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A SEQUENT CALCULUS FOR A NEGATIVE FREE LOGIC
Abstract
This article presents a sequent calculus for a negative free logic with identity, called N. The main
theorem (in part 1) is the admissibility of the Cut-rule. The second part of this essay is devoted to
proofs of soundness, compactness and completeness of N relative to a standard semantics for
negative free logic.
Keywords: Free Logic, Cut-elimination, Compactness, Completeness, Existence
1 INTRODUCTION
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K. Lambert introduced the term ‘free logic’ in the late 1960s. A standard
definition of free logic is the following: A logic S is a free logic iff (1) S is free of
existential presuppositions with respect to the singular terms of S, (2) S is free of
existential presuppositions with respect to the general terms of S and finally (3) the
quantifiers of S have existential import. There are three “families” of free logics:
positive, negative and neutral. For our purposes it suffices to define a negative free
logic: A negative free logic is a free logic, where each simple statement containing
at least one empty singular term is false. E.g. ‘Vulcan is (identical with) Vulcan’ is
false; and so is ‘Vulcan rotates’ (Cf. Lambert (1997, 81ff)). It should be mentioned
that there is no particular formal system called “the” negative free logic, but there
is a whole family of such systems. Since the sentence ‘Vulcan is (identical with)
Vulcan’ is false in negative free logics, a general feature of these is that identity
statements can only be true just in case they contain no non-denoting singular term
(cf. Hintikka (1964), Schock (1969)). So, negative free logics have a non-standard
identity theory, unlike positive free logics. Scales (1969) in his pioneering work in
negative free logic presents strong arguments in favor of this family of free logics
in general.
The history of negative free logic can be traced back to Aristotle; a clear
example for this is the following passage: “It might, indeed, very well seem that
the same sort of thing does occur in the case of contraries said with combination,
‘Socrates is well’ being contrary to ‘Socrates is sick’. Yet not even with these is it
necessary always for one to be true and the other false. For if Socrates exists one
will be true and one false, but if he does not both will be false; neither ‘Socrates is
sick’ nor ‘Socrates is well’ will be true if Socrates himself does not exist at all.”
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Research on this paper was funded by the FWF (P17392-G04).
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Aristotle (1984, 21)
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Scott (1967) argued explicitly that free logic is relevant in some fields of
mathematics and computer science. Though, as Gumb (2000, 2001) stresses the
term ‘free logic’ is not in use in the literature of mathematics and computer
science.
G. Gentzen (1934/35) introduced sequent calculi as a new kind of formal
system. These are not as close to actual reasoning as natural deduction systems,
but they have very interesting metalogical properties. The most famous one is the
Hauptsatz or the normal form theorem. In this essay we introduce a sequent
calculus for negative free logic with identity and prove the Hauptsatz for this
system. Consistency and the theorem concerning the sub-formula property are
easily provable, given the proof of the Hauptsatz. The sub-formula property is one
salient feature of sequent calculi because it follows from it that the information
required for a derivation of a sequent is contained in the sequent itself. For this
reason sequent calculi are particularly interesting for automated theorem proving.
In contrast to this kind of formal system Hilbert-style formal systems lack the sub-
formula property. This is so, because in general modus ponens deprives formulas
of the necessary information for their derivation. So far there is no sequent
calculus fpr negative free logic in the literature on free logic. In section 1 a
corresponding sequent calculus – called N – is introduced. In the second part
soundness, compactness and completeness is proved for the system in question
relative to a standard semantics for negative free logics. The completeness
theorem provides us with another proof of the Hauptsatz but it is not constructive,
i.e. this proof does not give us a procedure for effectively eliminating cuts.
Before we present the formal system we have to state the language and some
definitions.
Language L
The alphabet of L consists of
(i) a denumerably infinite set of free individual variables (FV)
(ii) a denumerably infinite set of bound individual variables
(iii) an at most denumerably infinite set of (for any n) n-ary predicate
symbols
(iv) ¬, →, ⇒, ∀, E!, =
We use as syntactical variables (with and without indices):
a, b for free individual variables
x, y for bound individual variables
s, t for terms
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F
n
, G
n
for n-ary predicate symbols
D, G for prime formulas
A, B, C, … for formulas
Γ, Δ, Ψ, Φ are finite (possibly empty) sequences of formulas
Terms, formulas and sequents
(1) Every free individual variable is a term.
(2) If t
1
, …, t
n
are terms and F
n
is an n-ary predicate, then F
n
t
1
…t
n
is a formula.
(3) If t is a term, then E!t is a formula.
(4) If s and t are terms, then s = t is a formula.
(5) If A and B are formulas, then ¬A, A → B are formulas.
(6) If A[a] is a formula, s.t. in A the bound variable x does not occur, then
∀xA[x] is a formula.
(7) If A
1
, …, A
n
and B
1
, …, B
m
are formulas, then A
1
, …, A
n
⇒ B
1
, …, B
m
is a
sequent. (The left part of A
1
, …, A
n
⇒ B
1
, …, B
m
is called the antecedent
and the right part succedent. Intuitively, the antecendent is interpreted as a
conjunction of the formulas occurring in the antecedent, i.e. A
1
∧ … ∧ A
n
;
and the succedent is interpreted as a disjunction of the formulas occurring
in the succedent, i.e. B
1
∨ … ∨ B
m
)
The notation A[a] is explained in the following way: the free variable a occurs in
A in several distinguished places. A[x] is the formula which is obtained from A[a]
by substituting the bound variable x for the distinguished occurrences of the free
variable a in A[a].
∧, ∨, ↔, ∃ are defined as usual. Prime formulas are all formulas which are
constructible by the clauses (2) – (4).
For the proof of the Hauptsatz it is important to have the notion of the ‘degree of a
formula’:
Inductive definition of the degree of a formula
(G1) g(A) = 0, if A is prime.
(G2) g(A→B) = g(A) + g(B) + 1
(G3) g(¬A) = g(A) + 1
(G4) g(∀xA[x]) = g(A) + 1
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1 THE FORMAL SYSTEM N
Axioms
(Ax1) D, Γ ⇒ Δ, D
(Ax2) E!t, Γ ⇒ Δ, t = t
(Ax3) s = t, D[s], Γ ⇒ Δ, D[t]
(Ax4) D[t], Γ ⇒ Δ, E!t
Rules
Structural rules
(P⇒) Γ, A, B, Δ ⇒ Ψ (⇒P) Γ ⇒ Δ, A, B, Ψ
Γ, B, A, Δ ⇒ Ψ Γ ⇒ Δ, B, A, Ψ
Logical rules
(¬⇒) Γ ⇒ Δ, A (⇒¬) A, Γ ⇒ Δ
¬A, Γ ⇒Δ Γ ⇒ Δ, ¬A
(→⇒) Γ ⇒ Δ, A B, Γ ⇒ Δ (⇒→) A, Γ ⇒ Δ, B
A → B, Γ ⇒ Δ Γ ⇒ Δ, A → B
(∀⇒) A[t], ∀xA[x], Γ ⇒ Δ ∀xA[x], Γ ⇒ Δ, E!t
∀xA[x], Γ ⇒ Δ
(⇒∀) E!a, Γ ⇒ Δ, A[a] (V!)
Γ ⇒ Δ, ∀xA[x]
(I-Cut) Γ ⇒ Δ, s = t s = t, Γ ⇒ Δ
Γ ⇒ Δ
Remarks
• (V!) means that the free variable a must not occur below the inference line.
• One distinctive feature of free logic is that although it allows for its singular
terms to be empty, the quantifiers of free logic retain their existential import. That
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the quantifiers have existential import in this logic is seen by the rules (∀⇒) and
(⇒∀).
• The last rule – (I-Cut) – termed inessential cut by Takeuti (Takeuti (1987, p. 40).
It is ‘inessential’ because the only cut-formulas admitted by (I-Cut) are equality
formulas.
Inductive definition of derivability in N:
(i) Every axiom is derivable in N.
(ii) If the premise(s) of a (basic) rule of inference is (are) derivable in N,
so is the conclusion of this (basic) inference derivable in N.
A formula A is provable in N iff the sequent ⇒ A is derivable in N (D A).
An application of a logical rule is a logical inference. If Γ ⇒ Δ is a derivable
sequent in N, then h(Γ ⇒ Δ) denotes the minimal number of logical inferences that
are necessary for a derivation of Γ ⇒ Δ. We call h(Γ ⇒ Δ) the height of the
sequent Γ ⇒ Δ.
Definitions
A rule is called admissible in N provided that if the premises of the rule are
derivable in N then the conclusion of this rule is derivable in N.
A rule is called directly admissible in N if it is admissible and each application of
the rule can be replaced by a finite number of basic rules.
A rule is called indirectly admissible in N if it is admissible but not directly
admissible.
The most important difference between directly and indirectly admissible rules is
that indirectly admissible rules are not invariant to extensions of the system.
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1.1 Some directly admissible rules
(∨⇒) A, Γ ⇒ Δ B, Γ ⇒ Δ (⇒∨) Γ ⇒ Δ, A, B
A ∨ B, Γ ⇒ Δ Γ ⇒ Δ, A ∨ B
(∧⇒) A, B, Γ ⇒ Δ (⇒∧) Γ ⇒ Δ, A Γ ⇒ Δ, B
A ∧ B, Γ ⇒ Δ Γ ⇒ Δ, A ∧ B
(∃⇒) E!a, A[a], Γ ⇒ Δ (V!)
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For further details (and examples) on directly/indirectly admissible rules see Schütte (1960, pp. 44f.).