The simplest protoalgebraic logic
Josep Maria Font
Mathematical Logic Quarterly∗ 59 (2013) 435–451
Abstract
The logic I is the sentential logic defined in the language with just implication → by the axiom
of reflexivity or identity “ ϕ → ϕ” and the rule of Modus Ponens “from ϕ and ϕ → ψ to infer ψ”.
The theorems of this logic are exactly all formulas of the form ϕ → ϕ. I argue that this is the simplest
protoalgebraic logic, and that in it every set of assumptions encodes in itself not only all its consequences
but also their proofs. In this paper I study this logic from the point of view of abstract algebraic logic,
and in particular I use it as a relatively natural counterexample to settle some open problems in this
theory. It appears that this logic has almost no properties: it is neither equivalential nor weakly
algebraizable; it does not have an algebraic semantics; it does not satisfy any form of the Deduction
Theorem, other than the most general parameterized and local one that all protoalgebraic logics satisfy;
it is not filter-distributive; and so on. It satisfies some forms of the interpolation property but in a rather
trivial way. Very few things are known about its algebraic counterpart, save that its “intrinsic variety”
is the class of all algebras of the similarity type.
1
Introduction
One of the most important features of the emergence of abstract algebraic logic in recent decades is
the classification of sentential logics in two hierarchies, the Leibniz hierarchy and the Frege hierarchy1 ,
according to the strength of several aspects of their algebraic behaviour. The Leibniz hierarchy is the
more complicated of the two and it has been more extensively studied. It is a historical fact that good
classification schemes have played a key role in the evolution of many fields of mathematics and of
science in general; and progress in structuring and understanding very general theories is almost always
the result of parallel advances both in obtaining general results and in scrutinizing examples that are
situated near the frontiers of the field (whether considered to be within it or not). The logic I study in
this paper lives right on the edge of what is certainly one of the largest classes in the Leibniz hierarchy:
the class of protoalgebraic logics [1, 7].
One of the most interesting features of protoalgebraic logics is that they can be defined and characterized from several, in principle unrelated, points of view. Here I focus on the following syntactic
approach. According to a theorem of B LOK and P IGOZZI [3], a logic2 L is protoalgebraic when there is a
set ∆( x, y) of formulas in at most two variables satisfying the following two very elementary properties:
`L ∆( x, x )
x , ∆( x, y) `L y
∗ This
(R∆)
(MP∆)
author-produced version does not incorporate some of the stylistic changes made by MLQ editorial office.
a compact introduction to the two hierarchies in the general context of the evolution of algebraic logic, see [8], especially
Sections 2.6 and 3.4; other general reference works are [7, 10, 12].
2 A (sentential) logic L is here understood as a substitution-invariant (or structural) consequence relation ` on the algebra of
L
formulas of some (sentential) language. Notice that finitarity is not assumed as part of the definition of a logic: some logics are
finitary and some are not.
1 For
1
In this case it is said that the set ∆ witnesses protoalgebraicity of L . When the logic L is finitary3 this set
can always be taken to be finite. The case ∆ = ∅ is not excluded, but corresponds only to the so-called
trivial4 logics. Clearly, the weakest5 non-trivial logic in a given language for which a given set ∆ 6= ∅
witnesses protoalgebraicity is the logic axiomatized by just (R ∆ ) and (MP∆ ). It is easy to see that for this
to be non-trivial, one needs to have at least one connective of arity 2 or greater in the language.
Thus, arguably, the simplest (non-trivial) protoalgebraic logic is one where the language is the simplest
one possible, i.e., it consists of just one binary connective, which it seems reasonable to write as → , and
where the set ∆ consists of just the simplest binary formula, i.e., ∆( x, y) = { x → y} . Formally:
D EFINITION 1.1. The logic I is the logic in the language h→i of type h2i axiomatized in the usual Frege-Hilbert
style by
the axiom
and the rule
x→x
x , x → y y.
(I)
(MP)
The symbol will be used in this paper as a kind of “separator” when expressing formal rules of
inference in Hilbert-style axiomatic systems, and to write sequents in Gentzen-style calculi, in order not
to use the symbols ` or → or ⇒ , which have other roles and whose use here could therefore lead to
misunderstanding.
I named this logic I following the conventions of the area of BCI logic and related systems6, where
axiom (I) appears with this name; in other contexts it appears as (R). This logic seems to have been
considered only occasionally in the literature; it appears in passing in [15, page 6]7 and in [4, page 169]8
as a counterexample to particular points.
The consequence relation of this logic will be denoted by `I , and its associated closure operator9 by
CI , so that the theory generated by a set Γ of formulas will be denoted by CI Γ. The set of theorems of
this logic is CI ∅, and I will write `I ϕ instead of ∅ `I ϕ.
The properties of implication included in the definition of I are arguably prototypical properties of
an implication connective, found whenever such a connective is considered. They are, however, very
basic or minimal; and it may seem very little can be deduced from them. Still, I think the properties of
this logic are worth studying, even if the conclusion of such study may be that “this logic has almost no
properties”. Furthermore, this logic can serve as a relatively natural counterexample to certain points in
abstract algebraic logic, as happens twice herein. The study is presented here as a kind of revision of the
main notions and techniques in abstract algebraic logic, and in some sense it is a reflection on the (lack of)
strength of the Laws of Identity (or Reflexivity) and Modus Ponens when they are not accompanied by
other properties.
It is worth observing that the “simplicity” requirement made when defining I has somehow broken the previous reasoning on “weakness” that led to the definition: while I is clearly the weakest
protoalgebraic logic for which the set { x → y} witnesses protoalgebraicity, it is clearly not the weakest
protoalgebraic logic, in an absolute sense, in the same language. As an example, it is easy to see that the
logic axiomatized by the axiom x → x and the rule x, x → y, y → x y is strictly weaker than I ; but its
protoalgebraicity is witnessed by another set, namely { x → y, y → x } .
The layout of the paper is as follows. Section 2 establishes the main syntactic or proof-theoretic
properties of I . I show that in some sense, every set of assumptions encodes in itself all its consequences
3A
logic L is finitary when Γ `L ϕ implies Γ0 `L ϕ for some finite Γ0 ⊆ Γ.
logic L is trivial when it satisfies x `L y for two distinct variables x, y; or, equivalently, α `L β for any two formulas α, β.
In each language there are exactly two trivial logics: the inconsistent logic and the almost inconsistent logic, which are the logics
that have only one theory (the set of all formulas) and two different theories (the empty set and the set of all formulas), respectively.
5 For logics L and L0 over the same language, L is weaker than L0 when Γ ` ϕ implies Γ ` ϕ; then L0 is said to be stronger
L
L0
than L .
6 This name for I was suggested by C LINT VAN A LTEN .
7 I owe this information to A LEXEI M URAVITSKY .
8 See also footnote 15.
9 The relation ` and the operator C are mutually interdefinable by the equivalence: ϕ ∈ C Γ if and only if Γ ` ϕ.
I
I
I
I
4A
2
and also their proofs. As a consequence, several properties are derived which are then used in the rest
of the paper. This section ends with a presentation of a Gentzen-style calculus that defines I and has
good proof-theoretic properties (cut elimination and an enhanced subformula property). In Section 3, a
simple set of congruence formulas with parameters for I is found, as a consequence of a general result
for protoalgebraic logics. This set is used to prove that the Leibniz congruence of many theories is the
identity relation; this fact has several interesting consequences in the paper, in particular it allows to
answer a question of F ÉLIX B OU about a property concerning the commutativity of the Leibniz operator
with substitutions. Sections 4 and 5 contain the main results of the paper. In the first, I show that I
does not have any of the properties usually considered in the literature that relate to all kinds of logical
connectives, such as conjunction, disjunction, implication (notably, I show that it does not satisfy the
Parameterized Deduction-Detachment Theorem, being—I believe—the first explicit example of this
in the literature) or, indirectly, to negation (through R AFTERY’s abstract version of the Inconsistency
Lemma) and modality. I also analyse some interpolation properties. Finally, I show that I , while finitary,
is not logically compact in the related but non-equivalent sense that every inconsistent set contains a
finite inconsistent subset. In Section 5, the last in the paper, I show that I does not belong to any of
the classes of the Leibniz hierarchy other than that of protoalgebraic logics, or to any of the classes of
the Frege hierarchy, and that it is strictly weaker than other logics of pure implication considered in the
literature. The paper ends with some general semantical considerations, although this is the aspect of I
that seems most difficult to study, and really nice results have not so far been obtained.
Another peculiar feature of I , the structure of its lattice of theories, is studied in [9].
This paper assumes familiarity with the notions and notations of abstract algebraic logic as given in
[7, 10, 12]. In order to aid the flow and comprehension of the text, however, many of the central notions
will be recalled at appropriate place.
2
Basic syntactic properties
To start with, the theorems of this logic are easily determined:
P ROPOSITION 2.1. A formula ϕ is a theorem of I if and only if ϕ = α → α for some α.
P ROOF. The set of formulas of the form α → α for arbitrary α, which actually consists of all substitution
instances of the only axiom of I , is easily seen to be closed under the rule (MP); therefore, it is the
smallest theory, that is, the set of theorems of the logic.
This result might be viewed as categorizing I as an “extremely relevant” logic: β follows from α, in
the classical sense that α → β is a theorem, if and only if the assumption α is the most relevant possible
with respect to the conclusion β, namely when it equals β.
Since the set of theorems is neither empty nor the set of all formulas, we have:
C OROLLARY 2.2. The logic I is decidable and neither inconsistent nor almost inconsistent.
N OTATIONS AND TERMINOLOGY.
• The set of atomic formulas or variables is denoted by Var. For any formula α, the set of the variables
S
appearing in it is denoted by varα. If Γ is a set of formulas, then var Γ := {varα : α ∈ Γ } . The letters
x, y, z, t, u, w will denote variables. Variables represented by different letters are different, unless the
contrary is explicitly admitted.
• The algebra of formulas is denoted by F m = h Fm, →i , where Fm denotes the set of all formulas
of this similarity type. Lowercase Greek letters except σ, with or without subindices, will denote
formulas. Non-boldface uppercase Greek letters, also with or without subindices, will denote sets
of formulas. As usual, a notation such as α( x, y, z) means that varα ⊆ { x, y, z} and so on; and then
α( ϕ, ψ, δ) denotes the result of replacing the variables x, y, z in α by the formulas ϕ, ψ, δ, respectively.
3
• If α = γ → δ then γ is the head of α and δ is its tail. All formulas except the variables have one (and
only one) head and one (and only one) tail. Two formulas that are not variables are equal if and only if
they have the same head and the same tail.
• In an instance α, α → β β of (MP) the formula α is the short assumption, and the formula α → β is
the long assumption.
• Vectors such as ~z or ~δ will denote (finite or infinite) sequences of variables or formulas, respectively;
this is indicated by writing simply ~z ∈ Var or ~δ ∈ Fm, leaving the length of the sequence unspecified
unless needed. For convenience, the members of non-empty finite sequences will be indexed starting
with 1.
• Let ~α be a finite sequence of formulas and β a formula. Then ~α → β denotes the nested formula
α1 → α2 → (α3 → . . . (αn → β)...) when ~α = hα1 , . . . , αn i for some n > 1, and denotes the formula β
when ~α is the empty sequence. This is easily defined by induction.
• A formula β is a final subformula of a formula ϕ when ϕ = ~α → β for some finite sequence ~α
of formulas; this sequence is called the prefix (of β in ϕ). A final variable of a formula is a final
subformula that is a variable.
• If ~α = hα1 , . . . , αn i is a finite, non-empty sequence of formulas, then its final segments are denoted by
~α[ k) := hαk , . . . , αn i ; we also need to set ~α[ n+1) := ∅ for notational convenience. Observe that ~α[ 1) = ~α.
Note that, according to the definition of ~α → β, any formula is a final subformula of itself, with the
empty prefix. A formula may have several final subformulas, but it always has one and only one final
variable; every formula can be written in the form ~α → x in a unique way, where x is its final variable,
and its final subformulas are the formulas ~α[ k) → x for 1 6 k 6 n + 1, where n is the length of ~α.
The notation ~α → β extends the notation αn → β, which is very often used in the literature; while it is
sufficiently self-explanatory, it is also potentially confusing, because it tends to suggest that ~α and β are
the head and the tail, respectively, of ~α → β, which is not the case. In order to avoid misunderstanding, it
may be useful to formulate the following, fairly obvious property:
L EMMA 2.3. Let ~α = hα1 , . . . , αn i and ~γ = hγ1 , . . . , γm i be finite, non-empty sequences of formulas, and let
β, δ be formulas. If ~α → β = ~γ → δ then one of the following, mutually exclusive situations holds:
(1) n = m, that is, ~α and ~γ have the same length, and αi = γi for all i ∈ {1, . . . , n} and β = δ.
(2) n < m, that is, ~α is strictly shorter than ~γ , and αi = γi for all i ∈ {1, . . . , n} and β = ~γ[ n+1) → δ.
(3) n > m, that is, ~α is strictly longer than ~γ , and αi = γi for all i ∈ {1, . . . , m} and δ = ~α[ m+1) → β.
The extreme simplicity of the axiomatic system defining I allows for a first, naïve examination of
what non-trivial proofs could be like. This analysis reveals that, in some sense, each set of assumptions
encodes in itself not only the formulas that can be inferred from it, but also the proofs themselves. Indeed, a
non-trivial application of (MP) in a proof of a formula β from a set of assumptions Γ must have the
form α, α → β β with α 6= β, so that by Proposition 2.1, α → β is not an instance of the axiom, and
hence either belongs to Γ or is the result of another non-trivial application of (MP). In the latter case, this
would have the form α0 , α0 → (α → β) α → β, again with α0 6= α → β, and so on. Since proofs are finite,
working upwards and focusing on the long assumptions of the applications of (MP), which can never be
instances of the axiom, we see that the proof must necessarily begin this branch with a formula from Γ,
which will be of the form ~α → β, where the αi are the short assumptions of the same applications of (MP),
here written as α, α0 , etc. But on the other hand, all these short assumptions must in turn be derived from
Γ as well, so the same analysis could be performed for each of them. That is, one informally guesses the
following:
4
N ON -P ROPOSITION 2.4. Γ `I β if and only if β satisfies one of the following conditions:
(a) β is a theorem.
(b) β ∈ Γ.
(c) There is a finite, non-empty sequence ~α of formulas such that ~α → β ∈ Γ and each αi satisfies one of the
conditions (a), (b) or (c).
As a simple example where the three situations for the αi appear, consider:
Γ `I t
(1)
where
Γ := z , x → (y → y) → (z → t) , x → (y → y) → (z → t) → x .
Here we see that the formula from Γ having t as its final subformula is the formula x → (y → y) →
(z → t) , with prefix x , y → y , z . In a proof of t from Γ, each of the formulas in this prefix appears as
the short assumption in one application of (MP), and each in turn follows from Γ by a different method,
namely the three situations described in Non-Proposition 2.4: z belongs to Γ, y → y is a theorem, and x
follows from Γ by an application of (MP) to two members of Γ, hence x is also a final subformula of
another formula in Γ, and this time the prefix is a single formula directly belonging to Γ.
From Non-Proposition 2.4, it would follow that from every finite set of assumptions one can effectively
construct the proofs of all its consequences; notice that those that are not theorems are finite in number, as
will be shown in Corollary 2.9.3. This implies not only decidability for theoremhood, but also for finite
consequences.
I am not going to ask you to rely on these considerations, nor on Non-Proposition 2.4 itself; instead, I
will directly prove some of its consequences: the ones that will be effectively used in the sequel.
The particular case of the assumption in Non-Proposition 2.4 where Γ consists of a single formula
can be formulated in a more precise and simpler way, and can be formally proved:
P ROPOSITION 2.5. α `I β if and only if β satisfies one of the following conditions:
(a) β is a theorem.
(b) β = α.
(c) α = ~τ → β for some finite, non-empty sequence ~τ of theorems.
P ROOF. ( ⇐ ) is obvious, so let us prove ( ⇒ ) by induction on the length of a proof of β from α. The
only thing to be checked is that the property applying to β is preserved under (MP), so we assume that
γ , γ → δ satisfy it, and prove that δ satisfies it. We first examine the three cases concerning γ → δ.
(a) If γ → δ is a theorem, then γ = δ. Since by assumption γ satisfies the property, δ satisfies it.
(b) If γ → δ = α then we show, by looking at the conditions satisfied by γ, that γ must be a theorem, so
that δ satisfies (c). If γ is a theorem by (a), then δ directly satisfies (c). Since γ 6= α, condition (b) is not
possible. Finally, in the case of (c), α = ~τ → γ for some finite, non-empty sequence ~τ of theorems; then
α = γ → δ = ~τ → γ which by Lemma 2.3 implies that γ = τ1 , again a theorem.
(c) If α = ~ρ → (γ → δ) for some sequence ~ρ of theorems, then as before we show, by looking at the
conditions satisfied by γ, that γ must be a theorem, so that δ satisfies (c). If γ is a theorem by (a), then
again δ directly satisfies (c). Since γ 6= α, condition (b) is not possible. Finally, in case (c), α = ~τ → γ for
some finite, non-empty sequence ~τ of theorems; then ~τ → γ = ~ρ → (γ → δ) . Looking at Lemma 2.3, we
see that ~τ and ~ρ cannot have the same length, for this would imply that γ = γ → δ, which is absurd;
and ~ρ cannot be strictly longer than ~τ , for then γ = ~ρ[ k) → (γ → δ) for some k, which is absurd as well.
Thus, ~ρ must be strictly shorter than ~τ and γ → δ = ~τ[ k) → γ for some k, and this implies that γ = τk , a
theorem.
No similar characterizations have been found of the theory generated by more than one formula.
However, there are cases where it is possible to prove that the theory consists of just the generators
5
plus the theorems; these theories are more thoroughly studied in [9], and some of them appear in Corollary 2.9.1. Some consequences of Proposition 2.5 can already be obtained. The interderivability relation
a`L of a logic L is defined as α a`L β if and only if both α `L β and β `L α; for an arbitrary L this is
always an equivalence relation (but need not be a congruence).
P ROPOSITION 2.6. α a`I β if and only if α and β are both theorems, or α = β.
P ROOF. ( ⇐ ) is trivial and holds for any logic. ( ⇒ ) If α `I β and β `I α, then in each of these cases
one of conditions (a), (b) or (c) from Proposition 2.5 must hold. Clearly, possibility (c) cannot hold in both
cases, neither can we have (b) in one case and (c) in the other. Now assume we have (a) β is a theorem,
and (c) β = ~τ → α for some finite, non-empty sequence of theorems ~τ . Since β is a theorem and so are
all the τi , after applying (MP) a finite number of times, we find that α is a theorem as well. Thus, in this
case the two formulas are theorems. The remaining possibilities are contained in those of the statement
(which are not mutually exclusive).
Thus, the interderivability relation of I is the smallest possible one, for the two situations here
∼
described must be included in a`L for any logic L . The Tarski congruence Ω L is the largest con∼
gruence of the formula algebra below the interderivability relation a`L ; the quotient F m/ Ω L is the
Lindenbaum-Tarski algebra of L .
∼
P ROPOSITION 2.7. The Tarski congruence of I , Ω I , is the identity relation.
∼
∼
P ROOF. Take any two formulas α, β with α 6= β, and assume that α ≡ β ( Ω I ) . Since Ω I is a
∼
congruence, for any variable z appearing neither in α nor in β, also α → z ≡ β → z ( Ω I ) , and this
implies that α → z a`I β → z. But α → z 6= β → z, α 6= z and β 6= z, so neither α → z nor β → z are
theorems; by Proposition 2.6 these facts imply that α → z a6 `I β → z: a contradiction.
In several applications, the following property of consequence in I , containing part of the information
in Non-Proposition 2.4, will be sufficient. Note that here the cases (b) and (c) of 2.4 are merged into one
condition.
P ROPOSITION 2.8. If Γ `I β, then β is a theorem or β is a final subformula of some formula in Γ.
P ROOF. By induction on the length of the proof of β from Γ. If β ∈ Γ ∪ CI ∅, then it satisfies the property.
Now assume there is some α such that both α and α → β have shorter proofs from Γ and hence, by the
induction hypothesis, satisfy the property. If α → β is a theorem, then α = β and hence β satisfies the
property. If α → β is a final subformula of some formula in Γ then so is β, hence it also satisfies the
property.
Note, however, that in general the prefix of the formula from Γ containing β as a final subformula
need not consist only of theorems, as in Proposition 2.5; this is shown by the example in (1). Note also that
the converse of the implication in Proposition 2.8 is not true, the reason being precisely Proposition 2.5.
Some useful consequences of Proposition 2.8 are:
C OROLLARY 2.9.
1. If Γ is a set of variables then CI Γ = Γ ∪ CI ∅.
2. If x, y are two distinct variables, then CI { x } ∩ CI {y} = CI ∅.
3. A finitely generated theory contains only a finite number of non-theorems.
4. No finite set is inconsistent10 ; in particular, there does not exist an inconsistent formula.
5. If Γ `I β then β is a theorem or varβ ⊆ varΓ.
This section ends with Gentzen-style axiomatizability.
10 A
set of formulas Γ is inconsistent when the theory it generates, CL Γ, is the set of all formulas.
6
D EFINITION 2.10. The Gentzen calculus G is formulated in the language h→i and on sequents of the form
Γ β where Γ is a finite set of formulas and β is a formula, and has as its only structural rules
αα
( CUT )
Γα
∆, α β
Γ, ∆ β
∅ α→α
( G - MP )
Γα
∆, β γ
.
Γ, ∆, α → β γ
( AXIOM )
and as its only logical rules
(G-I)
Exchange and contraction rules are implicit in the fact that the antecedents of our sequents are sets
of formulas. Note also the absence of the weakening rule, which is coherent with the formulation
of Proposition 2.12, and the comment at the end of the section.
L EMMA 2.11. The sequent α, α → β β is derivable in G without the cut rule.
P ROOF.
αα
ββ
( G - MP )
α, α → β β
P ROPOSITION 2.12. Γ `I β if and only if there is a finite Γ0 ⊆ Γ such that the sequent Γ0 β is derivable in
the calculus G.
P ROOF. ( ⇒ ) By induction on the length of the proof of β from Γ in `I .
- If β ∈ Γ, then take Γ0 = { β} because by (AXIOM), β β is derivable in G.
- If β is an instance of (I), then take Γ0 = ∅ because by (G - R) ∅ β is derivable in G.
- If β is the result of an instance of (MP), then there is some α such that Γ `I α and Γ `I α → β with
shorter proofs, so that by the inductive hypothesis there are finite Γ0 , Γ1 ⊆ Γ such that the sequents
Γ0 α and Γ1 α → β are derivable in G. Then the proof tree
. . .. . .
...
Γ1 α → β
. . .. . .
...
Γ0 α
ββ
( G - MP )
Γ0 , α → β β
( CUT )
Γ0 , Γ1 β
shows that Γ0 ∪ Γ1 β is derivable in G.
( ⇐ ) It is enough to show that the relation `I satisfies the rules of G. This is obvious for the structural
rules, because `I is a consequence relation, and for (G - R) because ∅ `I α → α. As for (G - MP), assume
that Γ `I α and ∆, β `I γ for finite Γ, ∆. Then by (MP) applied to the first assumption, we obtain
Γ, α → β `I β, and from this and the second assumption it follows that Γ, ∆, α → β `I γ.
P ROPOSITION 2.13. The calculus G enjoys cut-elimination.
P ROOF. This can be shown by the standard technique (induction on the rank of the cut, and then on the
complexity of the cut formula), as in the implicative fragment of intuitionistic logic. Despite the presence
of the unusual initial sequents ( G - I ), everything works and the conclusion is reached without difficulty.
The details are left to the reader.
P ROPOSITION 2.14. If Γ γ is derivable in G and Γ 6= ∅, then all the formulas that occur in a cut-free proof
of Γ γ are subformulas of some formula in Γ.
P ROOF. By induction on the length of a cut-free proof of Γ γ. Since (CUT) is not applied, there are
only three possibilities:
7
1. The proof just consists of an instance of (AXIOM). Then this must be γ γ and Γ = {γ} , so that the
property is satisfied.
2. The proof just consists of an instance of the initial sequent (G - R). Then Γ = ∅ and the property does
not apply.
3. The proof ends with an application of (G - MP). Then Γ = Γ0 ∪ ∆ ∪ {α → β} and there are cut-free
proofs of Γ0 α and of ∆, β γ. If Γ0 = ∅, then note that the only cut-free proof of ∅ α is the
sequent itself, and the only formula occurring in it is a subformula of α → β ∈ Γ. Otherwise, the
inductive hypothesis applies and therefore all formulas in these proofs are subformulas of some
formula in Γ0 or in ∆ or of β, but these in turn will be subformulas of α → β; in all cases they will be
subformulas of some formula in Γ, so the desired conclusion is reached.
The restriction that Γ be non-empty in the statement is necessary for it to make sense. Moreover,
clearly ∅ γ is derivable if and only if γ = α → α, and its only cut-free derivation consists of a single
application of ( G - I ).
In some sense, this calculus is a “minimal” one for the sequents corresponding to the consequence
relation of I , as it only proves those having a real proof with logical rules and where all assumptions
are really used, and not others that are the product of applying weakening, which is implicit in the
definition of `I (as in all consequence relations); for instance by Proposition 2.14 this calculus will not
prove many instances of Γ γ when γ is a theorem and Γ is non-empty (for instance those where γ is
not a subformula of any formula in Γ).
3
Sets of congruence formulas with parameters
M ORE NOTATIONS AND TERMINOLOGY. In abstract algebraic logic, some points (such as the definition
used here of a protoalgebraic logic in terms of a set of formulas ∆( x, y) ) require us to focus on two
particular, arbitrary but fixed, distinct variables x and y. In this context, variables other than these
two that occur in other formulas are called parameters. In particular, one considers arbitrary formulas
and sets of formulas where such parameters can appear, which are indicated by the notations ϕ( x, y, ~z )
and ∆( x, y, ~z ) . Obviously, only a finite number of parameters can occur in a particular formula, but
all possible values might effectively occur in an infinite set of formulas; thus, in these notations the
sequence ~z of parameters may be infinite, sometimes even consisting of all the variables except x and y.
This happens for instance when all possible formulas are taken into consideration. In those situations,
notations such as ϕ( x, y, ~z ) or ψ( x, ~z ) do not refer to any particular sequence ~z of parameters but merely
indicate the (possible) presence, apart from x and y, of other variables; thereby ensuring that the notation
is adequate to allow some replacement to be performed in the formulas.
In many applications these parameters are replaced by arbitrary formulas, which in this context
will be represented by a sequence ~δ of the required length. In particular, given any set ∆( x, y, ~z ) one
often considers the set of formulas that results from performing all possible replacements; this set will be
represented in the following way
∆hα, βi :=
∆(α, β, ~δ ) : ~δ ∈ Fm
[
(2)
for any two formulas α and β.
A set of congruence formulas with parameters for a logic L is a set ∆( x, y, ~z ) of formulas containing
the variables x, y and possibly parameters ~z such that for all theories Γ of L and all α, β ∈ Fm,
α ≡ β (Ω Γ ) if and only if ∆hα, βi ⊆ Γ,
where Ω Γ is the Leibniz congruence of theory Γ; this is the largest congruence of the formula algebra11
that is compatible with Γ, that is, that does not identify formulas in Γ with formulas outside it. Such
11 Is
T
∼
is easy to prove that Ω L = {Ω Γ : Γ a theory of L} .
8
a set defines the Leibniz congruence Ω A F of a filter F of the logic12 over an arbitrary algebra A in a
similar way, but this will not be used here.
The fundamental set ΣL of a logic L is the set of all formulas ϕ( x, y, ~z ) such that `L ϕ( x, x, ~z ) .
In the case of I , the following lemma follows from Proposition 2.1:
L EMMA 3.1. ϕ( x, y, ~z ) ∈ ΣI if and only if ϕ( x, y, ~z ) = ϕ1 ( x, y, ~z ) → ϕ2 ( x, y, ~z ) for some ϕ1 , ϕ2 such that
ϕ1 ( x, x, ~z ) = ϕ2 ( x, x, ~z ) .
One of the fundamental results of the theory of protoalgebraic logics is that a logic is protoalgebraic
if and only if it has some set of congruence formulas with parameters; indeed, the fundamental set ΣL ,
which is a theory, is always such a set of congruence formulas with parameters for a protoalgebraic
logic [7, Theorem 1.2.7]. It is interesting that one can always take a certain subset, which in particular
cases can be much simpler:
P ROPOSITION 3.2. Let L be an arbitrary protoalgebraic logic with a set ∆( x, y) that satisfies the properties (R ∆ )
and (MP∆ ). Then each of the following sets of formulas
[
∆ 1 ( x, y, ~z ) :=
∆ ψ( x, ~z ) , ψ(y, ~z ) ∪ ∆ ψ(y, ~z ) , ψ( x, ~z ) : ψ( x, ~z ) ∈ Fm
[
∆ 2 ( x, y, ~z ) :=
∆ ψ( x, ~z ) , ψ(y, ~z ) : ψ( x, ~z ) ∈ Fm .
is a set of congruence formulas with parameters for L .
Observe that the set ∆ 2 is similar to ∆ 1 but “without symmetrizing”, and that ∆ ⊆ ∆ 2 ⊆ ∆ 1 ⊆ ΣL .
P ROOF. (∆ 1 ) We have to show that for any theory Γ and any two formulas α, β, α ≡ β (Ω Γ ) if and
only if ∆ 1 hα, βi ⊆ Γ :
( ⇒ ) Because ∆ 1 ⊆ ΣI and this is a set of congruence formulas with parameters.
( ⇐ ) If ∆ 1 hα, βi ⊆ Γ then, by (MP∆ ) and substitution-invariance, for every ψ( x, ~z ) we have that
ψ(α, ~z ) ∈ Γ if and only if ψ( β, ~z ) ∈ Γ. But this is Czelakowski’s characterization [7, Corollary 0.5.4] of
Ω Γ. So α ≡ β (Ω Γ ) .
(∆ 2 ) Clearly, any set of formulas that is L -interderivable with a set of congruence formulas for L is
itself one as well. Therefore, the property will follow from the fact that ∆ 2 h x, yi a`L ∆ 1 h x, yi , which
is slightly tricky, but not difficult to prove: ∆ 1 h x, yi `L ∆ 2 h x, yi because ∆ 2 ⊆ ∆ 1 . The formulas in
∆ 1 h x, yi which do not belong to ∆ 2 h x, yi have the form λ ψ(y, ~δ ) , ψ( x, ~δ ) for some λ ∈ ∆, some
ψ( x, ~z ) and some ~δ. Fix one of these, take a new variable w and define γ( x, ~z, w) := λ ψ( x, ~z ) , ψ(w, ~z )
and ~δ0 := h~δ, x i , considering the sequence h~z, wi as the parameters. Then ∆ γ( x, ~δ0 ) , γ(y, ~δ0 ) ⊆
∆ 2 h x, yi . But γ( x, ~δ0 ) = γ( x, ~δ, x ) = λ ψ( x, ~δ ) , ψ( x, ~δ ) , which is a theorem by (R ∆ ), therefore by
(MP∆ ) ∆ 2 h x, yi `L γ(y, ~δ0 ) = λ ψ(y, ~δ ) , ψ( x, ~δ ) . This shows that ∆ 2 h x, yi `L ∆ 1 h x, yi and hence that
∆ 2 h x, yi a`L ∆ 1 h x, yi .
The part of this result that concerns ∆ 2 is stated as Theorem 13.5 of [3]; however, the proof of that
theorem is incomplete and follows quite a different route, so it seemed sensible to prove it directly here.
Applying Proposition 3.2 to our particular case we obtain:
C OROLLARY 3.3. Each of the following sets of formulas
∆ 1 ( x, y, ~z ) := ψ( x, ~z ) → ψ(y, ~z ) , ψ(y, ~z ) → ψ( x, ~z ) : ψ( x, ~z ) ∈ Fm
∆ 2 ( x, y, ~z ) := ψ( x, ~z ) → ψ(y, ~z ) : ψ( x, ~z ) ∈ Fm
is a set of congruence formulas with parameters for I .
12 If A is an algebra and F ⊆ A , F is a filter of a logic L when Γ ` ϕ implies that h ϕ ∈ F for all h ∈ Hom(F m, A) such
L
that h Γ ⊆ F. Then it is said that the matrix hA, F i is a model of L . The Leibniz congruence of the matrix, Ω A F, is the largest
congruence of A compatible with F, that is, that does not identify points in F with points outside it.
9
For the desired applications either set is convenient. This section closes with two such miscellaneous
applications.
P ROPOSITION 3.4. If Γ is a set of formulas where not all variables occur, then Ω CI Γ is the identity relation;
i.e., the logical matrix hF m, CI Γ i is a reduced model of I .
P ROOF. By Corollary 3.3, if α ≡ β (Ω CI Γ ) then for all ψ( x, ~z ) , ψ(α, ~z ) → ψ( β, ~z ) ∈ CI Γ. However, if
α 6= β, these formulas are not theorems, so by Corollary 2.9.5 their variables should occur in Γ. But since
ψ( x, ~z ) ranges over all formulas, with all the variables, this would imply that all variables occur in Γ,
thus contradicting the assumption.
In particular, the Leibniz congruence of all finitely generated theories is the identity relation. Similar
arguments can be used to show that the Leibniz congruence of other theories is also the identity relation;
for instance, using Corollary 2.9.1, one can show that if Γ is the set of all the variables, then Ω CI Γ is the
identity relation as well.
Consider the following property, where σ is a substitution and Γ ⊆ Fm:
If α ≡ β (Ω CL Γ ) then σα ≡ σ β (Ω CL σ Γ )
∀ α, β ∈ Fm.
(3)
This property is considered in [7, § 1.2], where it is proved that protoalgebraic logics satisfy it for
surjective substitutions and arbitrary Γ (Corollary 1.2.11), and also for arbitrary substitutions and finite Γ
(Corollary 1.2.12). Moreover, it is easy to prove that equivalential logics13 and Fregean logics14 satisfy (3)
for all substitutions and all Γ. In connection with a discussion on the Leibniz interpolation property for
equivalential logics [6], F ÉLIX B OU asked whether actually protoalgebraic logics satisfy (3) without any
restrictions (because no counterexamples were given in [7]). The answer is negative, and I is such a
counterexample:
P ROPOSITION 3.5. The logic I does not satisfy the property (3) for all substitutions and all Γ.
P ROOF. Take Γ := ∆ 2 h x, yi . Then obviously x ≡ y (Ω CI Γ ) . Now take any substitution σ leaving
x, y fixed and such that its range does not contain some variable; for instance, σ x := x , σy := y
and σz := u for all variables z 6= x, y, where u is any fixed variable. Then σ Γ does not have all the
variables, and this fact, by Proposition 3.4, implies that Ω CL σ Γ is the identity relation, contradicting
that x ≡ y (Ω CI σ Γ ) .
4
Some metalogical results
In this section I look at what are commonly called “metalogical” properties of the logic I ; most of these
concern the behaviour of the logical connectives, mainly of conjunction, disjunction, and implication, all
in an abstract, generalized sense; in the case of implication this means considering all variants of the
Deduction-Detachment Theorem (DDT) that have appeared up to now in the literature.
P ROPOSITION 4.1. The logic I is not conjunctive.
P ROOF. A logic L is said to be conjunctive when there is a (binary) formula ϕ( x, y) such that { x, y} a`L
ϕ( x, y) . Such a formula, commonly denoted by x ∧ y, is called a conjunction for L . In the case of I ,
if { x, y} a`I ϕ( x, y) , then Corollary 2.9.1 implies that ϕ( x, y) must be either x, or y, neither of which
is possible because x 6`I y and y 6`I x, or a theorem, but then the two variables would be theorems
themselves, which is not the case.
P ROPOSITION 4.2. The logic I does not have a parameterized disjunction, not even in the weak sense.
13 See
14 See
[7, Chapter 3].
[7, Chapter 6], [8, Section 3.4] and [10, Definition 3.12].
10
P ROOF. A set ∇( x, y, ~z ) ⊆ Fm is a parameterized disjunction for a logic L when for all Γ ∪ {α, β} ⊆
Fm , CL Γ ∪ ∇hα, βi = CL ( Γ, α) ∩ CL ( Γ, β) , where ∇hα, βi is defined as in (2) of page 8. The same
concept in the weak sense means requiring the property to hold just for Γ = ∅.
Now assume such a set in the weak sense exists for I . In particular, for distinct variables x and y it
should satisfy CI ∇h x, yi = CI { x } ∩ CI {y} = CI ∅ by Corollary 2.9.2. Since ∇( x, y, ~z ) ⊆ ∇h x, yi , all
formulas in ∇( x, y, ~z ) should be theorems, and hence by structurality all formulas in ∇hα, βi as well,
for all α, β. Then for any formula α, we would have CI {α} = CI {α} ∩ CI {α} = CI ∇hα, αi = CI ∅;
that is, every formula would be a theorem, which is not the case. Thus, a parameterized disjunction for
I does not exist, not even in the weak sense.
It is interesting to note here that, in contrast, in [9] it is shown that I possesses a non-term-definable
weak disjunction, that is, a well-defined (but not term-defined) binary function f on formulas such that
CI { ϕ} ∩ CI {ψ} = CI f( ϕ, ψ) for all ϕ, ψ ∈ Fm. Finding this function involves determining the
atomic structure of the lattice of theories of I , which is the object of [9].
C OROLLARY 4.3. The logic I is not filter-distributive, does not satisfy any Contextual Deduction-Detachment
Theorem (CDDT), and does not satisfy any Deduction-Detachment Theorem (DDT).
P ROOF. For finitary protoalgebraic logics, being filter-distributive is equivalent to having a parameterized disjunction [7, Theorem 2.5.17]; by [17, Theorem 6.8], having a CDDT implies being filter-distributive.
Finally, the DDT is a stronger form of the CDDT.
The CDDT and the DDT are the strongest forms of the Deduction Theorem that have been studied so
far in the literature, although, much weaker versions also have been. This research has had an important
impact on the general theory of abstract algebraic logic. Another of the fundamental results of the theory
of protoalgebraic logics is that a logic L is protoalgebraic if and only if it satisfies the Parameterized
Local Deduction-Detachment Theorem (PLDDT): there is a family Φ( x, y, ~z ) of sets of formulas in two
variables and possibly parameters such that for all Γ ∪ {α, β} ⊆ Fm,
Γ, α `L β if and only if there is Σ ∈ Φ and there are ~δ ∈ Fm
such that Γ `L Σ(α, β, ~δ ).
(PLDDT)
The stronger, non-local version of this property, introduced in [7], has been much less studied: a logic
L satisfies the Parameterized Deduction-Detachment Theorem (PDDT) with respect to a set Σ( x, y, ~z )
when for any Γ ∪ {α, β} ⊆ Fm,
Γ, α `L β if and only if there are ~δ ∈ Fm such that Γ `L Σ(α, β, ~δ) .
(PDDT)
However, up to now no explicit example of a protoalgebraic logic that does not satisfy the PDDT has
been presented. I is such an example:
P ROPOSITION 4.4. The logic I does not satisfy any Parameterized Deduction-Detachment Theorem (PDDT).
P ROOF. I will use the bridge theorem of [7, Theorem 2.4.1], which states that a protoalgebraic logic has
PDDT if and only if its class of matrix models has “factor-determined finitely generated filters on direct
products”, i.e., any finitely generated filter of the logic on a product of two models is the product of two
finitely generated filters on the factors. I will show that I does not have PDDT by exhibiting a principal
filter on the product of two models that is not the product of any pair of filters. I should remark that the
filters involved in this property are filters “on the model”, that is, filters of the logic containing the base
filter of the model.
Denote by T (a capital τ) the set of all theorems of I ; then hF m, T i is a matrix model of I . Consider the
product hF m × F m , T × T i = hF m, T i × hF m, T i , and the principal filter F on this product generated
by the element h x, x i for a variable x. It is easy to see that actually F = T × T ∪ {h x, x i} , because this
last set contains the base filter T × T (hence a fortiori it contains all images of the axioms), contains the
desired point h x, x i , and is closed under (MP). But this set cannot be equal to the product of any two
11
filters on hF m, T i , i.e., to the product of two theories: if the product of two theories contains h x, x i then
each of these theories has to contain x; but since they will contain also all theorems, their product has to
contain all pairs h ϕ, x i and h x, ϕi for all theorems ϕ, and none of these pairs belongs to F.
C OROLLARY 4.5. The property PLDDT does not imply the property PDDT.
Many protoalgebraic logics satisfy another strengthened version of the PLDDT, specifically that in
which parameters are not present but locality remains. However, this is not the case for I :
P ROPOSITION 4.6. The logic I does not satisfy any Local Deduction-Detachment Theorem (LDDT).
P ROOF. I satisfies the Local Deduction-Detachment Theorem (LDDT) when there is an LDD family for
it; that is, a family Φ( x, y) of sets of formulas in at most two variables such that for all Γ ∪ {α, β} ⊆ Fm,
Γ, α `I β if and only if there is some Σ ∈ Φ such that Γ `I Σ(α, β).
(LDDT)
Assuming such a family exists, we apply (LDDT) to two known facts:
(a) Since x 0I y, for every Σ ∈ Φ there is some γ ∈ Σ such that 0I γ( x, y) .
(b) Since y → x , y `I x, there is some Σ0 ∈ Φ such that y `I Σ0 (y → x, x ) .
Applying (a) to the set found in (b) shows that there is some γ0 ∈ Σ0 such that
0I γ0 ( x, y) and y `I γ0 (y → x, x ) .
(c)
I will show that such a γ0 ( x, y) cannot exist. As a consequence of Proposition 2.5, and since y is the only
final subformula of itself, we see that y `I ϕ if and only if ϕ = y or ϕ is a theorem. This implies that γ0
cannot be a variable, because if it was, then γ0 (y → x, x ) would be either y → x or x, and the second
property in (c) would fail.
Thus, γ0 = α → β, and to satisfy (c) α 6= β and y `I α(y → x, x ) → β(y → x, x ) . As we have seen before,
this last fact can only be the case if the formula is a theorem, that is, α(y → x, x ) = β(y → x, x ) . We are
going to see that this is contradictory. More precisely, I show by induction on the complexity of α( x, y)
that for all β( x, y) , if α(y → x, x ) = β(y → x, x ) then α = β:
(1) If α( x, y) = x, then α(y → x, x ) = y → x = β(y → x, x ) . The only possibility for this is β( x, y) = x,
that is, α = β; for otherwise either β( x, y) = y which implies β(y → x, x ) = x 6= y → x against the
assumption, or β( x, y) = β 1 → β 2 which implies β 1 (y → x, x ) = y, which is impossible.
(2) If α( x, y) = y, then α(y → x, x ) = x = β(y → x, x ) and the only way to obtain this is with β( x, y) = y,
that is, with α = β.
(3) If α = α1 → α2 and both αi satisfy the property, then the assumption α(y → x, x ) = β(y → x, x )
implies that β(y → x, x ) is not a variable. Then clearly β 6= y, but also β 6= x because in this
case β(y → x, x ) = y → x = α1 (y → x, x ) → α2 (y → x, x ) which implies α1 (y → x, x ) = y which
is impossible. Thus, β = β 1 → β 2 , and so αi (y → x, x ) = β i (y → x, x ) for i = 1, 2. The induction
hypothesis implies that αi = β i for i = 1, 2 and then α = β as was to be proved.
This completes the proof that the formula γ0 satisfying (c) cannot exist. Therefore, an LDD family Φ for
I cannot exist: The logic I cannot satisfy any kind of LDDT.
Thus I does not satisfy any of the standard forms of the Deduction Theorem for any set of implication
formulas. It does satisfy (MP) for → , which is half of any Deduction Theorem, but does not satisfy even
the weakest of such theorems, called the “Weak Deduction Theorem” in [21]. Here I am going to express
it as a Gentzen-style rule; when doing so, one says that a logic L satisfies a Gentzen-style rule when the
metalogical implication obtained by replacing the symbol in the sequents appearing in any instance of
the rule by the symbol `L is true.
P ROPOSITION 4.7. I does not satisfy the Weak Deduction Theorem for → , which can be formalized as the
Gentzen-style rule
αβ
.
(WDT)
α→β
12
P ROOF. Clearly ( x → x ) → y `I y, but 0I ( x → x ) → y → y by Proposition 2.1.
One can also try to see whether the same rule is true for another binary formula δ( x, y) :
αβ
.
δ(α, β)
(WDT-δ)
A trivial solution to this is to take any theorem as δ; and it turns out this is the only possible solution:
P ROPOSITION 4.8. The logic I satisfies the Weak Deduction Theorem ( WDT-δ ) for a binary formula δ( x, y) if
and only if δ( x, y) is a theorem.
P ROOF. Clearly I satisfies the ( WDT-δ ) for any theorem δ. Now assume I satisfies it for some binary
formula δ( x, y) . Obviously δ cannot be a variable, for otherwise `I α for all α (because α `I α always
holds). Thus, δ = δ1 → δ2 , and we have to show that δ1 = δ2 . Now take two different variables x, y; since
x `I y → y, we should have that `I δ1 ( x, y → y) → δ2 ( x, y → y) , that is, that δ1 ( x, y → y) = δ2 ( x, y → y) .
Now reasoning by induction on the complexity of δ1 ( x, y) we can see that, for any two formulas δ1 ( x, y)
and δ2 ( x, y) , the assumption that δ1 ( x, y → y) = δ2 ( x, y → y) implies that δ1 = δ2 .
(1) If δ1 = x, then x = δ2 ( x, y → y) which can only be the case when δ2 ( x, y) = x and hence δ1 = δ2 .
(2) If δ1 = y, then y → y = δ2 ( x, y → y) which can only be the case when δ2 ( x, y) = y and hence δ1 = δ2
as well.
(3) If δ1 = α → β then α( x, y → y) → β( x, y → y) = δ2 ( x, y → y) and we now consider the possibilities
for the structure of δ2 . Obviously, δ2 6= x, and if δ2 = y then α( x, y → y) → β( x, y → y) = y → y,
which implies that α( x, y → y) = β( x, y → y) = y, which is clearly impossible. Thus forcefully
δ2 = α0 → β0 so that α( x, y → y) → β( x, y → y) = α0 ( x, y → y) → β0 ( x, y → y) . But this implies that
α( x, y → y) = α0 ( x, y → y) and β( x, y → y) = β0 ( x, y → y) , and by the induction hypothesis this
implies that α = α0 and β = β0 ; that is, δ1 = δ2 .
This result also implies that I does not satisfy the “Graded Deduction-Detachment Theorem” of [11,
13], called the “General Deduction Theorem” in [20].
P ROPOSITION 4.9. The logic I satisfies neither the Inconsistency Lemma, nor the Classical Inconsistency
Lemma, nor the Properties of Intuitionistic Reductio ad Absurdum or of plain Reductio ad Absurdum.
P ROOF. According to [18], a logic L satisfies the Inconsistency Lemma if there is a denumerable sequence
hΨn : n ∈ ω i of finite sets of formulas in respectively n variables, i.e., Ψn ( x1 , . . . , xn ) , such that for all
Γ ∪ {α1 , . . . , αn } ⊆ Fm,
Γ ∪ {α1 , . . . , αn } is inconsistent if and only if Γ `L Ψn (α1 , . . . , αn ).
Note that if such a sequence exists, then for each n the set Ψn ( x1 , . . . , xn ) ∪ { x1 , . . . , xn } must be inconsistent. But this is a finite set, and by Corollary 2.9.4 we know that in the case of I no finite set is
inconsistent. Therefore, I cannot satisfy any Inconsistency Lemma. The Classical Inconsistency Lemma
includes the foregoing, so I cannot satisfy it either.
The properties called in [10] Property of Intuitionistic Reductio ad Absurdum (PIRA) and Property of Reductio
ad Absurdum (PRA) are the particular cases of the Inconsistency Lemma and the Classical Inconsistency
Lemma considered above, limited to the case n = 1 and for a single formula which reflects the properties
of intuitionistic (respectively, classical) negation ¬ x. The same argument shows that I cannot satisfy
them.
The abstract property of Introduction of a Modality (IM) is studied in [10]. A logic L satisfies IM
with respect to a unary formula δ( x ) when for any formulas ϕ1 , . . . , ϕn , ψ, if ϕ1 , . . . , ϕn `L ψ, then
δ( ϕ1 ) , . . . , δ( ϕn ) `L δ(ψ) . That is, when it satisfies the set of Gentzen-style rules:
ϕ1 , . . . , ϕ n ψ
δ ( ϕ1 ) , . . . , δ ( ϕ n ) δ ( ψ )
13
for all n ∈ ω; for n = 0 the antecedents of the two sequents are empty. In almost all modal logics these
are among the most basic Gentzen-style rules satisfied by unary formulas involving modalities, such as
x and 3x. It turns out that I may satisfy this property only when it is trivial:
P ROPOSITION 4.10. The logic I satisfies the property IM for a unary formula δ( x ) if and only if either δ( x ) = x
or δ( x ) is a theorem.
P ROOF. Clearly I satisfies IM for said formulas. Now assume I satisfies IM for a unary formula
δ( x ) 6= x, so that δ( x ) = δ1 ( x ) → δ2 ( x ) . Since `I x → x, we have that `I δ( x → x ) , therefore
δ1 ( x → x ) = δ2 ( x → x ) . An easy inductive argument on the length of δ1 ( x ) quickly shows that δ1 ( x ) =
δ2 ( x ) , which means that δ( x ) is a theorem. This completes the proof of the statement.
Now consider several interpolation properties. Some of them are true for I in a rather trivial way, as
the interpolant can be chosen to be one of the two formulas to be interpolated; indeed, some interpolation
properties hold in the so-called uniform version:
1. The implicative form: If `I α → β then α = β so α itself can be taken as the interpolant, and therefore
this is uniform interpolation.
2. Leibniz Interpolation Property [6]: if α ≡ β (Ω CI Γ ) , then, by Corollary 3.3, for every ψ( x, ~z ) and every
~δ, ψ(α, ~δ ) → ψ( β, ~δ ) ∈ CI Γ. If we exclude the trivial case α = β, those formulas are not theorems,
therefore by Corollary 2.9.5 their variables must be in some formula in Γ, and as a consequence Γ
will contain all variables. In this case, var( Γ, α) ∩ var( Γ, β) is the set of all variables, hence either α or
β can be taken as interpolant. If one takes α as the interpolant, then one gets uniform interpolation as
well.
As for Craig’s or the deductive form, it is also satisfied, in a (less) trivial way:
P ROPOSITION 4.11. The logic I satisfies the “deductive” interpolation property: if Γ `I β and varβ ∩ varΓ 6=
∅, then there is some α with varα ⊆ varβ ∩ varΓ such that Γ `I α and α `I β.
P ROOF. If β is not a theorem, then by Corollary 2.9.5 varβ ⊆ varΓ, so β itself can be taken as the
interpolant. If β is a theorem, then it suffices to take any theorem in the variables of varβ ∩ varΓ as the
interpolant.
Finally, I close this section with a property whose formulation does not involve explicitly the grammatical structure of formulas.
P ROPOSITION 4.12. The logic I is not “logically compact” in the sense that every inconsistent set of formulas
has a finite inconsistent subset.
P ROOF. This is a consequence of Corollary 2.9.4, which shows that no finite set is inconsistent; clearly
there are infinite inconsistent sets, for instance the set of all formulas or the set (α → α) → β : α, β ∈
Fm .
5
Classification and semantic issues
In this section I look into the classification of I in the hierarchies of abstract algebraic logic. In general,
locating a logic in these hierarchies gives information about the kind of relation between the logic and
its algebraic models. However, in this case this gives very little information. To begin with, I is totally
outside the Frege hierarchy, as it does not even belong to its lowest level:
P ROPOSITION 5.1. The logic I is not selfextensional.
P ROOF. By definition, a selfextensional logic [21] is one whose interderivability relation is a congruence.
This is not the case here: trivially, x → x a`I y → y, but by Proposition 2.6 ( x → x ) → z a6`I (y → y) →
z.
14
Although it does not belong to either hierarchy, the class of logics having an algebraic semantics has
its importance in abstract algebraic logic, as it represents a very natural form of relation between a logic
and a class of algebras. A logic has an algebraic semantics, in the technical sense of the term introduced
in [2], when it is complete with respect to a class of matrices where the filter is the set of points satisfying
a fixed set of equations in one variable; these equations are called “defining equations”. However:
P ROPOSITION 5.2. The logic I does not have an algebraic semantics.
P ROOF. This was proved in [4, Theorem 2.19]15, but here is a more direct argument. If I had an algebraic
semantics, then by Theorem 2.16 and Proposition 2.17 of [4], for each defining equation δ ≈ ε one should
have δ ≡ ε (Ω CI { x }) . But by Proposition 3.4, Ω CI { x } is the identity relation, therefore δ = ε. Thus all
defining equations have the form δ ≈ δ. Since any algebra satisfies the equation δ( ϕ) ≈ δ( ϕ) , it would
follow that `I ϕ for all ϕ; that is, I would be the inconsistent logic, which is not the case.
Concerning the Leibniz hierarchy, of course I is protoalgebraic by its very definition, but we will see
that it does not belong to any other class in the Leibniz hierarchy. Since these classes are defined by properties
of the Leibniz operator (the map F 7→ Ω A F defined on all filters F of the logic, for any algebra A), direct
proofs using these properties are preferable.
C OROLLARY 5.3. The logic I is not truth-equational. As a consequence, it does not belong to any class in the
Leibniz hierarchy with “algebraizable” in its name: it is not weakly algebraizable, not algebraizable, not finitely
algebraizable, etc. And it is not implicative.
P ROOF. The first statement is a corollary to Proposition 5.2, because all truth-equational logics have an
algebraic semantics. But it is also a corollary to Proposition 3.4, which shows that the Leibniz operator
is not injective on the theories of I . However, by Theorem 28 of [16], the Leibniz operator of a truthequational logic is always injective on the filters of the logic over any algebra, and in particular on the
theories (which are the filters over the formula algebra). Therefore I is not truth-equational. It can also
be proved semantically, see Example 5.5 below. The rest follows because all the other classes mentioned
are contained in that of truth-equational logics.
C OROLLARY 5.4. The logic I is not equivalential.
P ROOF. This can be seen as a corollary to Proposition 3.5, because all equivalential logics satisfy property (3) for all substitutions, but can also be proved semantically, as shown in the following example. E XAMPLE 5.5. Let A = h A, →i be the algebra on the set A = {0, a, b, 1} with the binary operation
given by the following table:
→ 0 a b 1
0 1 1 1 1
a 0 1 b 1
b a a 1 1
1 0 a b 1
It is straightforward to check that the sets {1} and {1, b} are I -filters, and that Ω A {1} = Ω A {1, b} = Id.
This shows that the Leibniz operator is not injective on I -filters, therefore I is not truth-equational.
Moreover, if one considers the subalgebra B of A with universe {1, b} , then {1, b} is obviously an
I -filter
on B, butthis time Ω B {1, b} is the total relation, hence it is not the identity
relation. Thus,
the
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matrix A, {1, b} is a reduced model of I which has a non-reduced submatrix B , {1, b} . But by
Theorem 3.2.1 of [7], the class of reduced models of any equivalential logic is closed under the formation
of submatrices. Therefore the logic I cannot be equivalential.
15 The
logic appears in [4] only as an example of a logic without an algebraic semantics, and is not named in any particular way.
matrix hA, F i is reduced when Ω A F is the identity relation. For a logic L , Alg∗L denotes the class of algebra reducts of all
reduced models of L .
16 A
15
Observe however that B ∈ Alg∗I , for Ω B {1} is the identity relation.
It makes sense to compare I to other weak logics of implication existing in the literature, especially to
the few already known to be neither equivalential nor weakly algebraizable. The fact that the theorems
of I are characterized in such a neat way in Proposition 2.1 makes it easy to see that it is strictly weaker
than many. The following are some examples.
• The weakest implicative logic I L in the language h→i . The well-known class of implicative logics is
introduced in [19]; the weakest one is not explicitly defined in the literature, but it exists because the
intersection of any family of logics over the same language is a logic, and being implicative is obviously
preserved under intersections. From [19] we know that by definition all implicative logics satisfy
properties (I) and (MP), therefore I is weaker than I L ; and it is strictly so because by definition all
implicative logics satisfy y `I L x → y, while y 0I x → y by Proposition 2.5. There is also a difference
in the theorems of the two logics: x → (y → y) is a theorem of I L (replace y by y → y in the previous
rule) but not of I .
• The logic G 0 , the logic in the language h→i defined by the derivable sequents of the Gentzen system
with the structural rules and the rules ( WDT ) of Proposition 4.7 and ( G - MP ) of Definition 2.10. This
logic satisfies (I) and (MP), so I is weaker than it, but they are different because of Proposition 4.7.
Again, there are differences in the theorems: the ( WDT ) implies x → (y → y) is a theorem of G 0 , but
we know it is not a theorem of I .
• The weak logics of implication G 1 and H1 studied in [5]. They are related to other weaker forms of
the Deduction Theorem, and algebraically to the quasi-variety of quasi-Hilbert algebras. Since they
are slightly stronger than G 0 , I is also strictly weaker than them. Note that in [5, Theorem 3.17] it is
proved that G 1 is neither equivalential nor weakly algebraizable, so G 0 is also neither.
• All the logics from the family BCI and related logics such as BCK , and most of their weakened
versions considered in the literature, are also stronger than I as far as they are axiomatically defined
with a set of axioms containing (I) and (MP). For instance, axiom B is ( x → y) → (z → x ) → (z → y) ,
which is not a theorem of I .
• A family of logics that are protoalgebraic but neither equivalential nor weakly algebraizable is that of
D A C OSTA’s paraconsistent systems Cn , of which the weakest is Cω and the strongest is C1 . The latter
was shown not to be algebraizable in [14]; that paper actually shows that it is not weakly algebraizable,
and the same counterexample can be used to show that it is not equivalential either. Since being
equivalential and being weakly algebraizable are properties that are transmitted from a weaker logic
to stronger ones, it follows that neither Cn nor Cω can be either. All these logics are expansions of the
positive fragment of intuitionistic logic, so their implicative fragments are all stronger than I L , and
hence strictly stronger than I .
Finally, just a few facts about the semantical aspect of I .
P ROPOSITION 5.6. The intrinsic variety of I , in symbols VI , is the class of all algebras of the similarity type.
P ROOF. By definition the intrinsic variety VL of a logic L is the variety generated by its Lindenbaum∼
∼
Tarski algebra F m/ Ω L . In the case of I , by Proposition 2.7 Ω I is the identity relation, therefore the
variety is generated by a formula algebra, hence it is the class of all algebras of the type.
Since I is protoalgebraic, its algebraic counterpart is defined by the general theory to be the class
Alg∗I of all the algebra reducts of its reduced models (see footnote 16). Research into this class has
borne little fruit so far, and Proposition 5.6 does not help much either (actually, Alg∗L also generates VL ,
so that in general Alg∗L ⊆ VL ). Some things we know of Alg∗I are:
• By Proposition 3.4, it contains all formula algebras of the similarity type (i.e., of all cardinalities).
Hence, it cannot be characterized in a way that involves equations, and the variety it generates is the
variety of all algebras of the type.
16
• In particular, it contains algebras where x → x ≈ y → y does not hold; which implies that the smallest
I -filter on them is not a one-element set (the formula algebras are instances of this property).
• It strictly contains all implicative algebras [19] and all quasi-Hilbert algebras [5] (strictly because it
contains the formula algebras).
• It is not the class of all algebras of the type. In particular it does not even contain all algebras A having
an element 1 such that the matrix A, {1} is a model of I , i.e., such that a → a = 1 and that 1 → a = 1
implies a = 1; which means that there are models of this form which are not reduced. Example 5.7
below is one of these.
E XAMPLE 5.7. Let A = h A, →i be the algebra on the set A = { a, b, 1} with the binary operation given
by the following table:
→ a b 1
a 1 1 b
b 1 1 b
1 a a 1
The set {1} is an I -filter, indeed the smallest one, but Ω A {1} is the congruence that identifies a and
b. Since the logic is protoalgebraic, the Leibniz operator must be monotonic on the filters of the logic,
which implies there is no filter of I whose Leibniz congruence is the identity relation; anyway, neither
{1, a} nor {1, b} are I -filters. Therefore A ∈
/ Alg∗I .
Thus, the algebraic counterpart of I must be a strange class; actually, since the logic is protoalgebraic
but not weakly algebraizable, it is not enough to consider algebras as its models, one has to consider
matrices. It is clear that a lot remains to be done in this direction.
Acknowledgements. I thank F ÉLIX B OU for asking questions that stimulated research (though I could
not answer all of them), A LEXEI M URAVITSKY and C LINT VAN A LTEN for several suggestions, and the
anonymous referee for many wise observations and improvements. While working on this paper I was
partially funded by the research project MTM2011-25747 from the government of Spain, which includes
FEDER funds from the European Union, and the research grant 2009SGR-1433 from the government of
Catalonia.
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Departament de Probabilitat, Lògica i Estadística
Universitat de Barcelona (UB)
Gran Via de les Corts Catalanes 585, E-08007 Barcelona, Spain
[email protected]
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