

$\sim[A]_{\bar{x}}=[B]_{\bar{y}}$, $\bar{x}$ and $\bar{y}$ of different
lengths

  * •

$[A]_{\bar{x}}=[A^{\prime}]_{\bar{x}^{\prime}}$ ($\alpha$-equivalence)

  * •

(B)
$[A]_{\bar{x}}=[B]_{\bar{x}}\leftrightarrow\square\forall\bar{x}.(A\leftrightarrow
B)$

  * •

(T) $\square A\rightarrow A$

  * •

(K) $\square(A\rightarrow B)\rightarrow(\square A\rightarrow\square B)$

  * •

(S5) $\lozenge A\rightarrow\square\lozenge A$

  * •

(MP) From $A$ and $A\rightarrow B$ infer $B$

  * •

(N) From $A$ infer $\square A$

  * •

(Gen) From $A$ infer $\forall v.A$

Here, as usual, $\lozenge A=_{df}\neg\square\neg A$.

###  1.16 The Logic of Necessity T1

The following system is equivalent to T1:

  * •

All Tautologies

  * •

(Ins) $\forall v.A(v)\rightarrow A(t)$, $t$ free for $v$ in A.

  * •

(QImp) $\forall v.(A\rightarrow B)\rightarrow(A\rightarrow\forall v.B)$, $v$
not free in $A$

  * •

(Id) $v=v$

  * •

(L’) $v=w\rightarrow(A(v,v)\rightarrow A(v,w))$, $w$ free for $v$ in the
positions it replaces $v$, $A$ atomic.

  * •

$\sim[A]_{\bar{x}}=[B]_{\bar{y}}$, $\bar{x}$ and $\bar{y}$ of different
lengths

  * •

$[A]_{\bar{x}}=[A^{\prime}]_{\bar{x}^{\prime}}$ ($\alpha$-equivalence)

  * •

(B1) $\square(A\leftrightarrow B)\leftrightarrow[A]=[B]$

  * •

(B2) $\forall
v.[A(v)]_{\bar{x}}=[B(v)]_{\bar{x}}\leftrightarrow[A(v)]_{\bar{x}v}=[B(v)]_{\bar{x}v}$

  * •

(T) $\square A\rightarrow A$

  * •

(K) $\square(A\rightarrow B)\rightarrow(\square A\rightarrow\square B)$

  * •

(S5’) $\sim x=y\rightarrow\square\sim x=y$

  * •

(MP) From $A$ and $A\rightarrow B$ infer $B$

  * •

(N) From $A$ infer $\square A$

  * •

(Gen) From $A$ infer $\forall v.A$

###  1.17 The Logic of Necessity T1

To prove these results we use the following

###### Lemma 1.3

In T1 we have:

i) $\vdash x=y\rightarrow\square x=y$

ii) $\vdash\sim\square x=y\leftrightarrow\sim x=y$

The following standard results follow from the modal axioms:

###### Lemma 1.4

i) $\square A\rightarrow\square\lozenge A$

ii) $A\rightarrow\square\lozenge A$ and $\lozenge\square A\rightarrow A$

iii) We have $\vdash\lozenge A\rightarrow B$ iff $\vdash A\rightarrow\square
B$

iv) If $\vdash A\rightarrow B$ then we can derive $\vdash\lozenge
A\rightarrow\lozenge B$

v) $\vdash\square\forall v.A(v)\rightarrow\forall v.\square A(v)$

vi) The _Barcan Formula_ $\vdash\forall.v\square A(v)\rightarrow\square\forall
v.A(v)$

###### Lemma 1.5

We have $\forall s,t.[s=s]=[t=t]$ (There is only one necessary truth).

We can also consider the restricted case (with modal collapse) in which the
intensional operator is interpreted as set-theoretic extension

$[F(v_{1},...,v_{n})]_{v_{1}...v_{n}}\equiv\\{v_{1},...,v_{n}:F(v_{1},...,v_{n})\\}$

in which case (B) is an axiom of extensionality.

###  1.18 The Logic of Intention T2

  * •

All Tautologies

  * •

(Ins) $\forall v.A(v)\rightarrow A(t)$, $t$ free for $v$ in $A$.

  * •

(QImp) $\forall v.(A\rightarrow B)\rightarrow(A\rightarrow\forall v.B)$, $v$
not free in $A$

  * •

(Id) $v=v$

  * •

(L) $v=w\rightarrow(A(v,v)\leftrightarrow A(v,w))$, $w$ free for $v$ in the
positions it replaces $v$.

  * •

$\sim[A]_{\bar{x}}=[B]_{\bar{y}}$, $\bar{x}$ and $\bar{y}$ of different
lengths

  * •

$[A]_{\bar{x}}=[A^{\prime}]_{\bar{x}^{\prime}}$ ($\alpha$-equivalence)

  * •

$[A]_{\bar{x}}=[B]_{\bar{x}}\rightarrow(A\leftrightarrow B)$

  * •

$[A]_{\bar{x}}\neq[B]_{\bar{y}}$, non-elementary terms belonging to different
syntactic domains.

  * •

Let $t^{\prime}=b(t)$ and $r^{\prime}=b(r)$ where $b$ is the first operation
in the Bealer decomposition of $t^{\prime}$ and $r^{\prime}$. Then
$t=r\leftrightarrow t^{\prime}=r^{\prime}$

  * •

Let $t=b(t^{\prime},r^{\prime})$ and $r=b(t^{\prime\prime},r^{\prime\prime})$
where $b$ is the first operation in the Bealer decomposition of $t$ and $r$.
The $t=t^{\prime}\leftrightarrow
t^{\prime}=t^{\prime\prime}\&r^{\prime}=r^{\prime\prime}$.

  * •

If $t=s$ and $t$ is elementary and $F$ occurs in $t$ and $G$ in $s$ then $\sim
u=v$ whenever $u$ is elementary containing $G$ and $F$ occurs in $v$ (Non-
circularity)

  * •

(MP) From $A$ and $A\rightarrow B$ infer $B$

  * •

(TGen) Suppose that $F$ does not occur in $A(v)$, $t$ containing $F$ is
elementary and $t^{\prime}$ is free for $v$ in $A$. Then from $\vdash A(t)$ we
infer $\vdash A(t^{\prime})$.

  * •

(Gen) From $A$ infer $\forall v.A$

###  1.19 Model Structures

Let $F=\emptyset$ and $T=\\{\emptyset,\\{\emptyset\\}\\}$. A model
$\mathcal{M}$ consists of a set $\mathcal{D}$ with a decomposition into the
union of disjoint sets $\mathcal{D}_{i}$ for $i\geq-1$, a distinguished
element $id\in\mathcal{D}_{2}$, a set $\mathcal{H}$ of functions $H$ whose
domain is $\mathcal{D}$ and such that $H$ is the identify on
$\mathcal{D}_{-1}$ and on $\mathcal{D}_{0}$ has range $\\{T,F\\}$ and on
$\mathcal{D}_{i}$ has range $\mathcal{P}(\mathcal{D}^{i})$ where
$\mathcal{D}^{i}=\mathcal{D}\times...\times\mathcal{D}$($i$ terms) for $i\geq
1$. All these functions must satisfy $H(id)={(x,y):x=y}$. There is a
distinguished function $\mathcal{G}$ representing the _actual extension_.
Furthermore we have unary (partial) operations $n,e,u,a,i,r$ and binary
(partial) operations $a,p_{n}$ with $n\geq 0$. These operations must satisfy:

  * •

$n(\mathcal{D}_{i})\subset\mathcal{D}_{i},i\geq 0$

  * •

$e(\mathcal{D}_{i})\subset\mathcal{D}_{i+1},i\geq 0$

  * •

$u(\mathcal{D}_{i})\subset\mathcal{D}_{i-1},i\geq 1$ and
$exis(\mathcal{D}_{0}\subset\mathcal{D}_{0})$.

  * •

$c(\mathcal{D}_{i})\subset\mathcal{D}_{i},i\geq 2$

  * •

$i(\mathcal{D}_{i})\subset\mathcal{D}_{i},i\geq 3$

  * •

$r(\mathcal{D}_{i})\subset\mathcal{D}_{i-1},i\geq 2$

  * •

$a(\mathcal{D}_{i}\times\mathcal{D}_{i})\subset\mathcal{D}_{i},i\geq 0$

  * •

$p_{n}(\mathcal{D}_{i}\times\mathcal{D_{j}})\subset\mathcal{D}_{i+n-1},i\geq
1,j\geq n$

We define inductively
$p_{0}^{n}(d,x_{1},...,x_{n})=p_{0}(p_{0}^{n-1}(d,x_{2},...,x_{n-1}),x_{1})$
and $p_{0}^{1}(d,x_{1})=p_{0}(d,x_{1})$.

###  1.20 Model Structures

Furthermore we have the important constraint on how the functions in
$\mathcal{H}$ relate to these operations. Here $d$ denotes a suitable member
of $\mathcal{D}^{i}$. For all $\mathcal{H}\in H$ we must have:

  * •

$d^{\prime}\in\mathcal{H}n(d)\text{ iff }d^{\prime}\notin\mathcal{H}(d)$

  * •

$(x_{1},...,x_{i},x_{i+1})\in\mathcal{H}e(d)\text{ iff
}(x_{1},...,x_{i})\in\mathcal{H}(d)$

  * •

$(x_{1},...,x_{i-1})\in\mathcal{H}u(d)\text{ iff there is a $x_{i}$ such that
}(x_{1},...,x_{i})\in\mathcal{H}(d)$

  * •

$(x_{1},...,x_{i-1},x_{i})\in\mathcal{H}c(d)\text{ iff
}(x_{2},...,x_{i},x_{1})\in\mathcal{H}(d)$

  * •

$(x_{1},...,x_{i-1},x_{i})\in\mathcal{H}i(d)\text{ iff
}(x_{1},...,x_{i},x_{i-1})\in\mathcal{H}(d)$

  * •

$(x_{1},...,x_{i-1},x_{i})\in\mathcal{H}r(d)\text{ iff
}(x_{1},...,x_{i},x_{i})\in\mathcal{H}(d)$

  * •

$d^{\prime\prime}\in\mathcal{H}a(d,d^{\prime})\text{ iff
}d^{\prime\prime}\in\mathcal{H}(d)\text{ and }d\in\mathcal{H}(d^{\prime})$

  * •

$(x_{1},...,x_{n})\in\mathcal{H}p_{0}(d,d^{\prime})\text{iff}(x_{1},...,x_{n},d^{\prime})\in\mathcal{H}(d)$

  * •

$(x_{1},...,x_{i-1},y_{1},..,y_{n})\in\mathcal{H}p_{n}(d,d^{\prime})\text{ iff
}(x_{1},...,x_{i},p_{0}^{n}(d^{\prime},y_{1},...,y_{n}))\in\mathcal{H}(d)$

###  1.21 Model Structures

  * •

A model $\mathcal{M}$ is called type 1 if forall $i\geq-1$,
$x,y\in\mathcal{D}_{i}$ and for all $\mathcal{H}\in H$ we have
$\mathcal{H}(x)=\mathcal{H}(x)$ then $x=y$.

  * •

A model $\mathcal{M}$ is called type 2 if the operations are one-to-one,
disjoint on their ranges and non-cycling. This means that each element has a
unique decomposition in terms of these operations.

  * •

Given a model $\mathcal{M}$, an _interpretation_ $\mathcal{I}$ assigns $i$-ary
predicate symbols to elements of $\mathcal{D}_{i}$ and $=$ to $id$. An
_assignment_ $\mathcal{A}$ assigns the variables to elements in $\mathcal{D}$.

  * •

Given $\mathcal{M},\mathcal{I}$ and $\mathcal{A}$ an we define the
_denotation_ $D_{\mathcal{I},\mathcal{A},\mathcal{M}}(t)$ of a term $t$ as
follows. If $t$ is a variable then it is $\mathcal{A}(t)$. If $t$ is an
elementary term $[F(v_{1},...,v_{n})]_{v_{1}...v_{n}}$ then it is
$\mathcal{I}(F)$. Otherwise, consider the Bealer decomposition $\mathbf{B}(t)$
of $t$. Then in the formal expression we replace $A,E_{v},N,U,I,C,R,P_{n}$ by
$a,e,n,u,i,c,r,p_{n}$ and the elementary abstracts and variables which appear
by their interpretations and assignments to obtain a well-defined element of
$\mathcal{D}$.

  * •

Given a formula $A$ we define

$T_{\mathcal{I},\mathcal{A},\mathcal{M}}(A)\textit{ iff
}\mathcal{G}D_{\mathcal{I},\mathcal{A},\mathcal{M}}([A])=T$

  * •

A formula $A$ is T1-valid iff $T_{\mathcal{I},\mathcal{A},\mathcal{M}}(F)$ for
all type 1 model $\mathcal{M}$, interpretation $\mathcal{I}$ and assignment
$\mathcal{A}$.

  * •

A formula $A$ is T2-valid iff $T_{\mathcal{I},\mathcal{A},\mathcal{M}}(F)$ for
all type 2 model $\mathcal{M}$, interpretation $\mathcal{I}$ and assignment
$\mathcal{A}$.

###  1.22 Soundness of T1 and T2

###### Theorem 1.6

A formula $A$ is T1-valid if $\vdash_{T1}A$

###### Theorem 1.7

A formula $A$ is T2-valid if $\vdash_{T2}A$

###  1.23 Soundness of T1 and T2

To prove these soundness results we use the following important auxiliary
lemmas

  * •

Let $v_{k}$ be an externally quantifiable variable in $[B(v_{k})]_{\bar{x}}$
and let $t_{k}$ be free for $v_{k}$ in $[B(v_{k})]_{\bar{x}}$. Consider any
model structure $\mathcal{M}$ and any interpretation $\mathcal{I}$ and
assignment $\mathcal{A}$. Let $\mathcal{A}^{\prime}$ be an assignment which is
just like $\mathcal{A}$ except that
$\mathcal{A}^{\prime}(v_{k})=D_{\mathcal{I}\mathcal{A}\mathcal{M}}(t_{k})$.
Then

$D_{\mathcal{I}\mathcal{A^{\prime}}\mathcal{M}}([B(v_{k})]_{\bar{x}})=D_{\mathcal{I}\mathcal{A}\mathcal{M}}([B(t_{k})]_{\bar{x}})$

  * •

For all $\mathcal{M}$ and $\mathcal{I}$, $\mathcal{A}$ we have
$D_{\mathcal{I}\mathcal{A}\mathcal{M}}([A]_{v_{1}...v_{k}})\in\mathcal{D}_{k}$

  * •

For all $\mathcal{M}$ of type 1 and $\mathcal{I}$, $\mathcal{A}$ and terms
$t,t^{\prime}$ we have

$D_{\mathcal{I}\mathcal{A}\mathcal{M}}([t=t])=D_{\mathcal{I}\mathcal{A}\mathcal{M}}([t^{\prime}=t^{\prime}])$

  * •

Let $v_{k}$ be free in $[A(v_{k})]_{\bar{x}}$. Then for all $\mathcal{M}$ of
type 1 and $\mathcal{I}$, $\mathcal{A}$

$D_{\mathcal{I}\mathcal{A}\mathcal{M}}([A(v_{k})]_{\bar{x}})=p_{0}(D_{\mathcal{I}\mathcal{A}\mathcal{M}}([A(v_{k})]_{\bar{x}v_{k}}),\mathcal{A}(v_{k}))$

  * •

For all $\mathcal{I}$,$\mathcal{A}$,$\mathcal{M}$,
$T_{\mathcal{I}\mathcal{A}\mathcal{M}}(F(t_{1},...,t_{n}))$ iff
$(D_{\mathcal{I}\mathcal{A}\mathcal{M}}(t_{1}),...,D_{\mathcal{I}\mathcal{A}\mathcal{M}}(t_{n}))\in\mathcal{I}(F)$.

  * •

$T_{\mathcal{I}\mathcal{A}\mathcal{M}}(A\&B)$ iff
$T_{\mathcal{I}\mathcal{A}\mathcal{M}}(A)$ and
$T_{\mathcal{I}\mathcal{A}\mathcal{M}}(B)$. Also
$T_{\mathcal{I}\mathcal{A}\mathcal{M}}(\neg A)$ iff it is not the case that
$T_{\mathcal{I}\mathcal{A}\mathcal{M}}(A)$.

  * •

Let $\mathcal{M}$ be type 1. Then
$T_{\mathcal{I}\mathcal{A}\mathcal{M}}(\exists v_{k}A)$ iff there is an
assignment $\mathcal{A}^{\prime}$ like $\mathcal{A}$ except perhaps for what
it assigns to $v_{k}$ and such that
$T_{\mathcal{I}\mathcal{A^{\prime}}\mathcal{M}}(A(v_{k}))$.

###  1.24 Completeness of T1

###### Theorem 1.8

If a formula $A$ is T1-valid then $\vdash_{T_{1}}A$.

Since $\vdash_{T_{1}}A$ iff $\\{\neg A\\}$ is inconsistent it is enough to
show

###### Theorem 1.9

A set formulas $\Gamma$ in $T1$ is consistent iff it has a T1-model.

Only one direction needs to be shown. For if $\Gamma$ has a model then it must
be consistent.

###### Definition 1.10

Let $L^{*}_{\omega}$ be an extension of $L_{\omega}$. A set $\mathcal{A}$ of
sets of sentences in $L^{*}_{\omega}$ is $T1$-_perfect_ if

  * •

Every set in $\mathcal{A}$ is maximal, consistent and $\omega$-complete.

  * •

If an identity sentence $t=t^{\prime}$ is in one of the sets of $\mathcal{A}$
it is in all of them.

  * •

If a sentence $[A]_{\bar{x}}=[B]_{\bar{x}}$ belongs to one set of
$\mathcal{A}$ there is a set in $\mathcal{A}$ which contains the sentence
$\exists\bar{x}.\sim A\leftrightarrow B$.

  * •

For every closed term $[B]_{x_{1}...x_{n}}$ there is an atomic predicate $F$
such that $[B]_{x_{1}...x_{n}}=[F(x_{1},...,x_{n})]_{x_{1}...x_{n}}$ belongs
to a set in $\mathcal{A}$.

###  1.25 Completeness of T1

The theorem follows from the following two lemmas:

###### Lemma 1.11

Let $\Gamma$ be consistent. Then there is a countable extension of
$L_{\omega}$ relative to which there is a $T1$-perfect set $\mathcal{A}$
containing $\Gamma$ in one of its members.

###### Lemma 1.12

Every $T1$-perfect set relative to an extension of $L_{\omega}$ has a type 1
model.

###  1.26 Completeness of T2

###### Theorem 1.13

If a formula $A$ is T2-valid then $\vdash_{T_{2}}A$.

Since $\vdash_{T_{2}}A$ iff $\\{\neg A\\}$ is inconsistent it is enough to
show

###### Theorem 1.14

A set formulas $\Gamma$ in $T2$ is consistent iff it has a T2-model.

Only one direction needs to be shown. For if $\Gamma$ has a model then it must
be consistent.

###### Definition 1.15

Let $L^{*}_{\omega}$ be an extension of $L_{\omega}$. A set $\mathcal{A}$ of
sets of sentences in $L^{*}_{\omega}$ is $T2$-_perfect_ if

  * •

Every set in $\mathcal{A}$ is maximal, consistent and $\omega$-complete.

  * •

For every closed term $[B]_{x_{1}...x_{n}}$ there is an atomic predicate $F$
such that $[B]_{x_{1}...x_{n}}=[F(x_{1},...,x_{n})]_{x_{1}...x_{n}}$ belongs
to a set in $\mathcal{A}$.

###  1.27 Completeness of T2

The theorem follows from the following two lemmas:

###### Lemma 1.16

Let $\Gamma$ be consistent. Then there is a countable extension of
$L_{\omega}$ relative to which there is a $T2$-perfect set $\mathcal{A}$
containing $\Gamma$ in one of its members.

###### Lemma 1.17

Every $T2$-perfect set relative to an extension of $L_{\omega}$ has a type 2
model.

###  1.28 Fregean style Arithmetic in T2

Let $\Delta$ be a binary predicate in T2, where we write $x\Delta y$ for
$\Delta(x,y)$.

  * •

$0=_{df}[\neg\exists v(v\Delta y)]_{y}$

  * •

$NNx\equiv_{df}\forall z(I(z)\rightarrow x\Delta z)$

  * •

$I(z)\equiv_{df}0\Delta z\enspace\&\enspace(\forall y(y\Delta z\rightarrow
S(y)\Delta z))$

  * •

$S(x)\equiv_{df}[\exists u(u\Delta x\enspace\&\enspace\exists v(\neg v\Delta
u\enspace\&\enspace y\cong[w=v\vee w\Delta u)]_{w})]_{y}$

  * •

$x\cong y\equiv_{df}\forall w(w\Delta x\leftrightarrow w\Delta y)$

###  1.29 Fregean Arithmetic in T2

###### Lemma 1.18

Bealer, _Quality and Concept_ , p122.

  * •

$NN0$

  * •

$NNx\rightarrow NN(S(x))$

  * •

$\neg\exists x.0=S(x)$

  * •

$x=y\rightarrow S(x)=S(y)$

  * •

$\forall z(I(z)\rightarrow\forall x(NNx\rightarrow x\Delta z))$

Let $A(x)\equiv_{df}NNx\enspace\&\enspace\exists y.(NNy\enspace\&\enspace
y=S(x))$. Consider the following _predication axiom_

$x\Delta[A(u)]_{u}\leftrightarrow A(x)$

###### Theorem 1.19

_Quality and Concept_ , p127. If we add the above predication axiom the
complete theory for the structure $<NN,=,S,0>$ is derivable in T2.

Frege’s Law V is generally blamed for the paradoxes. But the paradoxes can be
avoided if the law is retained but higher-order variables and quantifiers are
dropped, for then we obtain the sound and complete system of first-order logic
with extensional abstraction (T1 with modal collapse), law V corresponding to
axiom B.

### Future Work

  * •

Applications to philosophical logic: defining a truth predicate, solving the
paradoxes, explaining incompleteness, etc.

  * •

Alternative presentation: introduce a single term forming operator $[A]$
(Quine operator) combined with $\lambda$ abstraction so that
$[A]_{x_{1}...x_{n}}=\lambda x_{1}...\lambda x_{n}[A]$. Problem: how do we
interpret $\lambda x.x$ ?

  * •

Prove soundness and completeness of T2’, Bealer’s proposal for a combination
of logics T1 and T2.

  * •

Prove the conjectured soundness and completenes of Bealer’s logic of qualities
and concepts T3 (_Q. and C._ p.192-195 ).

  * •

Natural Deduction presentations of the logic and intuitionistic, type-
theoretic versions wherein various axioms are interpreted as rules.

  * •

Combinatory Logic style presentation without variables ?

  * •

Connection to other formulations of Fregean Arithmetic based on fragments of
second-order logic: intensional abstraction corresponds to the abstraction
operator of Prawitz’s type 2 presentation of second-order logic.

  * •

Generalisation of subobect classifiers in Topos Theory to account for
intentional abstracts.

  * •

Applications to the formal analysis of natural language.

## Some theorems in T1 and alternative axioms

###### Lemma 1.20

In T1 we have:

i) $\vdash x=y\rightarrow\square x=y$

ii) $\vdash\sim\square x=y\leftrightarrow\sim x=y$

###### Proof.

For i), by the identity axiom we have $x=x$. Hence using the necessitation
rule we get

$\square x=x$

which is

$[(x=x)]=[[x=x]=[x=x]]$

which we shall denote by $B(x,x)$. Use $B(x,x)$ in the classical axiom scheme

$v=w\rightarrow A(v,v)\leftrightarrow A(v,w)$

with $v=x$, $w=y$, swap the two right-most implications and use MP to obtain
as desired.

For ii), take the converse of the $T$ instance $\square x=y\rightarrow x=y$ to
obtain $\neg x=y\rightarrow\neg\square x=y$. For the other direction simply
take the converse of i)

###### Lemma 1.21

We obtain an equivalent system if in T1 we replace S5 with

(*) $\sim x=y\rightarrow\square\sim x=y$.

###### Proof.

Assuming (*), since $\lozenge A$ is by definition $\neg([\neg A]=[[\neg
A]=[\neg A]])$ we get S5 immediately.

In the other direction take the S5 instance

$\sim\square\neg\neg x=y\rightarrow\square\sim\square\sim\sim x=y$

that is,

$\sim\square x=y\rightarrow\square\sim\square x=y$

Now by ii) of lemma 1.20 we have

$\sim\square x=y\rightarrow\sim x=y$

Applying necessitation and using $K$ we get

$\square\sim\square x=y\rightarrow\square\sim x=y$

and combining with (**) we get as desired.

We have the well-known ”adjunction” and ”functorial” relations of S5.

###### Lemma 1.22

In T1 we have

i) $\vdash A\rightarrow\lozenge A$

ii) $\vdash A\rightarrow\square\lozenge A$

iii) $\lozenge\square A\rightarrow A$

iv) If $\vdash A\rightarrow B$ then we can derive $\vdash\lozenge
A\rightarrow\lozenge B$

v) We have $\vdash\lozenge A\rightarrow B$ iff $\vdash A\rightarrow\square B$

###### Proof.

For i) take the converse of the T instance $\square\sim A\rightarrow\sim A$.
For ii) combine i) and S5. For iii) take the converse of the S5 instance
$\sim\square\lozenge\sim A\rightarrow\sim\lozenge\sim A$ and use $T$. For iv)
take the converse of the premise, apply T and take the converse again. Finally
v) follows easily from T, iv), iii) and ii).

###### Lemma 1.23

$\vdash\square\forall v.A(v)\rightarrow\forall v.\square A(v)$

###### Proof.

Take the Ins instance

$\forall v.A(v)\rightarrow A(v)$

Applying necessitation, $K$ and modus ponens yields:

$\square\forall v.A(v)\rightarrow\square A(v)$

Applying generalization:

$\forall v.(\square\forall v.A(v)\rightarrow\square A(v))$

Since $v$ is not free on the left side of the implication we can apply QImp
and MP to obtain

$\square\forall v.A(v)\rightarrow\forall v.\square A(v)$

###### Lemma 1.24

We have the _Barcan Formula_ $\vdash\forall v.\square
A(v)\rightarrow\square\forall v.A(v)$.

###### Proof.

We have by Ins

$\forall v.\square A\rightarrow\square A$

Applying $\lozenge$, lemma 1.22 iv) yields

$\lozenge\forall v.\square A\rightarrow\lozenge\square A\rightarrow A$

Hence by generalization, QImp and modus ponens:

$\lozenge\forall v.\square A\rightarrow\forall v.A$

which by adjunction yields

$\forall v.\square A\rightarrow\square\forall v.A$

###### Lemma 1.25

We obtain an equivalent system to T1 when the identity axiom L is restricted
to atomic predicates only and is restricted to the form (L*)
$x=y\rightarrow(A(x,x)\rightarrow A(x,y))$.

###### Proof.

L obviously implies L* as a particular case. In the other direction we first
show that $x=y\rightarrow(A(x,x)\leftarrow A(x,y))$. We have that
$x=y\rightarrow(A(x,x)\rightarrow A(x,y))$ and $A(x,y)$ arises from $A(x,x)$
by replacing some occurrences of $x$ by $y$. So $A(x,x)$ arises from replacing
some occurrences of $y$ in $A(x,y)$ by $x$. Hence we have
$y=x\rightarrow(A(x,y)\rightarrow A(x,x))$. Also from L* we can derive
$x=y\rightarrow y=x$ and using this we get as desired. We can now proceed
inductively on the structure of the formula $A$. The case of negation is
immediate. For conjunction we use the tautology $(A\rightarrow
B)\rightarrow(C\rightarrow D)\rightarrow(A\&C\rightarrow B\&D)$ and for
quantification we use generalisation, QImp and modus ponens.

###### Lemma 1.26

We also obtain an equivalent system to T1 if instead of B take

$\square(A\leftrightarrow B)\leftrightarrow([A]=[B])$

$\forall
v.([A(v)]_{\bar{x}}=[B(v)]_{\bar{x}})\leftrightarrow[A(v)]_{\bar{x}v}=[B(v)]_{\bar{x}v}$

###### Proof.

We show that

$[A]_{\bar{x}}=[B]_{\bar{x}}\leftrightarrow\square\forall\bar{x}.(A\leftrightarrow
B)$

Let $\bar{x}=\bar{x^{\prime}}v$.

By B’2 $[A]_{\bar{x^{\prime}}v}=[B]_{\bar{x^{\prime}}v}\leftrightarrow\forall
v.([A(v)]_{\bar{x^{\prime}}}=[B(v)]_{\bar{x^{\prime}}})$ and using again B’2
repeatedly we get

$[A]_{\bar{x}v}=[B]_{\bar{x}v}\leftrightarrow\forall\bar{x},v.([A(v)]=[B(v)])$

Using B’1 we get

$[A]_{\bar{x}}=[B]_{\bar{x}}\leftrightarrow\forall\bar{x}.\square(A\leftrightarrow
B)$

Using repeatedly Barcan’s formula and its converse we get B as desired. The
other direction is similar.

###### Lemma 1.27

We have $\forall x,y.[x=x]=[y=y]$.

###### Proof.

We have $x=x\leftrightarrow y=y$. We apply necessitation and B’1 and then
generalisation to get as desired.


# Revisiting log-periodic oscillations

Jean-Marc Luck  Université Paris-Saclay, CNRS, CEA, Institut de Physique
Théorique, 91191 Gif-sur-Yvette, France<EMAIL_ADDRESS>
###### Abstract

This work is inspired by a recent study of a two-dimensional stochastic
fragmentation model. We show that the configurational entropy of this model
exhibits log-periodic oscillations as a function of the sample size, by
exploiting an exact recursion relation for the numbers of its jammed
configurations. This is seemingly the first statistical-mechanical model where
log-periodic oscillations affect the size dependence of an extensive quantity.
We then propose and investigate in great depth a one-dimensional analogue of
the fragmentation model. This one-dimensional model possesses a critical
point, separating a strong-coupling phase where the free energy is super-
extensive from a weak-coupling one where the free energy is extensive and
exhibits log-periodic oscillations. This model is generalized to a family of
one-dimensional models with two integer parameters, which exhibit essentially
the same phenomenology.

###### keywords:

Log-periodic oscillations , Oscillatory critical amplitudes , Fragmentation
models , Finite-size scaling , Non-linear recursions , Discrete scale
invariance

##  1 Introduction

From the very beginning of the renormalization-group era, it has been
suggested that discrete real-space renormalization-group transformations might
result in the modulation of critical behavior by a log-periodic function of
the distance $t=|T-T_{c}|/T_{c}$ to the critical point [1, 2, 3]. Oscillatory
critical amplitudes have been later observed and investigated for models
defined on hierarchical lattices [4, 5, 6, 7, 8, 9, 10]. These lattices are
self-similar geometric structures that exhibit discrete scale invariance, and
therefore admit exact renormalization-group transformations. Similar features
are shared by other fractal structures, on which various models have been
shown to exhibit log-periodic oscillations (see e.g. [11, 12, 13, 14, 15, 16,
17]). The same phenomenon has also been evidenced as a consequence of the
fractal spectra of some aperiodic structures (see e.g. [18, 19, 20, 21]). The
first ever mention of log-periodic oscillations seems to date back to 1948,
long before the renormalization group was applied to phase transitions. In a
work devoted to the theory of branching processes, Harris suggested that
iterating a discrete mapping might yield log-periodic oscillations [22]. These
oscillations have then been observed and studied in a more detailed
investigation of a germane combinatorial problem, namely the enumeration of a
class of binary trees [23].

Log-periodic oscillations have since then been reported in a broad variety of
situations, as testified by the overview by Sornette [24]. In many settings
including turbulence, fracture, earthquakes, financial crashes, and quantum
gravity, the occurrence of oscillations is attributed to the emergence of an
approximate discrete scale invariance. The associated scaling factor is often
around two. Various physical mechanisms, including the period-doubling route
to chaos, have been invoked to explain this phenomenon. In most of the above
situations, the observed amplitude of oscillations is sizeable, i.e., of the
order of 10 percent. This figure is in stark contrast with the historical
example of critical phenomena, where log-periodic oscillations are typically
tiny, with a magnitude of order $10^{-5}$, and therefore hard to observe.

The singular behavior of physical quantities might also be modulated by
periodic amplitudes for reasons that are unrelated to discrete scale
invariance. Confining the discussion to statistical physics, one-dimensional
disordered systems provide a breadth of examples of interest. There, periodic
oscillations are eventually due to a discrete translation invariance,
reflecting the atomic nature of the underlying lattice [25, 26]. In some
models exhibiting anomalous biased diffusion, the growth law of the mean
displacement is modulated by a log-periodic function of time [27, 28]. In
electron and phonon spectra of disordered chains, the density of states has
exponentially small Lifshitz tails, whose amplitudes are modulated by periodic
functions of $(\Delta E)^{-1/2}$, where $\Delta E$ is the distance to the band
edge [29, 30].

A last example, which motivated the present work, consists of a stochastic
fragmentation model introduced in [31]. A rectangular sample with size $(m,n)$
drawn on the square lattice is randomly cut into four smaller ones. The
process is then repeated and stops when the system reaches a jammed
configuration where all parts are sticks, i.e., the smaller size of each part
equals unity. Consider for definiteness a square sample of size $n$. In the
first step of the fragmentation process, the square is cut into four parts
whose linear sizes are of the order of $n/2$. This observation suggests the
emergence of some weak form of discrete scale invariance with scaling factor
two, that is somehow reminiscent of what is claimed to occur in more complex
phenomena such as turbulence or diffusion-limited aggregation [24]. The main
advantage of the above fragmentation model, which triggered our interest, is
the existence of an exact recursion formula for the numbers $Z_{m,n}$ of
jammed configurations.

The setup of the present paper is as follows. In Section 2 we revisit the
stochastic fragmentation model introduced in [31]. By exploiting the exact
recursion recalled above, we demonstrate that the configurational entropy
based on the numbers $Z_{n,n}$ of jammed configurations on square samples
exhibits tiny log-periodic oscillations in the sample size $n$. We then
introduce and investigate in detail a one-dimensional (1D) toy model capturing
the essential features of the combinatorics of the fragmentation model
(Sections 3 to 6). In spite of its relative simplicity, the 1D model turns out
to have a rich phenomenology, with a critical point (Section 4) separating a
strong-coupling phase (Section 5) from a weak-coupling one (Section 6). In
Section 7 we consider several other examples of 1D non-linear recursions. A
brief discussion of our findings is given in Section 8.

##  2 The fragmentation model revisited

In this section we revisit the second fragmentation model introduced in [31].
This irreversible stochastic model is defined recursively as follows. The
sample is a rectangle with size $(m,n)$ drawn on the square lattice. The first
step of the fragmentation consists in cutting the rectangle into four smaller
ones, of respective sizes $(i,j)$, $(i,n-j)$, $(m-i,j)$, $(m-i,n-j)$ (see
Figure 1). The integers $i$ and $j$ are chosen uniformly in the ranges $1\leq
i\leq m-1$ and $1\leq j\leq n-1$. The process is repeated and stops when the
system reaches a jammed configuration where all parts are sticks, i.e., the
smaller size of each part equals unity. Figure 2 shows a jammed tiling thus
obtained on a square of size 50.

Figure 1: First step of the fragmentation of a rectangular sample with $m=5$,
$n=3$, $i=2$ and $j=1$. Figure 2: A jammed tiling of a square sample of size
50 (courtesy Paul Krapivsky).

This fragmentation model enjoys the property that the stochastic histories of
different rectangles are mutually independent from the epoch they are formed.
As a consequence, the numbers $Z_{m,n}$ of jammed tiling configurations on
rectangular samples of size $(m,n)$ obey the recursion formula [31]

$Z_{m,n}=\sum_{i=1}^{m-1}\sum_{j=1}^{n-1}Z_{i,j}Z_{i,n-j}Z_{m-i,j}Z_{m-i,n-j},$
(1)

with initial conditions $Z_{m,1}=Z_{1,n}=1$ for all $m$ and $n$. These
‘initial’ values actually express the ‘final’ condition that a jammed tiling
is reached when all parts are sticks, i.e., the smaller size of each part
equals unity. The recursion formula (1) determines all the numbers of jammed
tilings, which obey the symmetry $Z_{m,n}=Z_{n,m}$. The first few of them are
listed in Table 1.

$\begin{array}[]{|c|rrrrrrrr|}\hline\cr
m\hbox{\large{$\backslash$}}n&1&2&3&4&5&6&7&8\\\ \hline\cr
1&1&1&1&1&1&1&1&1\\\ 2&1&1&2&3&4&5&6&7\\\ 3&1&2&4&10&20&36&60&94\\\
4&1&3&10&33&98&258&618&1\,379\\\ 5&1&4&20&98&436&1\,676&5\,848&18\,906\\\
6&1&5&36&258&1\,676&9\,524&48\,296&225\,938\\\
7&1&6&60&618&5\,848&48\,296&354\,224&2\,387\,112\\\
8&1&7&94&1\,379&18\,906&225\,938&2\,387\,112&23\,097\,969\\\
\hline\cr\end{array}$ Table 1: Numbers $Z_{m,n}$ of jammed tilings on
rectangular samples of size $(m,n)$ for $m$ and $n$ up to 8.

The numbers $Z_{m,n}$ of tiling configurations grow rapidly with the sample
size $(m,n)$. It is indeed to be expected that their logarithm is extensive,
in the sense that it obeys an area law of the form

$\ln Z_{m,n}\approx Smn,$ (2)

where $S$ is the bulk configurational entropy of the model per unit area [32,
33], for which the estimate $S\approx 0.2805$ is given in [31]. There are
therefore some $10^{304}$ different configurations of jammed tilings on a
square of size 50, only one of which is shown in Figure 2. A more complete
thermodynamical expression for $\ln Z_{m,n}$, including the contributions
$S_{e}$ of the edges of the sample and $S_{c}$ of its corners, reads

$\ln Z_{m,n}\approx Smn+2S_{e}(m+n)+4S_{c}.$ (3)

The asymptotic behavior of $\ln Z_{m,n}$ emerges as a global property of the
solution to the recursion (1), that has resisted all our attempts at analysis.
As it turns out, the entropies $S$, $S_{e}$ and $S_{c}$ are modulated by log-
periodic oscillations (see below). Moreover, at variance with the
thermodynamical expectation (3), the edge and corner entropies $S_{e}$ and
$S_{c}$ have a complex dependence on the aspect ratio $r=m/n$ of the sample
(not described here). None of these peculiar features was mentioned in [31].

In order to pursue, we have recourse to a numerical solution of the recursion
(1). For the sake of definiteness, we focus our attention onto square samples.
As already put forward in Section 1, a square of size $n$ is cut into four
parts whose sizes are of the order of $n/2$ in the first step of the
fragmentation process. This suggests the emergence of some weak form of
discrete scale invariance with scaling factor two, that is somehow reminiscent
of what is claimed to occur in turbulence or in diffusion-limited aggregation
[24]. This intuitive line of reasoning therefore opens the possibility that
the area law (2) might be modulated by a 1-periodic oscillatory function of
the variable

$x=\frac{\ln n}{\ln 2}.$ (4)

Our goal is to demonstrate that log-periodic oscillations are indeed present
in the configurational entropy of the fragmentation model. For methodological
reasons, we begin by considering the ratio

$R_{n}=\frac{Z_{n,n}}{(Z_{n/2,n/2})^{4}}$ (5)

for even $n$. The denominator of (5) is nothing but the central term in the
expression (1) for $Z_{n,n}$, corresponding to $i=j=n/2$. It is therefore
natural to interpret $R_{n}$ as the effective number of terms contributing to
the recursion (1) for $Z_{n,n}$.

Figure 3: Logarithmic plot of the ratio $R_{n}$ introduced in (5) against the
logarithmic variable $x$ defined in (4).

Figure 3 shows a logarithmic plot of $R_{n}$ against the logarithmic variable
$x$ defined in (4). If the numbers $Z_{n,n}$ of tiling configurations were
exactly given by the thermodynamic formula (3), we would have

$\ln R_{n}\approx-4S_{e}n-12S_{c}.$ (6)

It is obvious from Figure 3 that the edge configurational entropy $S_{e}$
vanishes for square samples. Indeed $\ln R_{n}$ would otherwise exhibit a
linear growth in $n$ with slope $4|S_{e}|$, i.e., an exponential growth in
$x$.

The quantity $\ln R_{n}$ plotted in Figure 3 clearly exhibits 1-periodic
oscillations in $x$. Here and throughout the following, we characterize an
oscillatory periodic function $f(x)$ by its average $f_{\rm ave}$ over one
period and by the relative magnitude of oscillations,

$f_{\rm osc}=\frac{f_{\rm max}-f_{\rm min}}{f_{\rm ave}}.$ (7)

The data shown in Figure 3 yield

$(\ln R)_{\rm ave}\approx 3.060,\qquad(\ln R)_{\rm osc}\approx 9.8\;10^{-3}.$
(8)

Pre-asymptotic corrections to log-periodic behavior are negligible in the
range shown in Figure 3.

The corner configurational entropy of a large square can be estimated as

$S_{c}=-\frac{(\ln R)_{\rm ave}}{12}\approx-0.255,$ (9)

whereas the effective number $R_{n}$ of terms entering the recursion (1)
approaches the rather large but finite limit

$R_{\rm eff}=\exp\left((\ln R)_{\rm ave}\right)\approx 21.32.$ (10)

A more accurate definition of the bulk configurational entropy of square
samples is given by

$S_{n}=\frac{\ln Z_{n,n}-4S_{c}}{n^{2}},$ (11)

where $S_{c}$ is taken from (9). Figure 4 shows that this quantity exhibits
very small but very clear 1-periodic oscillations in the logarithmic variable
$x$. This is seemingly the first instance where log-periodic oscillations are
reported for the size dependence of an extensive quantity in a statistical-
mechanical model.

The subtraction of the corner entropy $4S_{c}$ has drastically improved
convergence, so that corrections are again negligible in the range shown in
Figure 4. From a quantitative viewpoint, we have

$S_{\rm ave}\approx 0.280481,\qquad S_{\rm osc}\approx 1.44\;10^{-4}.$ (12)

The numerical value of $S_{\rm ave}$ is in full agreement with the estimate
$S\approx 0.2805$ given in [31]. The magnitude $S_{\rm osc}$ of oscillations
is within the range commonly observed, e.g. in the critical properties of
models on hierarchical lattices.

Figure 4: Configurational entropy $S_{n}$ defined in (11), plotted against the
logarithmic variable $x$ defined in (4).

##  3 1D model: generalities

In this section we introduce a one-dimensional (1D) analogue of the
combinatorics of jammed configurations in the fragmentation model investigated
in Section 2. This 1D toy model captures the main features of the
fragmentation model, whereas it is simple enough to be studied in great depth
(see Sections 4 to 6). Furthermore, the 1D model has a richer phenomenology
than the original fragmentation model, with a critical point separating a
weak-coupling phase, where the free energy is extensive and exhibits
oscillations, from a strong-coupling one, where the free energy is super-
extensive and does not manifest oscillations.

The 1D model is defined by the recursion relation

$Z_{n}=\sum_{k=1}^{n-1}Z_{k}^{2}Z_{n-k}^{2}\qquad(n\geq 2).$ (13)

This recursion keeps the essential characteristics of its two-dimensional
analogue (1), including chiefly the global degree four of the right-hand side,
and the symmetric roles of $Z_{k}$ and $Z_{n-k}$. The recursion (13) requires
only one initial condition, namely $Z_{1}$. We set

$Z_{1}=a.$ (14)

Hereafter the parameter $a$ is chosen to be positive, and viewed as a coupling
constant. The $Z_{n}$ are then positive, and interpreted as partition
functions.

The simplest of all initial conditions, $a=1$, yields integer values for the
$Z_{n}$:

$\displaystyle Z_{1}=1,\qquad Z_{2}=1,\qquad Z_{3}=2,\qquad Z_{4}=9,\qquad
Z_{5}=170,$ $\displaystyle Z_{6}=57\,978,\qquad Z_{7}=6\,722\,955\,416,$ (15)

and so on. These numbers are listed as sequence number A053294 in the OEIS
[34], where they are defined as the solution to the very recursion (13),
without any motivation nor any useful result.

For a generic initial condition, we have

$\displaystyle Z_{2}=a^{4},$ $\displaystyle Z_{3}=2a^{10},$ $\displaystyle
Z_{4}=a^{16}+8a^{22},$ $\displaystyle
Z_{5}=8a^{28}+2a^{34}+32a^{40}+128a^{46},$ $\displaystyle
Z_{6}=18a^{40}+32a^{46}+128a^{52}+128a^{58}+64a^{64}+1032a^{70}$
$\displaystyle{\hskip 12.0pt}+4352a^{76}+3072a^{82}+16384a^{88}+32768a^{94},$
(16)

and so on. The partition function $Z_{n}$ is a polynomial in $a$ of the form

$Z_{n}=A_{n}a^{\alpha_{n}}+\cdots+B_{n}a^{\beta_{n}}.$ (17)

The terms $A_{n}a^{\alpha_{n}}$ of lowest degrees and $B_{n}a^{\beta_{n}}$ of
largest degrees will be investigated in detail hereafter (see (36), (37),
(49), (61)). Furthermore, the degrees of successive powers of $a$ differ by
six, and so

$Z_{n}=a^{\alpha_{n}}\,P_{n}(z),$ (18)

where

$P_{n}(z)=A_{n}+\cdots+B_{n}z^{\Delta_{n}}$ (19)

is a polynomial in the variable

$z=a^{6}$ (20)

with degree

$\Delta_{n}=\frac{\beta_{n}-\alpha_{n}}{6}.$ (21)

We have

$\displaystyle P_{1}(z)=1,$ $\displaystyle P_{2}(z)=1,$ $\displaystyle
P_{3}(z)=2,$ $\displaystyle P_{4}(z)=1+8z,$ $\displaystyle
P_{5}(z)=8+2z+32z^{2}+128z^{3},$ $\displaystyle
P_{6}(z)=18+32z+128z^{2}+128z^{3}+64z^{4}+1032z^{5}$ $\displaystyle{\hskip
24.0pt}+4352z^{6}+3072z^{7}+16384z^{8}+32768z^{9},$ (22)

and so on. Table 2 gives the degrees $\alpha_{n}$, $\beta_{n}$ and
$\Delta_{n}$ up to $n=14$.

$\begin{array}[]{|r|r|r|r||r|r|r|r|}\hline\cr
n&\alpha_{n}&\beta_{n}&\Delta_{n}&n&\alpha_{n}&\beta_{n}&\Delta_{n}\\\
\hline\cr 1&1&1&0&8&64&382&53\\\ 2&4&4&0&9&88&766&113\\\
3&10&10&0&10&112&1534&237\\\ 4&16&22&1&11&136&3070&489\\\
5&28&46&3&12&160&6142&997\\\ 6&40&94&9&13&184&12286&2017\\\
7&52&190&23&14&208&24574&4061\\\ \hline\cr\end{array}$ Table 2: Degrees
$\alpha_{n}$, $\beta_{n}$ and $\Delta_{n}$ up to $n=14$.

Figure 5 shows a plot of the 237 zeros (or roots) of the polynomial
$P_{10}(z)$ in the complex $z$-plane. Most zeros sit near the circle with
radius $R=1/2$, shown in black (see (41)). The zeros pinch the positive real
axis at $z_{c}\approx 0.411$ (red symbol), corresponding to $a_{c}\approx
0.862$. This observation points toward the emergence of a non-trivial phase
diagram driven by the initial condition (14), where a critical point at
$a_{c}$ separates a strong-coupling phase ($a>a_{c}$) from a weak-coupling one
($a<a_{c}$). The detailed analysis performed in Sections 4 to 6 corroborates
this picture.

Figure 5: Blue symbols: the 237 zeros of the polynomial $P_{10}(z)$ in the
complex $z$-plane. Red symbol: accumulation point of the zeros on the positive
real axis at $z_{c}\approx 0.411$. The black circle has radius $R=1/2$.

##  4 1D model: critical region

The 1D model has a critical point at the following value

$a_{c}=0.862\,322\,847\,096\,001\,235\,198\,066\dots$ (23)

of the coupling constant. This critical point is somewhat similar to a
separatrix in nonlinear dynamics. The accuracy of the above number will be
commented on at the end of this section.

Right at $a=a_{c}$, the partition functions should obey a power law of the
form

$Z_{n}\approx C\,n^{\gamma}.$ (24)

Inserting this asymptotic behavior into (13), and using a continuum setting
where the sum over $k$ is replaced by an integral over $y=k/n$, we
successively obtain $\gamma=4\gamma+1$, hence

$\gamma=-\frac{1}{3},$ (25)

and

$C=C^{4}\int_{0}^{1}(y(1-y))^{-2/3}{\rm d}y,$ (26)

hence

$C=\left(\frac{\Gamma(2/3)}{\Gamma^{2}(1/3)}\right)^{1/3}=0.573\,557\,545\dots$
(27)

Near $a=a_{c}$, along the lines of finite-size scaling theory [35, 36], we
expect a scaling behavior of the form

$Z_{n}\approx C\,n^{-1/3}\bigl{(}1+D(a-a_{c})n^{\sigma}+\cdots\bigr{)},$ (28)

where the crossover exponent

$\sigma=\frac{1}{\nu}$ (29)

is the inverse of the correlation length exponent $\nu$. Inserting the scaling
behavior (28) into (13), working to first order in $a-a_{c}$, and using the
same continuum setting as above, we obtain the following ‘quantization
condition’ for the crossover exponent:

$\frac{\Gamma(\sigma+1/3)\Gamma(2/3)}{\Gamma(\sigma+2/3)\Gamma(1/3)}=\frac{1}{4},$
(30)

hence

$\sigma=8.260\,875\,323\dots,\quad\nu=0.121\,052\,547\dots$ (31)

The above scaling analysis yields the exact values of $\gamma$, $C$, $\sigma$
and $\nu$, but it neither predicts $a_{c}$ nor the crossover amplitude $D$.

As a consequence of the very large crossover exponent $\sigma$, the recursion
(13) is very unstable in the vicinity of $a_{c}$, i.e., very sensitive to
deviations from $a_{c}$, so that the very accurate numerical value of $a_{c}$
given in (23) can be obtained. This level of accuracy will be needed at some
places hereafter.

##  5 1D model: strong-coupling phase

In this section we consider the strong-coupling phase $(a>a_{c})$ of the 1D
model, which is simpler to analyze than its weak-coupling partner.

The partition functions $Z_{n}$ grow very fast in this phase, so that the sum
in (13) is expected to be dominated by its extremal terms ($k=1$ and $k=n-1$).
Forgetting about prefactors, this reads

$Z_{n}\sim Z_{n-1}^{2}.$ (32)

Several heuristic equations of this kind will be met hereafter in the context
of 1D models. The recursion (32) does not point toward the existence of
oscillations. It yields an exponentially super-extensive growth law of the
form

$\ln Z_{n}\approx K(a)\,2^{n}.$ (33)

The quantity $K(a)$ is referred to as the generalized free energy of the model
in its strong-coupling phase.

###  5.1 $a\to\infty$ regime

Let us begin by considering the situation where $a\to\infty$. In this regime,
we have (see (17))

$Z_{n}\approx B_{n}a^{\beta_{n}}.$ (34)

The above assumption that the sum in (13) is dominated by its two extremal
terms is consistent and translates to the recursion relations

$\beta_{n}=2(\beta_{n-1}+1),\qquad B_{n}=2B_{n-1}^{2}\qquad(n\geq 3),$ (35)

yielding

$\beta_{n}=3\,2^{n-1}-2\qquad(n\geq 1)$ (36)

and

$B_{n}=2^{2^{n-2}-1}\qquad(n\geq 2).$ (37)

These results agree with the form (33) and yield the asymptotic form of $K(a)$
at large $a$, namely

$K_{\rm asy}(a)=\frac{3}{2}\ln a+\frac{1}{4}\ln 2.$ (38)

The first correction term to the above asymptotic form is in $1/a^{6}$. The
expression (38) vanishes at

$a_{0}=2^{-1/6}=0.890\,898\,718\dots,$ (39)

whereas $K(a)$ vanishes at $a_{c}$ (see (23)). The small difference (3
percent) between $a_{c}$ and its strong-coupling approximation $a_{0}$ will be
demonstrated in Figure 6. It can be alternatively illustrated in terms of the
zeros of the polynomials $P_{n}(z)$, shown in Figure 5 for $n=10$. The form
(19) implies that the product of all zeros $z_{k}$ of $P_{n}(z)$ reads

$\prod_{k=1}^{\Delta_{n}}z_{k}=(-1)^{\Delta_{n}}\frac{A_{n}}{B_{n}}.$ (40)

It will be shown below (see (49) and (61)) that $\alpha_{n}$ and $A_{n}$ grow
much less rapidly than $\beta_{n}$ and $B_{n}$. As a consequence, we have
$\Delta_{n}\approx\beta_{n}/6\approx 2^{n-2}$ and

$R=\lim_{n\to\infty}\left(\frac{A_{n}}{B_{n}}\right)^{1/\Delta_{n}}=\frac{1}{2}.$
(41)

This number represents the modulus $|z_{k}|$ of a typical zero of $P_{n}(z)$
for large $n$. It is therefore to be expected that most zeros lie near the
circle with radius $1/2$. This prediction is corroborated by Figure 5.
Finally, $z_{0}=a_{0}^{6}=1/2$ sits right on the above circle, whereas
$z_{c}=a_{c}^{6}$, shown by a red symbol in Figure 5, is slightly inside that
circle.

###  5.2 Generic values $(a>a_{c})$

A numerical iteration of the recursion (13) demonstrates that the super-
exponential behavior (33) holds throughout the strong-coupling phase. Figure 6
shows the generalized free energy $K(a)$ thus obtained and its asymptotic form
$K_{\rm asy}(a)$, plotted against $a$ in some range above the critical point.
The values $a_{c}$ and $a_{0}$ where these functions vanish are shown by
arrows. The relatively small difference between these numbers, emphasized just
above, goes hand in hand with the observation that $K_{\rm asy}(a)$ provides a
good overall representation of the true $K(a)$, except in the immediate
vicinity of the critical point.

Figure 6: Generalized free energy $K(a)$ (red) and its asymptotic form $K_{\rm
asy}(a)$ (blue), plotted against $a$ in some range above the critical point.
Arrows: values $a_{c}$ and $a_{0}$ where these functions respectively vanish.

The critical behavior of the generalized free energy $K(a)$ can be predicted
by means of the following scaling argument. If $a-a_{c}$ is very small, the
exponential growth (33) only sets in when the sample size $n$ has exceeded a
crossover length diverging as $(a-a_{c})^{-\nu}$. This yields an exponentially
small essential critical singularity of the form

$K(a)\sim\exp\left(-C_{1}(a-a_{c})^{-\nu}\right).$ (42)

This functional form is corroborated by Figure 7, showing $-\ln K(a)$ against
$(a-a_{c})^{-\nu}$. The slope of the blue line yields

$C_{1}\approx 9.3.$ (43)

The rightmost plotted point is the deepest into the critical region, with
$a-a_{c}\approx 2.05\;10^{-15}$ and $K(a)\sim 10^{-233}$. These numbers call
for two remarks. First, only the numerical evaluation of exact equations such
as the recursion (13) can achieve such accuracy. Second, it is crucial to use
a very accurate value of $a_{c}$ itself (see (23)).

Figure 7: Plot of $-\ln K(a)$ against $(a-a_{c})^{-\nu}$, corroborating the
critical singularity (42) of the generalized free energy. The slope of the
blue line yields $C_{1}\approx 9.3$.

##  6 1D model: weak-coupling phase

We now turn to the weak-coupling phase of the 1D model $(a<a_{c})$. There, it
will be shown that the free energy is extensive, growing as $n^{2}$, and
exhibits log-periodic oscillations as a function of the sample size $n$,
corresponding to a discrete scaling factor two, just as the two-dimensional
fragmentation model studied in Section 2.

The partition functions $Z_{n}$ fall off to zero as $n$ increases, so that the
sum in (13) is expected to be dominated by its central terms ($k\approx n/2$).
Forgetting about prefactors, this reads

$Z_{n}\sim Z_{n/2}^{4}.$ (44)

This heuristic relation suggests a growth of $\ln Z_{n}$ of the form

$\ln Z_{n}\approx-F(a,x)\,n^{2},$ (45)

where $F(a,x)$ exhibits 1-periodic oscillations in the logarithmic variable
$x$ defined in (4). This quantity is hereafter referred to as the free energy
(density) of the model.

###  6.1 $a\to 0$ regime

Let us begin by considering the situation where $a\to 0$. In this regime, the
presence of log-periodic oscillations can be explained in simple terms. We
have (see (17))

$Z_{n}\approx A_{n}a^{\alpha_{n}}.$ (46)

We shall successively investigate the degrees $\alpha_{n}$ and the
coefficients $A_{n}$.

The degrees $\alpha_{n}$ entering (46) obey the recursion

$\alpha_{n}=2\min_{1\leq k\leq n-1}(\alpha_{k}+\alpha_{n-k}).$ (47)

These integers are given in Table 2 up to $n=12$. They are listed as sequence
number A073121 in the OEIS [34], together with useful formulas and references.
In particular, they obey the recursions

$\alpha_{2n}=4\alpha_{n},\qquad\alpha_{2n+1}=2(\alpha_{n}+\alpha_{n+1}).$ (48)

These relations exhibit a discrete scale invariance with scaling factor two.
In particular, $\alpha_{n}=n^{2}$ whenever $n=2^{m}$ is a power of two. For
$n$ in the interval $2^{m}\leq n\leq 2^{m+1}$, the degrees $\alpha_{n}$
exhibit an exact linear growth in $n$, of the form

$\alpha_{n}=2^{m}(3n-2^{m+1}),$ (49)

so that

$\alpha_{n}-n^{2}=(2^{m+1}-n)(n-2^{m}).$ (50)

It is then useful to split the logarithmic variable $x$ defined in (4) into
its integer and fractional parts according to

$x=\frac{\ln n}{\ln 2}=m+\xi\qquad(0\leq\xi\leq 1),$ (51)

and to introduce another reduced co-ordinate,

$\eta=\frac{n}{2^{m}}-1=2^{\xi}-1,$ (52)

that is also in the range $0\leq\eta\leq 1$. The result (49) can be exactly
recast as

$\alpha_{n}=n^{2}f(x),$ (53)

where $f(x)$ is a 1-periodic function of $x$, given by

$f(x)=2^{-\xi}(3-2^{1-\xi})=1+\frac{\eta(1-\eta)}{(1+\eta)^{2}}$ (54)

in terms of the co-ordinates $\xi$ or $\eta$. The function $f(x)$ will be
plotted in Figure 8. It reaches its minimum $f_{\rm min}=1$ at integer $x$ and
its maximum $f_{\rm max}=9/8$ for $\xi=\ln(4/3)/\ln 2$, i.e., $\eta=1/3$. We
have

$f_{\rm ave}=\frac{3}{4\ln 2}=1.082\,021\,280\dots,\qquad f_{\rm
osc}=\frac{\ln 2}{6}=0.115\,524\,530\dots$ (55)

We have thus established the validity of the scaling form (45), including log-
periodic oscillations, to leading order as $a\to 0$, and derived a first
estimate of the free energy in the small-$a$ regime,

$F(a,x)\approx f(x)|\ln a|.$ (56)

Let us now turn to the analysis of the coefficients $A_{n}$ entering (46). The
minimum in (47) turns out to be generically highly degenerate. More precisely,
for $n$ in the interval $2^{m}\leq n\leq 2^{m+1}$, i.e., $n=2^{m}+i$ with
$0\leq i\leq 2^{m}$, this minimum is reached for all integers $k$ of the form
$k=2^{m-1}+j$, where $j$ runs over the following range:

$\left\\{\matrix{0\leq i\leq 2^{m-1}:\hfill&0\leq j\leq i,\hfill\cr
2^{m-1}\leq i\leq 2^{m}:\quad&i-2^{m-1}\leq j\leq 2^{m-1}.\hfill}\right.$ (57)

Denoting by $K_{n}$ the set of integers $k$ defined above, the coefficients
$A_{n}$ obey the recursion

$A_{n}=\sum_{k\in K_{n}}A_{k}^{2}A_{n-k}^{2}.$ (58)

When $n=2^{m}$ is a power of two, the set $K_{n}$ consists of a single
element, $k=n/2$, and so $A_{n}=1$. When $n=3\;2^{m-1}$ is half-way between
two successive powers of two, the set $K_{n}$ is the largest, with $2^{m-1}+1$
elements, and the coefficient $A_{n}$ is (locally) maximal. We have

$A_{3}=2,\quad A_{6}=18,\quad A_{12}=113\,170,$ (59)

and so on. More generally, we have the symmetry

$A_{n}=A_{3\;2^{m}-n}.$ (60)

The coefficients $A_{n}$ obey the asymptotic growth law

$\ln A_{n}\approx n^{2}g(x),$ (61)

where $g(x)$ is a positive 1-periodic function of $x$. We have thus obtained a
more complete estimate of the free energy in the small-$a$ regime:

$F(a,x)\approx f(x)|\ln a|-g(x).$ (62)

The function $g(x)$ is positive, vanishes at integer $x$, and its average
reads

$g_{\rm ave}\approx 0.058\,187\,830.$ (63)

At variance with (47), the recursion (58) cannot be solved in closed form, so
that no analytic expression for $g(x)$ is available. As a consequence of (57),
$g(x)$ exhibits cusps at (presumably) all dyadic values $\eta=j/2^{k}$ of the
coordinate $\eta$, the strongest ones being at $\eta=2^{-k}$ and
$\eta=1-2^{-k}$. Figure 8 shows the 1-periodic functions $f(x)-1$ and $g(x)$
over one period. The cusp singularities of $g(x)$ are not visible at this
scale. Figure 9 shows the ratio

$R(x)=\frac{g(x)}{f(x)-1}$ (64)

against $\eta$. The formulas (50) and (60) imply that $R(x)$ is invariant
under the change of $\eta$ into $1-\eta$. The two main series of cusp
singularities are now clearly visible (red symbols).

Figure 8: The 1-periodic functions $f(x)-1$ (blue) and $g(x)$ (red), plotted
against $x$ over one period. Figure 9: Ratio $R(x)$ defined in (64), plotted
against $\eta$. Red symbols: main series of cusps at $\eta=2^{-k}$ and
$\eta=1-2^{-k}$.

###  6.2 Generic values $(a<a_{c})$

A numerical iteration of the recursion (13) demonstrates that the scaling
behavior (45) holds throughout the weak-coupling phase. We shall focus our
attention onto the average $F_{\rm ave}(a)$ and the relative magnitude of
oscillations $F_{\rm osc}(a)$ of the free energy $F(a,x)$.

In the $a\to 0$ regime, the estimate (62) shows that the average free energy
diverges logarithmically according to

$F_{\rm ave}(a)\approx f_{\rm ave}|\ln a|-g_{\rm ave},$ (65)

where $f_{\rm ave}$ and $g_{\rm ave}$ are respectively given in (55) and (63),
whereas $F_{\rm osc}(a)$ goes to the finite limit (see (55))

$F_{\rm osc}(0)=f_{\rm osc}=0.115\,524\,530\dots,$ (66)

whose value is quite sizeable.

In the critical regime $(a\to a_{c})$, scaling theory suggests that the
average free energy vanishes according to

$F_{\rm ave}(a)\approx C_{2}(a_{c}-a)^{2\nu},$ (67)

whereas the relative magnitude of oscillations has a finite limit $F_{\rm
osc}(a_{c})$. These expectations are corroborated by Figures 10 and 11. The
data plotted there have been extrapolated by means of very accurate 4th-degree
polynomial fits. There is no rationale behind the choice of the abscissas used
in these plots: the chosen powers of $a_{c}-a$ just turn out to allow for
accurate extrapolations. Figure 10 shows the combination
$(a_{c}-a)^{-2\nu}F_{\rm ave}(a)$ against $(a_{c}-a)^{\nu}$, yielding

$C_{2}\approx 0.0050.$ (68)

Comparing this number to the amplitude $C_{1}$ given in (43), we notice that,
even though $C_{1}$ is large and $C_{2}$ is small, the dimensionless
combination $C_{1}^{2}C_{2}\approx 0.43$ is of order unity. Figure 11 shows
$F_{\rm osc}(a)$ against $(a_{c}-a)^{2\nu}$, yielding

$F_{\rm osc}(a_{c})\approx 0.00111.$ (69)

Figure 10: Red symbols: $(a_{c}-a)^{-2\nu}F_{\rm ave}(a)$ against
$(a_{c}-a)^{\nu}$. Blue curve: polynomial fit yielding $C_{2}\approx 0.0050$.
Figure 11: Red symbols: $F_{\rm osc}(a)$ against $(a_{c}-a)^{2\nu}$. Blue
curve: polynomial fit yielding $F_{\rm osc}(a_{c})\approx 0.00111$.

As the coupling constant $a$ is increased from 0 to $a_{c}$, the average free
energy $F_{\rm ave}(a)$ decreases steadily from infinity to zero,
interpolating smoothly between the behaviors (65) and (67). In the same time,
the relative magnitude $F_{\rm osc}(a)$ of oscillations decreases by a factor
of order 100 from (66) to (69). Figure 12 shows that $F_{\rm osc}(a)$ does not
decay monotonically between those two limits. In particular, it exhibits a
cusp at $a_{*}=0.537\,750\dots$

The cusp in Figure 12 has to do with a change in the shape of the 1-periodic
function $F(a,x)$. Figure 13 shows the reduced oscillations $F(a,x)/F_{\rm
ave}(a)$ against the logarithmic variable $x$. Two consecutive periods are
shown for clarity. For the smaller values of $a$ (upper panel), the
oscillations strongly decrease in magnitude, while keeping essentially the
shape of $f(x)$ (see Figure 8), with a cusp around a single minimum at integer
$x$. For $a$ near $a_{*}$ (lower panel), the structure of the oscillations
becomes richer and changes rapidly with $a$. A second minimum with a cusp
develops at $\eta=1/2$, i.e., $\xi=\ln(3/2)/\ln 2=0.584\,962\dots$ This second
cusp and the original one have exactly the same height at $a=a_{*}$. This
degeneracy causes the structure observed in Figure 12. For larger $a$ (not
shown), the oscillations soon become very small and harmonic.

Figure 12: Logarithmic plot of $F_{\rm osc}(a)$ against $a$ all over the weak-
coupling phase. Blue symbols: limit values (66) at $a=0$ and (69) at
$a=a_{c}$. Green arrow: cusp at $a_{*}\approx 0.537750$.

Figure 13: Reduced oscillations $F(a,x)/F_{\rm ave}(a)$, plotted against the
logarithmic variable $x$ for several $a$ (see legends). Note the different
vertical scales in both panels.

##  7 Further 1D examples

In this section we propose four other 1D analogues of the recursion (13) for
the partition functions of the fragmentation model. In each case, we determine
whether the model may cross a phase transition when its initial conditions are
varied and whether is exhibits log-periodic oscillations in one of its phases.
These outcomes lead us to introduce, as our fifth example, what we can think
of as the most general recursion of this kind exhibiting log-periodic
oscillations.

### Example 1

This first example is obtained by suppressing the squares in the recursion
(13) defining the 1D model. We thus obtain the bilinear recursion

$Z_{n}=\sum_{k=1}^{n-1}Z_{k}Z_{n-k}\qquad(n\geq 2),$ (70)

with initial condition $Z_{1}=a$. In terms of the generating series

$G(z)=\sum_{n\geq 1}Z_{n}z^{n},$ (71)

this reads

$G(z)=az+G(z)^{2},$ (72)

hence

$G(z)=\frac{1-\sqrt{1-4az}}{2}.$ (73)

The partition functions therefore read

$Z_{n}=C_{n-1}a^{n}\approx\frac{(4a)^{n}}{4\sqrt{\pi n^{3}}},$ (74)

where

$C_{n-1}=\frac{(2n-2)!}{n!(n-1)!}$ (75)

is the $(n-1)$-st Catalan number. These integers have a panoply of
combinatorial interpretations. They are listed as sequence number A000108 in
the OEIS [34], together with many useful formulas and references. The
exponential growth rate of the $Z_{n}$ depends smoothly on the initial
condition. This example neither exhibits a phase transition nor oscillations.

### Example 2

This second example is an extension of the previous one, defined by the
trilinear recursion

$Z_{n}=\sum_{k+l+m=n}Z_{k}Z_{l}Z_{m}\qquad(n\geq 3),$ (76)

with initial conditions $Z_{1}=a$ and $Z_{2}=b$. For $a=b=1$, the $Z_{n}$ are
positive integers which count ternary trees. They are listed as sequence
number A019497 in the OEIS [34]. In full generality, the corresponding
generating series, defined in analogy with (71), obeys

$G(z)=az+bz^{2}+G(z)^{3}.$ (77)

We thus obtain the asymptotic behavior

$Z_{n}\approx\frac{C}{n^{3/2}\,{z_{0}}^{n}},$ (78)

where $z_{0}$ is the smallest zero of the discriminant of (77), namely

$\Delta=27z^{2}(a+bz)^{2}-4.$ (79)

In the range of physical relevance ($a$ and $b$ positive), the exponential
growth rate $1/z_{0}$ is positive and has a smooth dependence on the initial
conditions. This example therefore neither exhibits a phase transition nor
oscillations.

### Example 3

The third example is obtained by replacing each square in (13) by a higher
power, namely

$Z_{n}=\sum_{k=1}^{n-1}Z_{k}^{p}Z_{n-k}^{p}\qquad(n\geq 2),$ (80)

with $Z_{1}=a$. For any integer $p\geq 3$, the model exhibits the very same
phenomenology as the 1D model investigated above, with a critical point at
some $p$-dependent $a_{c}$. The strong-coupling phase $(a>a_{c})$ is captured
by the heuristic equation

$Z_{n}\sim Z_{n-1}^{p},$ (81)

generalizing (32) and yielding an exponentially growing free energy of the
form

$\ln Z_{n}\approx K(a)\,p^{n}.$ (82)

The weak-coupling phase $(a<a_{c})$ is captured by the heuristic equation

$Z_{n}\sim Z_{n/2}^{2p},$ (83)

generalizing (44) and yielding a free energy growing as

$\ln Z_{n}\approx-F(a,x)\,n^{d},$ (84)

where the effective dimension reads

$d=\frac{\ln 2p}{\ln 2},$ (85)

and $F(a,x)$ is a 1-periodic function of the logarithmic variable $x$ defined
in (4).

### Example 4

The fourth example is obtained by replacing the squares in (13) by two
different integer powers $p$ and $q$, such that $p>q\geq 1$, namely

$Z_{n}=\sum_{k=1}^{n-1}Z_{k}^{p}Z_{n-k}^{q}\qquad(n\geq 2),$ (86)

with $Z_{1}=a$. This is the first situation where $Z_{k}$ and $Z_{n-k}$ play
asymmetrical roles. The model has a critical point at some $a_{c}$ depending
on $p$ and $q$. The strong-coupling phase $(a>a_{c})$ is similar to that of
the previous example. In the weak-coupling phase $(a<a_{c})$, the free energy
grows as a power of the sample size, namely

$\ln Z_{n}\approx-F(a)\,n^{d},$ (87)

where the effective dimension $d$ is given by

$\bigl{(}p^{1/(d-1)}-1\bigr{)}\bigl{(}q^{1/(d-1)}-1\bigr{)}=1.$ (88)

The power-law (87) is modulated by slowly damped erratic oscillations. This
example therefore exhibits a continuous phase transition, but no everlasting
log-periodic oscillations.

### Example 5

This fifth example is meant to represent the most general recursion exhibiting
log-periodic oscillations, at least within the class of models under scrutiny.
The study of the previous examples suggests that the following ingredients are
necessary: each partition function enters non-linearly and all parts play
symmetric roles. We are thus led to consider the recursion

$Z_{n}=\sum_{k_{1}+\cdots+k_{r}=n}Z_{k_{1}}^{p}\dots Z_{k_{r}}^{p}\qquad(n\geq
r).$ (89)

The recursion (89) has to be supplemented with the $r-1$ initial conditions
$Z_{n}=a_{n}$ for $n=1,\dots,r-1$. This model has two integer parameters, the
degree $p\geq 2$ and the number $r\geq 2$ of parts. The recursion (13)
defining the 1D model is recovered for $p=r=2$, whereas the recursion (80)
defining Example 3 is recovered for $r=2$ and $p$ arbitrary.

For physically relevant, i.e., positive initial conditions, the simplest phase
diagram one can think of consists of a single critical surface in the space of
initial conditions, separating a strong-coupling phase from a weak-coupling
one. More complex scenarios allowing for other kinds of behavior in
intermediate regimes are however not entirely ruled out.

The strong-coupling phase is captured by the heuristic equation

$Z_{n}\sim Z_{n+1-r}^{p},$ (90)

generalizing (32) and (81), and yielding an exponentially growing free energy
of the form

$\ln Z_{n}\approx K\,{\rm e}^{n\mu},\qquad\mu=\frac{\ln p}{r-1}.$ (91)

The weak-coupling phase is captured by the heuristic equation

$Z_{n}\sim Z_{n/r}^{pr},$ (92)

generalizing (44) and (83), and yielding a free energy growing as a power of
the sample size modulated by periodic oscillations, of the form

$\ln Z_{n}\approx-F(x)\,n^{d},$ (93)

where the effective dimension reads

$d=\frac{\ln pr}{\ln r},$ (94)

and $F(x)$ is a 1-periodic function of the logarithmic variable

$x=\frac{\ln n}{\ln r}.$ (95)

The constant $K$ and the function $F(x)$ depend on the $r-1$ initial
conditions.

##  8 Discussion

The stochastic fragmentation model introduced in [31] sparked our interest in
revisiting the subject of log-periodic oscillations. The fragmentation model
is unique in that it combines the following two characteristics. On the one
hand, from a heuristic viewpoint, it can be expected to exhibit some weak
discrete scale invariance, in analogy with what occurs e.g. in turbulence or
in diffusion-limited aggregation. On the other hand, the numbers $Z_{m,n}$ of
jammed configurations obey an exact recursion formula. We have used the latter
property to demonstrate by numerical means that the model indeed exhibits the
log-periodic oscillations predicted by the former one. To our knowledge, this
is the first instance of a statistical-mechanical model where periodic
oscillations are reported in the size dependence of a physical quantity.

A 1D analogue of the fragmentation model has then been introduced and
investigated in detail. This 1D toy model has many appealing features. First
of all, it is simple enough to lend itself to a very detailed investigation.
In spite of this, it has a richer behavior than the 2D fragmentation model.
There is a critical value $a_{c}$ of the initial condition, interpreted as a
coupling constant, that is somewhat similar to a separatrix in nonlinear
dynamics. This critical point separates a strong-coupling phase where the free
energy is super-extensive and does not manifest oscillations, from a weak-
coupling one where the free energy is extensive and exhibits log-periodic
oscillations. The above characteristics can be established on much firmer
ground than for the 2D fragmentation model by means of so-called heuristic
equations such as (32) and (44). Finally, we have generalized the 1D model
into a family of models with two integer parameters, which exhibit essentially
the same phenomenology.

This work leaves a number of questions unanswered. The most pressing one
concerns what can be established by either rigorous or analytical means about
log-periodic oscillations in the fragmentation model. Possible answers range
from simply proving their existence to deriving explicit formulas for the log-
periodic functions that modulate the bulk configurational entropy and related
aspect-ratio-dependent quantities.

## Acknowledgments

It is a pleasure to thank Paul Krapivsky for very stimulating exchanges that
motivated this work, and for having graciously made available to me the graph
shown in Figure 2. A useful discussion with Jérémie Bouttier and Emmanuel
Guitter is also acknowledged.

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