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Journal ArticleDOI

A Sequent Calculus for a Negative Free Logic

01 Dec 2010-Studia Logica (Springer Netherlands)-Vol. 96, Iss: 3, pp 331-348

TL;DR: The main theorem is the admissibility of the Cut-rule and 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.

AbstractThis 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.

Topics: Cut-elimination theorem (70%), Sequent (69%), Natural deduction (68%), Noncommutative logic (67%), Proof calculus (64%)

Summary (1 min read)

Terms, formulas and sequents

  • (1) Every free individual variable is a term.
  • 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].
  • Prime formulas are all formulas which are constructible by the clauses (2) – (4).

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 the quantifiers have existential import in this logic is seen by the rules (∀⇒) and (⇒∀). .
  • It is ‘inessential’ because the only cut-formulas admitted by (I-Cut) are equality formulas.

1.2.3 Inversion rules

  • The authors keep track of the formula ∀xB[x] beginning at the bottom of the tree.
  • If the branch originally ends with an axiom then after the application of this procedure the corresponding branch ends again with an axiom.
  • The number of logical inferences does not increase.

1.3.3 Theorem (Hauptsatz).

  • Here and below the authors make use of the notations ‘(Γ ∪ Δ)’ and ‘(Γ ∪ Δ)E’; these notations should not be understood as sets but as strings of formulas.
  • This part is easy and is therefore omitted.
  • The induction step is analogous to the induction step for the proof of (I) and is omitted here.

2 SEMANTICS

  • There is a standard semantical approach to negative free logic (Morscher/Simons 2001, Lambert 2001, Burge 1991) and the authors shall follow it here.
  • It furthermore captures the underlying view from a standpoint of negative free logic, that every simple statement containing an empty singular term is false.

Theorems.

  • The authors mean by a “cut-free proof” a proof without any application of Cut, but there may be some applications of I-Cut. (1) states the completeness theorem and (2) the compactness theorem.
  • If the proof-search procedure does not terminate, then the sequent Γ ⇒ Δ is not valid (cf. Buss (1998, 34) and there is a “counter”-intpretation based on this proof-search procedure.
  • At each state the authors do the following: Loop: Let <Ai, tj> be the next pair in the enumeration.
  • If there are no active sequents remaining in P, exit from the loop; otherwise continue with the next loop iteration.
  • The authors shall state here the “critical” case for A = ∀xB[x].

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1
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
1
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.”
2
1
Research on this paper was funded by the FWF (P17392-G04).
2
Aristotle (1984, 21)

2
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

3
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(AB) = g(A) + g(B) + 1
(G3) g(¬A) = g(A) + 1
(G4) g(xA[x]) = g(A) + 1

4
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

5
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.
3
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!)
3
For further details (and examples) on directly/indirectly admissible rules see Schütte (1960, pp. 44f.).

Citations
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01 Jan 2017
TL;DR: This dissertation develops an inferentialist theory of meaning by developing a theory of quantification as marking coherent ways a language can be expanded and modality as the means by which the authors can reflect on the norms governing the assertion and denial conditions of their language.
Abstract: This dissertation develops an inferentialist theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. From this starting point the dissertation develops a theory of quantification as marking coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators. Second-Order Modal Logic

9 citations


Cites background from "A Sequent Calculus for a Negative F..."

  • ...The (1)A sequent calculus for free first-order quantification can be found in Gratzl [19] and Restall [57]....

    [...]

  • ...An atomic sentence is true when the (11)An account equivalent to the logic thus proposed as meaning conferring is given by Gratzl [19], and shown there to be cut eliminable....

    [...]

  • ...rules governing quantification are modified along the lines of Restall [57] or Gratzl [19] by using the following two instead of LD and RD: A, t : Γ, φrt{xs ñ Σ : B LDf A : Γ, Dxφñ Σ : B A : Γ ñ φrt{xs,Σ : B A : Γ ñ Σ : t, B RDf A : Γ ñ Dxφ,Σ : B...

    [...]


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Abstract: In this paper, I motivate a cut free sequent calculus for classical logic with first order quantification, allowing for singular terms free of existential import. Along the way, I motivate a criterion for rules designed to answer Prior’s question about what distinguishes rules for logical concepts, like conjunction from apparently similar rules for putative concepts like Prior’s tonk, and I show that the rules for the quantifiers—and the existence predicate—satisfy that condition.

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TL;DR: This article investigates the proof theory of the Quantified Argument Calculus as developed and systematically studied by Hanoch Ben-Yami and chooses a sequent calculus presentation, which allows for the proofs of a multitude of significant meta-theoretic results with minor modifications to the Gentzen’s original framework.
Abstract: This article investigates the proof theory of the Quantified Argument Calculus (Quarc) as developed and systematically studied by Hanoch Ben-Yami [3, 4]. Ben-Yami makes use of natural deduction (Suppes-Lemmon style), we, however, have chosen a sequent calculus presentation, which allows for the proofs of a multitude of significant meta-theoretic results with minor modifications to the Gentzen’s original framework, i.e., LK. As will be made clear in course of the article LK-Quarc will enjoy cut elimination and its corollaries (including subformula property and thus consistency).

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Cites background from "A Sequent Calculus for a Negative F..."

  • ...Note here that the rules of universal quantification bear a structural similarity to free logic [8]....

    [...]


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Abstract: The paper presents a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics. All free and quasi-free logics considered are formalised in the framework of sequent calculus, the latter for the first time. It is shown that in all cases remarkable simplifications of the starting systems are possible due to the special rule dealing with identity and existence predicate. Cut elimination is proved in a constructive way for sequent calculi adequate for all logics under consideration.

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TL;DR: A simple and unified system of abstract semantics is presented, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics (free and classical).
Abstract: Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free logics usually reject the claim that names need to denote in (ii), and of the systems considered in this paper, the positive free logic concedes that some atomic formulas containing non-denoting names (namely self-identity) are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject (i), are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics (free and classical). The final part of this paper is dedicated to extending the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework.

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