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Empty Logics

Federico Matias Pailos

Abstract

We will present a three-valued consequence relation for metainferences, called

MI, deﬁned in terms of ST and TS, two well known inferential consequence rela-

tions. MI has no valid metainference, and thus no valid inference either. Notwith-

standing, MI is informative about the meta-metainferential level. Afterwards, we

will introduce a hierarchy of consequence relations MInfor metainferences of level

n(for 1 ≤n < ω). Every metainference of level nor less is invalid in each MIn.

Nevertheless, each MInis informative about the metainferences of higher levels.

Finally, we will present a logic Lempty, based on the hierarchy of logics MIn, that

is genuinely empty, in the sense that it has no valid metainference of any level

whatsoever.

KEYWORDS: Logic, Metainferences, Metainferential Validity, Substructural

Logics, Empty Logic

1 Introduction

A logic is ordinarily presented as a consequence relation applied to a particular lan-

guage1. Informally, a consequence relation is a standard that an inference may, or may

not, live up to. The following quote by Rohan French assesses this issue.

Suppose that you... thought that logic was only the study of logical truths,

of determining which formulas are tautologies... Then one would be moved

to say that Strong Kleene logic -—the logic according to which a sentence is

a logical truth if it gets the value 1 on every valuation which assigns values

in the set {1, i, 0}to the propositional atoms and calculates the truth-values

of components... is not a logic... Of course, as we well know, while Strong

Kleene has no tautologies it does have many valid arguments... What I want

to propose here is that we make a similar move... While [the non-reﬂexive

sequent calculi] LKR does not have any valid sequents, it does have a great

many valid metasequents, higher-order sequents which have sets of sequents

as their premises and sequents as their conclusions. That is to say, while

LKR and other irreﬂexive systems have no valid inferences they do have a

great many valid metainferences. ([12]: p. 120.)

1As Priest notices in [17], logic is an ambiguous term. It can both refer to the theory of an investi-

gation, or to the subject matter of that survey. Throughout this work, we will use the word logic only

to refer to a logical theory

1

French2talks about metasequents. We will talk extensively about metainferences,

the semantic counterpart of metasequents.3In that quote, French suggests that being

informative about metainferences is enough for a consequence relation to specify a logic

(e.g., given a language). In fact, LKR−–e.g., LKR without the truth rules– has no

valid inference.4But if consequence relations don’t need to validate inferences to specify

a logic, do they really need to validate some metainferences? Could there be a logic

without valid metainferences?

We will give a positive answer to this question, and prove our point with an example.

Such logic, called MI, includes a consequence relation, a certain standard, that metain-

ferences, may or may not meet. In fact, no metainference will satisfy that standard.

Thus, no metainference will be valid in MI.

Notwithstanding, it seems reasonable to expect that a respectable logic establishes

that some things are right –or valid– and others are wrong -or invalid. This means that

such logic is informative about sentences, inferences or metainferences. More precisely,

a logic Lis informative about some set of things Sif and only if, for at least some

a∈S,Lvalidates a, and for at least some b∈S,Linvalidates b. Fortunately, we will

see that MI is informative about meta-metainferences, or (as we will also call them)

metainferences of level 2 -e.g., informally, inferences that have sets of metainferences as

premises, and a metainference as a conclusion.

Moreover, the path that French describes can be extended indeﬁnitely. It starts

with the Strong Kleene logic K3, a logic without tautologies, and follows with LKR−,

a logic without tautologies and without valid inferences that is sound and complete with

respect to the non-reﬂexive semantic logic TS.5The logic we will introduced, MI, not

only has no valid inferences, but also has no valid metainferences. We will see in what

precise sense MI looks a lot like TS.

So far, we have talked about metainferences as inferences from a set of inferences to

a particular inference. We have also talked about metametainferences as inferences that

have a set of metainferences as premises and a metainference as a conclusion. Thus,

it will be extremely advantageous to have a comprehensive label for all these things.

Thus, we will talk about metainferences as a general term applying to all those things,

and assign to each one of them a particular level. What we have called metainferences,

then, will be, strictly speaking, metainferences of level 1, or metainferences1.Meta-

2The non-reﬂexive logic LKR is the result of dropping the Identity axiom-scheme from a standard

presentation of Gentzen’s multiple-conclusion sequent calculus for classical logic, G1c, and adding the

bottom-up (e.g, invertible) versions of G1c’s operational rules, plus two rules for the truth predicate.

In this paper we will not be dealing with truth theories, but only with logics. Thus, out primary interest

will not be LKR, but the calculus obtained from LKR by dropping the rules for the truth predicate.

For a presentation of G1c, see [25]: p. 52).

3Metainferences can be obtained from metasequents by replacing the turnstile of the sequent calculus

for a double turnstile that indicates a semantic consequence relation. In Section 2 we will present a

formal deﬁnition of them.

4Though if propositional constants for truth values –with a suitable set of new rules for them– are

added, then the new system will have valid sequents. Every logic we will talk about in the rest of the

paper is based on a language that lacks these truth-value constants.

5We will explore TS in detail in the next Section.

2

metainferences will be, then, metainferences of level 2, or metainferences2, and so

on.

We will prove that, for every level nof metainferences, there is a consequence relation

that invalidates every sentence, inference and metainference of level nor less. Moreover,

that logic is informative about metainferences of higher levels.

This paper is structured as follows. In Section 2 we introduce the distinction between

inferences and metainferences, along with the notion of metainferential validity. We will

also present two substructural logics, ST and TS, that will be used to deﬁne the logic

MI. In Section 3 we present MI, a metainferential consequence relation that invalidates

every metainference (of level 1). In Section 4 we introduced MI2, a consequence relation

for metainference of level 2. No meta-metainference is valid in MI2. This result is

pushed one step forward in Section 5. There, we deﬁne a hierarchy of logics MIn. Each

MIninvalidates every metainference of level nor less. In Section 6, we introduce an

empty logic Lempty.Lempty is deﬁned with the resources provide by the hierarchy of

logics MIn. No metainference of any level is valid in Lempty. Finally, in Section 6,

we oﬀer some concluding remarks, and point to some directions in which the present

explorations can be further developed.

2 Metainferences, metainferential validity and two sub-

structural logics

In the next section, we will introduce the target consequence relation for metainferences,

MI.MI invalidates every sentence, inference and metainference of level 1. Nevertheless,

it is informative about metainferences of higher levels. But MI is not a standard

consequence relation. It does not relate sets of formulas of a language L, but sets of

inferences. But before introducing MI, it will be necessary to give some deﬁnitions.

In particular, we will need to make a clear distinction between standard inferences (as

relations between sets of sentences) and metainferences. Moreover, we must clarify

what we mean by metainferential validity. Finally, we will introduce in some detail two

important substructural logics: ST and TS.

2.1 Inferences and Metainferences

To understand and carry on our investigation, it will be essential to have an accurate

grasp of the received view about inferences, metainferences and consequence relations.

To analyze these matters, it will be useful to ﬁx some terminology. Let Lbe a

propositional language, such that FOR(L) is the absolutely free algebra of formulae

of L, whose universe we denote by F OR(L). As usual, we will let Γ,∆,and other

Greek capital letters represent ﬁnite sets of formulae. Finally, A,B,Cand Drepresent

formulae.6

6For the sake of simplicity, we will present single-conclusion logics. Moreover, the consequence

relation will be deﬁned for inferences with a ﬁnite set of premises. Nevertheless, we hope that the

results that we will introduce can be safely extended to multi-conclusion versions of these logics, that

3

Deﬁnition 2.1. A Tarskian consequence relation over a propositional language Lis

a relation ⊆℘(F OR(L)) ×F OR(L)obeying the following conditions for all A∈

F OR(L)and for all Γ,∆⊆F OR(L):

1. ΓAif A∈Γ(Reﬂexivity)

2. If ΓAand Γ⊆Γ′, then Γ′A(Monotonicity)

3. If ∆Aand ΓBfor every B∈∆, then ΓA(Cut)

Additionally, a (Tarskian) consequence relation is substitution-invariant whenever if

ΓA, and σis a substitution on FOR(L), then {σ(B)|B∈Γ}σ(A).

Deﬁnition 2.2. A Tarskian logic over a propositional language Lis an ordered pair

(FOR(L),), where is a substitution-invariant Tarskian consequence relation.

Notwithstanding, many philosophers have argued against the Tarskian conception

of logic. For example, [22], [2] and [21] argue for the generalization of the Tarskian

account multiple consequences. [2] and [13] give reasons for relaxing the Monotonicity

condition. Moreover, [15] and [11] argued for a generalization or liberalization that

allows logics to drop Reﬂexivity, Cut, or both of them. In this context is where logics

like the non-reﬂexive TS of the non-transitive ST where introduced.

These modiﬁcations, in turn, bring on the possibility of a shift in the nature of the

collection of formulae featured in the consequence relation. Thus, instead of treating

logical consequence to hold between (sets of) formulae, it may hold between labeled for-

mulae, multisets of formulae (where repetition matters), sequences of formulae (where

order makes a diﬀerence), etc. As we already mention, our target logic, MI, explicitly

moves from understanding logical consequence as a relation between collections of for-

mulae, to conceiving it as a relation between collections of inferences. Here, we should

understand inferences as an ordered pair whose ﬁrst member is a set (or multiset, or

sequence, etc.) of formulae, and a set of formulae as its second term. A metainference,

then, is an ordered pair whose ﬁrst member is a set (or multiset, etc.) of inferences, and

a single inference as its second term. This is a relatively standard move in the literature.

[1] , ﬁrst, and [4] , afterwards, discuss a generalization of the Tarksian account that al-

lows moving to logical consequence relations that do not hold only between collections

of formulae, but between objects of other nature.7Thus, more precisely:

Deﬁnition 2.3. A inference on Lis an ordered pair (Γ, A), where Γ⊆F OR(L)and

A∈F OR(L)(written Γ⇒A). SEQ0(L)is the set of all inferences or sequents on L.

Deﬁnition 2.4 ([8]).A metainference on Lis an ordered pair (Γ, A), where Γ⊆

SEQ0(L)and A∈SEQ0(L)(written Γ⇒1A). SEQ1(L)is the set of all metainfer-

ences on L.

admit also arguments with inﬁnite premises.

7Moreover, [1] suggested that the idea that logical consequence can be said to hold of relata other

than formulae is very reasonable to those used to sequent calculus systems —and, most prominently,

with substructural sequent calculi.

4

We will say, accordingly, that the following one on the left is a inference, whereas

the one on the right is a metainference

Γ⇒A

∆⇒BΣ⇒C

Γ⇒A

Therefore, we will distinguish two diﬀerent kinds of consequence relations: the in-

ferential and the metainferential.MI –our target logic– is a consequence relation of this

last type. In the next subsection we will give a precise characterization of this notion.

2.2 Metainferential validity

Put it informally, a metainference (or, as we will also call it, a metainference of level

1), is a pair of sets of inferences –as its ﬁrst member– and an inference –as its second

member. Every metainference will have the following structure:

Γ1∆1, ...Γn∆n

ΣΠ

We will call every Γi∆i, a premise of the metainference, while Σ Π will be its

conclusion.

Thus, a metainference is a inference-like structure. Thus, it makes sense to ask when

will a metainference Γ1∆1, ...Γn∆n≫ΣΠ be valid in a particular propositional

logic L.

Deﬁnition 2.5. A metainference Γ1∆1, ...Γn∆n≫ΣΠis valid in (a proposi-

tional logic) Lif and only if, for every valuation v, if vsatisﬁes every Γi∆iaccording

to L, then vsatisﬁes ΣΠaccording to L.

A valuation vsatisﬁes or conﬁrms a inference Γ ∆ in a speciﬁc logic Lif and only

if vis not a counterexample to Γ ∆’s validity in L.

Dicher and Paoli [8] have named this a local conception of metainferential validity.8

As it stands, the deﬁnition speciﬁes a standard of validity for metainferences –e.g.,

instances of metainferential schemes– in a particular logic. Nevertheless, it can –and

will– be also used to speciﬁed when a metainferential scheme is valid. In a nutshell, a

scheme is valid in Lif and only if every instance of it is valid.

Our target logic, MI, is a consequence relation for metainferences. But to under-

stand exactly how it works, we need ﬁrst to introduce two inferential but non-Tarskian

consequence relations: the non-reﬂexive TS and the non-transitive ST.9

8For more about this notion, and the diﬀerence between a local and a global metainferential validity,

see [8].

9Those readers that are already familiar with these logics may safely skip this part.

5

2.3 ST and TS: two substructural consequence relations

ST and TS are label as substructural because they give up at least one structural

feature of a Tarskian consequence relation. More speciﬁcally, ST drops Cut, while TS

abandons Reﬂexivity.

The logic ST can be portrayed as a p-logic. It was devised by Frankowski in [10] as

a means to characterize logical systems where valid derivations are such that the degree

of strength of the conclusions can be smaller than that of the premises.10

Deﬁnition 2.6 ([11]).Ap-consequence relation over a propositional language Lis

a relation ⊆℘(F OR(L))× ⊆ ℘(F OR(L)) obeying the following conditions for all

A∈F OR(L)and for all Γ,∆⊆F OR(L):11

1. ΓAif A∈Γ(Reﬂexivity)

2. If ΓAand Γ⊆Γ′, then Γ′A(Monotonicity)

Additionally, a p-consequence relation is substitution-invariant whenever if ΓA,

and σis a substitution on FOR(L), then {σ(B)|B∈Γ}σ(A).

Deﬁnition 2.7 ([11]).Ap-logic over a propositional language Lis an ordered pair

(FOR(L),), where is a substitution-invariant p-consequence relation.

In general, p-logics (q-logics) can be connected to p-matrices (q-matrices). ST can

be represented as a p-matrix logic associated to the 3-element Kleene algebra.

Deﬁnition 2.8 ([11]).For La propositional language, an L-p-matrix is a structure

hV,D+,D−,Oi, such that hV ,Oi is an algebra of the same similarity type as L, with

universe Vand a set of operations O, where D+,D−⊆ V and D+⊆ D−.

Deﬁnition 2.9. The 3-element Kleene algebra is the structure

K=h{1,1

2,0},{f¬

K, f ∧

K, f ∨

K}i

where the functions f¬

K, f ∧

K, f ∨

Kare as follows

f¬

K

1 0

1

2

1

2

10

f∧

K11

20

1 1 1

20

1

2

1

2

1

20

00 0 0

f∨

K11

20

1 1 1 1

1

211

2

1

2

011

20

Moreover, the functions →and ↔are deﬁnable via the usual deﬁnitions.

Deﬁnition 2.10 ([19]).A 3-valued ST-matrix is a p-matrix

MST =h{1,1

2,0},{1},{1,1

2},{f¬

K, f ∧

K, f ∨

K}i

such that h{1,1

2,0},{f¬

K, f ∧

K, f ∨

K}i is the 3-element Kleene algebra.

10For an extensive presentation of ST, see but also [6], [7] , [19] and [20].

11Once again, we will give a single-conclusion presentation of these logics.

6

Now, as is common practice, semantic structures such as p-matrices induce con-

sequence relations and, therefore, logics, with the help of valuation functions, e.g.,

homomorphisms from FOR(L) to the set of truth-values of the semantic structure

in question—in this case, the set {1,1

2,0}. Valuations are extended from propositional

variables to complex formulae with the help of the truth-functions for the connectives; in

this case the functions given by the 3-element Kleene algebra. In our particular case, we

can deﬁne what a valid inference or sequent is in any p-matrix logic—and, therefore, in

ST—in the following straightforward manner. Notice that, below, Mis a substitution-

invariant p-consequence relation, whence (FOR(L),M) is a a p-logic. In addition to

that, when some p-logic Lis induced by p- matrix M, we may interchangeably refer to

Mas L.

Deﬁnition 2.11. For Map-matrix, an M-valuation vsatisﬁes a sequent or inference

Γ⇒A(written vMΓ⇒A) if and only if, if v[Γ] ⊆ D+, then v(A)∈ D−. A sequent

or inference Γ⇒Ais M-valid (written MΓ⇒A) if and only if vMΓ⇒A, for all

M-valuations v.

But a more straightforward characterization of ST’s validity is the following one:

Deﬁnition 2.12. ST Γ⇒Aif and only if for every valuation v, if v(B)∈

{1}for all B∈Γ, then v(A)∈ {1,1

2}

Another12 13 interesting generalization of Tarskian consequence relations is the no-

tion of q-consequence relation, due to [15].14

Deﬁnition 2.14 ([15]).Aq-consequence relation over a propositional language Lis

a relation ⊆℘(F OR(L))× ⊆ (F OR(L)) obeying the following conditions for all

A, B ∈F OR(L)and for all Γ,∆⊆F O R(L):

12This deﬁnition can be adjusted to a multi-conclusion setting:

Deﬁnition 2.13. ST Γ⇒∆if and only if for every valuation v, if v(B)∈ {1}for all B∈Γ, then

for all A∈∆, v (A)∈ {1,1

2}

13Moreover, there is a third way to present ST’s validity, that requires to talk about strict and tolerant

satisfaction or truth. (In fact, ST stands for Strict-Tolerant, just as TS stands for Tolerant-Strict.) A

valuation vsatisﬁes tolerantly a formula Aif and only if v(A)∈ {1,1

2}, and satisﬁes it strictly if and

only if v(A)∈ {1}. A valuation vsatisﬁes an inference Γ ⇒Ain ST if and only if, if vstrictly satisﬁes

every B∈Γ, then vtolerantly satisﬁes A. Finally, an inference from Γ to Ais valid if and only if, for

every valuation v, if vsatisﬁes strictly every B∈Γ, then vsatisﬁes tolerantly A. Nevertheless, it is

worth mentioning that ST’s supporters themselves prefer to talk of strict and tolerant assertion rather

than talking about strict and tolerant satisfaction, or strict and tolerant truth. As [9] explains, the

reason why they use the idea of strict and tolerant assertion instead of either of the last two (pair of)

notions, is to avoid revenges paradoxes related to the notions of strictly true and strictly false.

14[23] claims that the relation of q-logic is devised to qualify as valid derivations of true sentences from

non-refuted premises (understood as hypotheses), whereas the notion of p-logic is devised to qualify as

valid derivations of conclusions whose degree of strength (understood as the conviction in its truth) is

smaller than that of the premises.

7

1. If ΓAand Γ⊆∆, then ∆A(Monotonicity)

2. Γ∪ {A|ΓA}Bif and only if ΓB(Quasi-closure)

Deﬁnition 2.15 ([15]).Aq-logic over a propositional language Lis an ordered pair

(FOR(L),), where is a substitution-invariant q-consequence relation.

Notice that while q-logics may fail to validate Reﬂexivity, p-logics might fail to

validate Cut and, thus, if they do, both are non-Tarskian or substructural logics.

Deﬁnition 2.16 ([15]).For La propositional language, an L-q-matrix is a structure

hV,D+,D−,Oi, such that hV ,Oi is an algebra of the same similarity type as L, with

universe Vand a set of operations O, where D+,D−⊆ V and D+∩ D−=∅.

A 3-valued q-matrix logics associated to the 3-element Kleene algebra that we would

like to present in connection to our ongoing investigation is the logic TS.

Deﬁnition 2.17 ([19] , [16]).A 3-valued TS-matrix is a q-matrix

MTS =h{1,1

2,0},{1},{0},{f¬

K, f ∧

K, f ∨

K}i

such that h{1,1

2,0},{f¬

K, f ∧

K, f ∨

K}i is the 3-element Kleene algebra.

TS is discussed by e.g. [6], and also by [5] in the context of the more general dis-

cussion of what represents a proper consequence relation between formulae. Moreover,

it was also discussed by [15] as a tool to model empirical inference with the aid of the

3-valued Kleene algebra. More recently, it was also stressed by Rohan French in [12] ,

in connection with the paradoxes of self-reference.

Now we need to clarify how q-matrix logics validate or invalidate both inferences and

metainferences —following e.g. [24] . Notice that, below, Mis a substitution-invariant

q-consequence relation, whence (FOR(L),M) is a q-logic. In addition to that, when

some q-logic Lis induced by a q-matrix M, we may interchangeably refer to Mas L.

Deﬁnition 2.18. For Maq-matrix, an M-valuation vsatisﬁes a sequent or inference

Γ⇒A(written vMΓ⇒A) if and only if, if v[Γ] ∩ D−=∅, then v(A)∈ D+. For

Maq-matrix, an inference Γ⇒Ais M-valid (written MΓ⇒A) if and only if

vMΓ⇒A, for all M-valuations v.

Deﬁnition 2.19. For Maq-matrix, an M-valuation vsatisﬁes a metainference Γ⇒1

A(written vMΓ⇒1A) if and only if, if vMB, for all B∈Γ, then vMA. A

metainference Γ⇒1Ais M-valid (written MΓ⇒1A) if and only if, if vMB, for

all B∈Γ, then vMA, for all M-valuations v.

The following is a simpler characterization of TS’s inferential validity:

Deﬁnition 2.20. TS Γ⇒Aif and only if for every valuation v, if v(B)∈ {1,1

2}

for all B∈Γ, then v(A)∈ {1}

8

From these deﬁnitions15 of ST and TS’s validities, the following facts follow.

Fact 2.22 ([7]). TS is a non-reﬂexive, and thus a substructural, logic.

Fact 2.23 ([7]). ST is a non-transitive, and thus a substructural, logic.

Fact 2.24 ([12]). TS has no valid inferences or sequents.

Fact 2.25 ([14], [19], [20]). ST and classical propositional logic CL have the same set

of valid inferences or sequents.

It is worth remembering that TS is sound and complete with respect to LKRinv

–e.g. LKR plus inversible rules for the operational part of the calculi, minus the truth

rules.16

3 MI, a logic for metainferences

We will now present MI, a logic that invalidates not only every inference –as TS does–,

but also every metainference (of level 1). To achieve this goal, MI’s consequence rela-

tion behaves pretty much as the inferential consequence relation TS does. Remember

that an inference is valid in TS if and only if, for every valuation v, if the premises

satisfy certain standard, then the conclusion meet some more demanding standard. In

particular, Γ ⇒Bis satisﬁed by a valuation if and only if, if for every premise A∈Γ,

v(A) = 1,1

2–e.g., if vtolerantly satisﬁes every premise–, then v(B) = 1 –e.g., vstrictly

satisﬁes the conclusion. Conversely, if vtolerantly satisﬁes every premise, but does not

strictly conﬁrms the conclusion, then vis a counterexample to the validity of that in-

ference. As an example, Identity –AA– is not valid in TS because there is at least

some formula Asuch that for some valuation v,v(A) = 1

2.

MI will work in a similar way, but in the metainferential level. Thus, MI’s standard

for the premises will be less demanding than the one for the conclusion.

Deﬁnition 3.1. A metainference Γ1A1, ...ΓnAn≫ΣBis valid in MI if and

only if, for every valuation v, if every ΓiAiis satisﬁed by vaccording to ST, then v

satisﬁes ΣBaccording to TS.

Fact 3.2. Therefore, no metainference Γ1A1, ...ΓnAn≫ΣBis valid in MI.

15This deﬁnition can also be adjusted to a multi-conclusion setting:

Deﬁnition 2.21. TS Γ⇒∆if and only if for every valuation v, if v(B)∈ {1,1

2}for all B∈Γ,

then for all A∈∆, v (A)∈ {1,1

2}

16About these logics we shall mention that in [3] , [8] and [18] it is shown that —through some

suitable translation— the set of valid inferences in LP coincides with the set of valid metainferences

in ST, while in [12] it is conjectured that —again, through some suitable translation— the set of valid

inferences in K3, e.g., Strong Kleene logic, coincides with the set of valid metainferences in TS.

9

Proof. Consider a valuation vsuch that, for every propositional letter pthat appears

in Γ1A1, ...ΓnAn≫ΣB,v(p) = 1

2. As MI matrices for the logical constants

are the ones of K3,vgives the value 1

2to every formula Ain the metainference. Thus,

every premise Γi⇒Aiis satisﬁed by vaccording to ST, but vdoes not conﬁrm the

conclusion Σ ⇒Baccording to TS.

Therefore, MI does not validate any metainference whatsoever, including those with

an empty set of premises –e.g., the inferences. Still, this does not mean that MI does

not validate metainferences of higher levels. Moreover, MI is informative about the

next metainferential level: MI validates some, but not every, meta-metainference –e.g.,

metainferences of level 2.

Ameta-metainference is an inference with the following structure: ΓΓ1, ..., ΓΓn≫

∆∆. Each ΓΓi, and also ∆∆, is a metainference, and ≫represents a consequence

relation between metainferences.

Deﬁnition 3.3. A meta-metainference ΓΓ1, ..., ΓΓn≫∆∆ is valid in MI if and only

if, for every valuation v, if if vsatisﬁes every ΓΓiaccording to MI, then vsatisﬁes ∆∆

according to MI.

Deﬁnition 3.4. Thus, a meta-metainference ΓΓ1, ..., ΓΓn≫∆∆ is valid in MI if

and only if, for every valuation v, if [for every ΓΓi, if vsatisﬁes every premise of ΓΓi

according to ST, then vsatisﬁes ΓΓi’s conclusion according to TS], then [if vsatisﬁes

every premise of ∆∆ according to ST, then vsatisﬁes ∆∆’s conclusion according to

TS].

Remark 3.5. More generally, if L1is an inferential consequence relation, a metain-

ference of level nis valid according to L1if and only if, for every valuation v, if every

premise is satisﬁed by vaccording to L1, then its conclusion is satisﬁed by vaccording

to L1. And for every i < j, if L2is a metainferential consequence relation of level i,

a metainference of level jis valid according to L2if and only if, for every valuation v,

if every premise is satisﬁed by vaccording to L2, then its conclusion is satisﬁed by v

according to L2.

With this deﬁnition at hand, we can prove the informativeness of MI.

Proof. We will give an example of a valid meta-metainference ΓΓ1, ..., ΓΓn≫∆∆,

and also an example of a invalid meta-metainference ΓΓ1∗, ..., ΓΓn∗ ≫ ∆∆∗, both

according to MI. Meta-metainference (1) is an MI’s invalid meta-metainference, while

metainferences (2) and (3) are examples of a valid meta-metainferences in MI.

AA

AA

(1) AA

AB

AA

AA

(2) AA

AA

10

AA

AA

(3) AA

AA∨ ¬A

To see why (1) is invalid it is enough to consider an instance of it such that every

formula that appears in the metainference is atomic. A valuation vsuch that v(A) = 1

and v(B) = 0 will do the rest of the job. Notice that vsatisﬁes the premise of the

meta-metainference according to MI. And this happens because vsatisﬁes A⇒A

according to ST and also in TS. Moreover, vsatisﬁes the premise of the conclusion

of the meta-metainference, because vsatisﬁes A⇒Aaccording to ST. But vdoes

not satisfy the conclusion of the conclusion of the meta-metainference in MI, because

vdoes not conﬁrm that inference according to TS.

To evaluate what happen with (2) and (3), we need to consider three diﬀerent cases:

(i) valuations vwhere v(A) = 0, (ii) valuations vwhere v(A) = 1

2, and (iii) valuations v

where v(A) = 1. In sub-cases (i) and (iii), vconﬁrms the conclusion of the conclusion

of the metainference (e.g., A⇒A, in (2), and A⇒A∨ ¬A, in (3)) in TS, and thus

satisﬁes the meta-metainference itself according to MI. In (ii), vdoes not satisfy the

premise of the meta-metainference, because vsatisﬁes the premise of the premise in ST,

but does not satisfy the conclusion of the premise according to TS. Thus, vconﬁrms

the meta-metainference according to MI.

4 One step forward: MI2

So far, we know that TS goes one step forward K3, because it not only invalidates

every formula, but also has no valid inference whatsoever. We have also shown that MI

goes beyond TS, because it also invalidates every metainference (of level 1). It seems

natural to wonder whether MI is the limit, or if there is some consequence relation that

not only invalidates every metainference, but also every meta-metainference.

We will present a consequence relation that works exactly like that. The logic that

we will call MI2validates no meta-metainference, but nevertheless is informative about

the meta-meta-metainferential level –e.g., about metainferences of level 3. As MI is

a consequence relation for metainferences –or metainferences of level 1–, MI2is a

consequence relation for meta-metainferences –or metainferences of level 2.

Ameta-metainference ΓΓ1∆∆1, ...ΓΓn∆∆n≫ΣΣ ΠΠ, then, is an inference

that has a set of metainferences as premises, and a metainference as its conclusion. Each

ΓΓi∆∆iis a premise of the meta-metainference, while ΣΣ ΠΠ is the conclusion.

The following is another –and more detailed– way to present the structure of meta-

11

metainferences, that will be used in the rest of the Section.

Γ1

1⇒∆1

1, ...Γ1

j⇒∆1

j, ...

Σ1

1⇒Π1

1

Γn−1

1⇒∆n−1

1, ...Γn−1

k⇒∆n−1

k

Σn−1

1⇒Πn−1

1

Γn

1⇒∆n

1, ...Γn

j⇒∆n

j

Σn

1⇒Πn

1

The metainferences above the double line are the premises of the meta-metainference,

while the metainference below the double line is the conclusion. We need now to intro-

duce a new notion. A stage of a metainference of any level nis a line in the metainfer-

ence. Thus, an inference has only one stage, a metainference –of level 1– has two stages

–e.g., one corresponding to the premises and one corresponding to the conclusion–, and a

meta-metainference has four stages –e.g., one for the premises of each premise, a second

one for the conclusions of the premises, a third one for the premises of the conclusion,

and a fourth one for the conclusion of the conclusion. More generally, a metainference

of level nhas 2nstages. Let us see, then, how MI2behaves.

Deﬁnition 4.1. A meta-metainference ΓΓ1, ..., ΓΓn≫∆∆ is valid in MI2if and only

if, for every valuation v, if vsatisﬁes every ΓΓiaccording to ST, then vsatisﬁes ∆∆

according to MI.

Fact 4.2. Therefore, no meta-metainference ΓΓ1⇒∆∆1, ...ΓΓn⇒∆∆n≫ΣΣ ⇒ΠΠ

is valid in MI2.

Proof. Consider a valuation vsuch that, for every propositional letter pthat appears

in ΓΓ1⇒∆∆1, ...ΓΓn⇒∆∆n≫ΣΣ ⇒ΠΠ, v(p) = 1

2. As MI2matrices for the

logical constants are K3’s ones, vwill give the value 1

2to every formula Ain the meta-

metainference. Thus, every premise ΓΓi⇒∆∆iwill be satisﬁed by vaccording to ST,

but vwill not satisfy the conclusion ΣΣ ⇒ΠΠ according to MI.

Nevertheless, MI2is informative about the next metainferential level. MI2makes

valid some, but not every, meta-meta-metainference –e.g. metainferences of level 3. A

meta-meta-metainference is an inference with the following structure: ΓΓΓ1, ..., ΓΓΓn≫

∆∆∆. ∆∆∆ and every ΓΓΓiare meta-metainferences, and ≫represents a consequence

relation between meta-metainferences.

Deﬁnition 4.3. A meta-meta-metainference of level 3 ΓΓΓ1, ..., ΓΓΓn≫∆∆∆ is valid

in MI2if and only if, for every valuation v, if vsatisﬁes every ΓΓΓiaccording to MI2,

then vsatisﬁes ∆∆∆ according to MI2.

Fact 4.4. It is not hard to show that MI2is informative about the meta-meta-metainferences.

Proof. There is at least one MIii’s valid meta-meta-metainference ΓΓΓ1, ..., ΓΓΓn≫

∆∆∆, and at least one MI2’s invalid meta-meta-metainference ΓΓΓ1∗, ..., ΓΓΓn∗ ≫

∆∆∆∗. We will give one example of each kind.

12

AA

AA

AA

AA

(*) AA

AA

AA

AA

AA

AA

AA

AB

(**) AA

AA

AA

AC

The meta-meta-metainferencential scheme (*) will be valid in MI2. Let ΓΓΓ be

any meta-metainference whatsoever. Thus, for every valuation v, either vsatisﬁes ΓΓΓ

according to MI2, or it does not. If it does, then vconﬁrms (*)’s conclusion according

to MI2, and then satisﬁes (*) itself. If it does not, then vdoes not satisﬁes (*)’s premise,

and then conﬁrms (*) according to MI2.

The meta-meta-metainferential scheme (**) will be invalid in MI2. Take a valuation

vsuch that v(A) = v(B) = 1 and v(C) = 0. vwill satisfy (**)’s premise, but will not

satisfy (**)’s conclusion according to MI2.

5 A general result: a hierarchy of logics MIn

In the previous section we have shown that MI is not the limit of the sequence of logics

that begins with K3–e.g., a logic that has no valid sentences– and follows with TS

–e.g., a logic that has no valid inferences. Though MI invalidates every metainference,

MI2also invalidates every metainference of level 2. Nevertheless, MI2is not the limit

of these kind of logics either. In fact, the sequent can be indeﬁnitively extended in

diﬀerent ways. We will sketched what seems to us a natural way to do it.

Fact 5.1. General Result

For any level nof metainferences, there is a consequence relation MIn for metain-

ferences of level n such that there are no valid MIn’s sentences, inferences an metain-

ferences of any level less or equal than n. Nevertheless, any MIn is informative about

metainferences of level n+ 1.

Let TS=MI0and MI=MI1. Both of them invalidate metainferences of level nor

less, but nevertheless they are informative about the next metainferential level. But we

still don’t know how logics MIn–for 2 < n– might look like. We will present a general

deﬁnition of logics MIn, for any nsuch that 2 ≤n.

Deﬁnition 5.2. Let Γ1, ...Γjand ∆be metainferences of level n. For any n, a metain-

ference of level n+ 1,Γ1, ...Γj≫∆is valid in MIn+1if and only if, for every valuation

v, if vsatisﬁes every Γ1, ...Γnin MIn−1, then vsatisﬁes ∆according to MIn.

13

Once the hierarchy is establish, the ﬁrst thing we need to prove is that each MIn

invalidates every metainference of level n.17

Proof. For any n, let Γ1, ...Γj≫∆ be a metainference of level n. Let vbe a valuation

such that, for every propositional letter pin Γ1, ...Γj≫∆, v(p) = 1

2. Then vis a

counterexample to Γ1, ...Γn≫∆’s validity in MIn+1.

The proof the previous assertion will be an induction on the level nof the logics

MIn. The inductive hypothesis says that for any metainference Γ1, ...Γn≫∆ of level

n, a valuation vsuch that for every propositional letter pin Γ1, ...Γn≫∆, v(p) = 1

2, is

a counterexample to the validity according of the metainference in MIn.

Base case: If n= 0, then MIn+1=MI1=MI. We have already shown that a

valuation vthat assigns the value 1

2to every propositional letter pin a metainference

of level 1, Γ1, ...Γn≫∆, is a counterexample to the metainference’s validity in MI.

Inductive step: If 1 < n, we need to prove that vwill not satisfy any metainference of

level n+ 1 in MIn+1. Take any metainference of level n+ 1, Γ1, ...Γn∆. By inductive

hypothesis, vwill not satisfy ∆ –a metainference of level n– according to MIn. We still

need to prove that vsatisﬁes every Γi(1 ≤i≤n) according to MIn−1. Remember

that each Γiis a metainference of level n. Thus, each of its premises is a metainference

of level n−1. By inductive hypothesis, vdoes not satisfy them according to MIn−1.

Thus, vsatisﬁes each Γi(because it does not satisfy any Γi’s premises) according to

MIn−1.

We will now prove that every MInis informative about the metainferences of level

n+ 1. As an example of a valid metainference of level n+ 1, take what may be called an

instance of Meta-Identity of level n+ 1, e.g., a metainference ΣΣ ≫ΣΣ, where ΣΣ is a

metainference of level n. For any valuation v, either vsatisﬁes ΣΣ according to MIn,

or it doesn’t. If it does, then it satisﬁes the conclusion of ΣΣ ≫ΣΣ, and thus satisﬁes

ΣΣ ≫ΣΣ itself. And if it doesn’t, then it does not satisﬁes the premise of ΣΣ ≫ΣΣ,

and so it satisﬁes the metainference itself.

The following one is an example of a metainference of level n+ 1 that is invalid

in MIn. Consider a metainference of level n+ 1, ΓΓ ≫∆∆, where both ΓΓ and

∆∆ are diﬀerent instances of Meta-Identity of level n. Moreover, at each stage of the

metainference, there will be only one inference, an instance of Identity, e.g., a case of

A⇒A. For the sake of simplicity, let Abe a propositional letter, and a diﬀerent one in

each case. Thus, the only inference that will appear in ΓΓ will be, for example, p⇒p;

17There are other ways to build the hierarchy. The following one is an alternative that may not be as

elegant as the one we have presented, but, nevertheless, achieves the same goals. (We leave the proofs

of these results as exercises for the reader.) The following one is another general deﬁnition of another

hierarchy of logics MIn∗

Deﬁnition 5.3. Let Γ1, ...Γnand ∆be metainferences of level n. For any n, a metainference of level

n+ 1,Γ1, ...Γn≫∆is valid in MI∗

n+1if and only if, for every valuation v, if every Γ1, ...Γnis satisﬁed

by vaccording to ST, then vconﬁrms ∆according to MI∗

n.

14

and the only inference that will be part of ∆∆ will be q⇒q. Take a valuation vsuch

that v(p) = 1 and v(q) = 1

2. That valuation will satisfy ΓΓ according to MIn, but

would not satisfy ∆∆ according to MIn.

5.1 An interesting result about the hierarchy, and a few open ques-

tions

Some questions arise at this point. For example, what is the exact relation between

an MInand the rest of the logics in the hierarchy? How do the diﬀerent MInbehave

with respect to the metainferences of a level higher than n+ 1? We will present one

intriguing result.

An interesting question is how two diﬀerent MIns, MIjand MIk(where j < k),

behave with respect to metainferences of a level higher than k. Do they validate exactly

the same metainferences? Is one included in the other (and which one)? Are they

incomparable?

Fact 5.4. Here is a partial answer. For any two diﬀerent MIns, MIjand MIk, such

that k=j+ 1,MIjand MIkare incomparable with respect to metainferences of level

k+ 1.

Proof. By induction on the level nof the metainferential consequence relation MIn.

Base case: n= 0. This is a question about how TS and MI behave with respect to

meta-metainferences (e.g., metainferences of level 2).

Therefore, there must be two meta-metainferences (1) and (2) such that: (i) (1) is

valid in TS, but invalid in MI, and (ii) (2) is valid in MI and invalid in TS. We will

begin with the proof of (i).

BB

BB

(1) AA

AA

As every valuation satisﬁes its conclusion, (1) is a valid TS meta-metainference. For

every valuation v, either v(A) = 1

2, or v(A) = 0, orv(A) = 1. If v(A) = 1

2, then vdoes

not satisfy the premise of the conclusion of the meta-metainference (e.g. AA) in TS.

Thus, vconﬁrms the conclusion of the metainference, and so satisﬁes (1). If v(A) = 0,1,

vsatisﬁes the conclusion of the conclusion in TS. Therefore, vconﬁrms the conclusion

of the meta-metainference, and thus the meta-metainference itself. Nevertheless, (1) is

an invalid MI meta-metainference. A valuation vsuch that v(A) = 1

2and v(B) = 0,1

satisﬁes the premise of the meta-metainference, but does not conﬁrms the conclusion

according to MI.

15

AA

AA

(2) BB∨C

AA

(2) is a valid MI’s meta-metainference. For every valuation v, either v(A) = 1

2,

or v(A) = 0, orv(A) = 1. If v(A) = 1

2, then vsatisﬁes the premise of the premise

of the meta-metainference (e.g., AA) according to ST, but does not conﬁrms the

conclusion of the premise of the meta-metainference (e.g. AA) in TS. Thus, vdoes

not satisfy the premise of the meta-metainference according to MI, and thus satisﬁes

the meta-metainference itself in MI. If v(A) = 0,1, vsatisﬁes the conclusion of the

conclusion of the meta-metainference according to TS. Thus, vconﬁrms the conclusion

of the meta-metainference, and therefore (2) itself.

Nevertheless, (2) is an invalid TS’s meta-metainference. A valuation vsuch that

v(A) = 1

2and v(B) = 0,1 conﬁrms the premise of the meta-metainference, but does not

satisfy the conclusion, both according to TS.

Inductive case: n > 0. Take any MInand MIn+1, such that n > 0. We will present

two metainferences of level n+ 2, (1*) and (2*), such that: (i) (1*) is valid in MIn, but

invalid in MIn+1, and (ii) (2*) is valid in MIn+1and MIn.

∆∆

∆∆

(1*) ΓΓ

ΓΓ

ΓΓ and ∆∆ are metainferences of level n. To simplify things, let every sequent that

belongs to them be an instance of Identity, but a diﬀerent one in each case. Thus, let

AAbe the only inference that is part of ΓΓ, and let BBbe the only inference in

∆∆. Aand Bare diﬀerent formulas, and to make things even more easy to handle, let’s

take them as (diﬀerent) propositional letters. ΓΓ and ∆∆ will have, in each case, 2n

stages, and each one will have only a single occurrence of the inference AA/BB).

(1*) is a valid MInmeta-metainference. Every valuation vsatisﬁes the conclusion

–e.g., a metainference of level n+ 1– in MIn, because, either vsatisﬁes the conclusion,

or does not satisfy the premise. Thus, every valuation vconﬁrms the metainference

according to MIn. Therefore, vconﬁrms (1*) according to MIn.

Nevertheless, (1*) is an invalid MIn+1metainference. Consider a valuation vsuch

that v(p) = 1

2for every propositional letter pthat is part of A, and v(q) = 1 to every

propositional letter qthat is part of B. The valuation vwill satisfy the premise, but

will not satisfy the conclusion. Thus, it will be a counterexample to the validity of (1*)

in MIn+1.

16

ΓΓ

ΓΓ

(2*) ∆∆

ΓΓ

(2*) is a valid MIn+1’s meta-metainference. The premise of (2*) is a metainference

of level n+ 1. Thus, a valuation vthat gives value 1

2to every propositional letter

pin Awill be a counterexample in MIn+1. Therefore, as vis a counterexample to

(2*)’s premise, it also conﬁrms (2*) itself according to MIn+1. Valuations v∗such that

v∗(p) = 0,1, satisfy ΓΓ according to MIn, and thus satisfy (2*)’s conclusion according

to MIn+1.

Nevertheless, (2*) is an invalid MIninference. Consider a valuation vsuch that

v(p) = 1

2and v(q) = 1. That valuation does not satisfy the premise of (2*)’s premise

according to MIn, and thus conﬁrms (2*)’s premise itself. Nevertheless, that valuation

does not satisfy the conclusion of (2*)’s conclusion according to MIn, but conﬁrms its

premise according to that same consequence relation. Then, vdoes not satisfy (2*)’s

conclusion itself in MIn. But, as vsatisﬁes (2*)’s premise, it is a counterexample in

MIn.

Still, this does not tell us nothing about how two diﬀerent MIns, MIjand MIk

–where j < k, but not necessarily k=j+ 1– behave about metainferences of a level

k+ 1. Moreover, we still don’t know which is the exact nature of the relation between

any two logics MIjand MIk–where j < k– with respect to metainferences of a level l

–where k < l, but not necessarily l=k+ 1. These are, at this point, open questions.

6 A truly empty logic

So far, what we have is a hierarchy of logics MIn, where each MIninvalidates every

metainference of level nor less –including every inference and every sentence. Is it

possible to follow this path until we found what might be called a truly empty logic

–e.g., a logic that invalidates any sentence, inference and metainference of any level

whatsoever?

It is not an easy thing to characterize such logic. For example, let EL1to be

a consequence relation with just two values, 1 and 0. EL1’s consequence relation is

understood classically as preservation of value 1 from premises to conclusion. What is

peculiar about it is that the only proper valuations are those that assign value 0 to every

formula. Thus, EL1will have no valid sentence. Still, any inference (with a non-empty

set of premises) Γ ⇒∆ will be valid in it, because no valuation vwill give value 1 to

each premise. Nevertheless, a slight modiﬁcation of EL1’s consequence relation might

do the work. Let EL2be as EL1, but with the following consequence relation: an

inference Γ ⇒∆ is valid in EL2if and only if, for every valuation v,v(γ) = 1 –for every

γ∈Γ– and for some δ∈∆, v(δ) = 1. As the only valuations are the ones of EL1, there

will be no valid inference, because no valuation will give value 1 to any formula. Still,

17

every metainference of level 1 (with a non-empty set of premises) will be valid, because

no valuation will satisfy the premises according to EL1.

We may go on modifying these logics in order to invalidate more and more metainfer-

ences. Ultimately, if we are lucky, what we might get is a series of consequence relations

that invalidates every metainference of level nor less, for every level n. Nevertheless,

these consequence relations seems a little bit too artiﬁcial to be really interesting. More-

over, we already have a similar hierarchy of logics, generated in a recursive way: the

hierarchy of MIns.

The next question we face is if we can build an empty logic from the hierarchy. In

a nutshell, we can. Let’s call an inference Σ ⇒Π a metainference of level 0. Moreover,

let Γ1, ...Γj≫∆ be a metainference of level n(1 ≤n < ω). We will call the target

logic, Lempty.

Deﬁnition 6.1. A metainference of level n(for any level n,0≤n < ω)Γ1, ...Γj≫∆

is valid in Lempty if and only if Γ1, ...Γj≫∆is valid in MIn, for every MIn. Thus,

Lempty is a truly empty logic, because no sentence, inference and metainference of level

n–for any level n– will be valid in Lempty.

Proof. Take any metainference whatsoever. That metainference will have a particular

level j. Then, that metainference will be invalid in MIj. Thus, it will be invalid in

Lempty.

7 Conclusion

TS is a logic that has no valid inferences. Nevertheless, TS is informative about the

metainferences (of level 1). It is a fair question to ask if TS is as empty as a logic

can be. In particular, could there be a logic without valid metainferences? We have

given a positive answer to this question. MI is that logic. A metainference is valid

in MI if and only if, for every valuation v, if vsatisﬁes every premise according to

ST, then vsatisfy the conclusion according to TS. This move is very similar to the

one that justify substructural logics like ST or TS. In those cases, the standards that

premises and conclusions of a sound inference should satisfy, are diﬀerent. The same

thing happens with MI. The main diﬀerence is that the sound inference, in this case,

is, strictly speaking, a metainference.

But MI is not the limit either. We have present a consequence relation for metain-

ferences of level 2, called MI2, which has no valid meta-metainferences, but is, nev-

ertheless, informative about the metainferences of level 3. Moreover, it is possible to

specify, for any level nof metainferences, a consequence relation MInfor metainferences

of level n, such that no metainferences of level nor less in valid in it. Nevertheless,

every MI is informative about the metainferences of level n+ 1.

Thus, every such MInwill validate some metainference –in particular, metainfer-

ences of level higher than n. No logic MIn, then, is really empty. Is there a truly

empty logic? Though it is not easy to design such a consequence relation, it is posible

18

to specify a truly empty logic using the hierarchy of logics MIn. A metainference of

level n, for any level n, will be valid in that logic, Lempty, if and only if it is valid in

every MIn. As any metainference has a particular level j, that metainference is invalid

in MIj. Thus, it will be invalid in Lempty. Therefore, Lempty is a truly empty logic.

At this point, there are many open questions about the hierarchy we have presented.

For example, what is the exact nature of the relation between two members of the

hierarchy, MIjand MIk–where j≤k– with respect to metainferences of level higher

thatn k? Is one included in the other? Are they incomparable? Moreover, is it possible

to design sequent calculi for these logics? How should such a calculus look like? We

have present a partial answer to the ﬁrst of these question in Section 5. But we have

remain silent with respect to the rest of them. These questions have no easy answers,

and we will leave them for future work.

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