A Sequent Calculus for a Negative Free Logic
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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Citations
11 citations
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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9 citations
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]....
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...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....
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...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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References
5 citations
"A Sequent Calculus for a Negative F..." refers background in this paper
...( 5 ) If A and B are formulas, then ¬A, A → B are formulas....
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...(4) There is a name d in L (thereby L is extended by those new individual free individual variables) for each object d in D The function ϕ will be recursively extended as follows: ( 5 ) ϕ(F n t 1 ... t n )= t iff t i (1 ≤ i ≤ n) is in the domain of ϕ and ∈ ϕ(F n )....
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...From (7) and (4) and ( 5 ) with two applications of Cut we finally get Γ ⇒ Δ. Case 4 By assumption N � DΓ , (Γ∪ Δ)E ⇒ Δ, such that (Γ ∪ Δ)E ⊇{ D[t]...
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5 citations