Proof-Theoretic Analysis of the Quantified Argument Calculus
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1 Proof-Theoretic Analysis of the Quantified Argument Calculus Edi Pavlovic Central European University, Budapest #IstandwithCEU PhDs in Logic IX May , RUB Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 1 / 23
2 Quantified Argument Calculus Hanoch Ben-Yami (2014). The Quantified Argument Calculus. The Review of Symbolic Logic 7(1), pp Quarc is a system of quantified logic which does away with variables and unrestricted predicates, but employs quantifiers applied directly to predicates which appear as arguments of other predicates (hence the name), along with operators that attach directly to predicates, and anaphors. It is in this respect (arguably) closer to natural language, while also achieving results similar to that of the Predicate Calculus. The aim of this paper is to demonstrate the latter of these claims. Edi Pavlovic and Norbert Gratzl (2016). Proof-Theoretic Analysis of the Quantified Argument Calculus. Submitted for review. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 2 / 23
3 Quarc Language of Quarc Quarc-Specific Language 1 Basic formula: -name a and predicate P combine into a formula: (a)p - a is P. 2 Formula: -universal quantifier applies to unary predicate S to form Quantified Argument S, - S combines with P to form a formula: ( S)P - All S are P. -particular quantifier applies to unary predicate S to form Quantified Argument S, - S combines with P to form a formula: ( S)P - Some S are P. 3 Also contains: -anaphoric expressions: ( S α )P (α)q - All S that are P are Q, -reordered predicates: (j, m)o 2,1 - John is seen by Mary, -negative predication: (j) H - John isn t happy. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 4 / 23
4 Quarc Value Assignments Value Assignments (Almost) standard, plus: Instantiation. Every unary predicate has a true instance: for every unary predicate S there is a name t such that (t)s is true. Observation ( S)P ( S)P is a theorem of Quarc. Given ( S)S, it follows that ( S)S is likewise a theorem. But note that is a particular quantifier, not to be conflated with the existential construction. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 5 / 23
5 Quarc Value Assignments Division of Quarc Subsystems of Quarc Quarc Quarc 2 Quarc 3 = + + Ins Quarc B Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 6 / 23
6 LK-Quarc LK-Quarc Proof system of Quarc is a Lemmon-Suppes style natural deduction. In our system we use a modification of Gentzen s LK. We divide our system into LK-Quarc B, LK-Quarc 2, LK-Quarc 3 and full LK-Quarc, and demonstrate a series of results for each one in turn, starting with LK-Quarc B. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 8 / 23
7 LK-Quarc LK-Quarc B LK-Quarc B Propositional LK plus 1 Quantifier rules: (L ): (R ): A [t/ M], Γ = Γ =, tm A [ M], Γ = tm, Γ =, A [t/ M] Γ =, A [ M] (L ): tm, A [t/ M], Γ = A [ M], Γ = (R ): Γ =, tm Γ =, A [t/ M] Γ =, A [ M] Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 9 / 23
8 LK-Quarc LK-Quarc B LK-Quarc B And also: 2 Special rules: (LA): A [...a 1...a n...], Γ = A [...a α /a 1...α/a n...], Γ = (LRd): (t 1,..., t n )R, Γ = (t π1,..., t πn )R π, Γ = (LNP): (t 1,..., t n )P, Γ = (t 1,..., t n ) P, Γ = (RA): Γ =, A [...a 1...a n...] Γ =, A [...a α /a 1...α/a n...] (RRd): Γ =, (t 1,..., t n )R Γ =, (t π1,..., t πn )R π (RNP): Γ =, (t 1,..., t n )P Γ =, (t 1,..., t n ) P Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 10 / 23
9 LK-Quarc LK-Quarc B LK-Quarc B - Results Theorem (Deductive Equivalence) Quarc B and LK-Quarc B are deductively equivalent. Lemma (LK-Quarc B to Quarc B ) Every endsequent Γ of some derivation in LK-Quarc B is, given standard translation, provable in Quarc B : Γ. Lemma (Quarc B to LK-Quarc B ) For every line L, i, A, R of any proof in Quarc B there exists a corresponding segment of a derivation in LK-Quarc B with the endsequent L A derivable from trivial lemmas and standard translations of lines R relies on. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 11 / 23
10 LK-Quarc LK-Quarc B LK-Quarc B - Results Theorem (Cut Elimination) LK-Quarc B enjoys the Cut Elimination property. Theorem (Subformula Property) LK-Quarc B enjoys the Subformula property. Corollary (Consistency) LK-Quarc B is consistent. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 12 / 23
11 LK-Quarc LK-Quarc 2 Quarc 2 The identity rules in Quarc are as follows: Identity Introduction, =I (i) a = a =I Identity Elimination, =E L 1 (k) A [b 1,..., b n ] L 2 (m) a = b L 1, L 2 (i) A [a/b 1,..., a/b n ] =E, k, m Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 13 / 23
12 LK-Quarc LK-Quarc 2 LK-Quarc 2 To expand LK-Quarc B into LK-Quarc 2 we add the following rules: a = a, Γ =1 Γ A [b], a = b, A [a/b], Γ =2 a = b, A [a/b], Γ where A is a basic formula and A [a/b] is a formula produced by substituting any number of occurrences of the singular argument b by a. The rule = 2 generalizes to any A. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 14 / 23
13 LK-Quarc LK-Quarc 2 LK-Quarc 2 - Results Theorem (Cut Elimination) LK-Quarc 2 enjoys the Cut Elimination property. Theorem (Weak Subformula Property) LK-Quarc 2 enjoys the Weak Subformula property. Corollary (Consistency) LK-Quarc 2 is consistent. Theorem (Conservativity) LK-Quarc 2 is a conservative expansion of LK-Quarc B. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 15 / 23
14 LK-Quarc LK-Quarc 3 LK-Quarc 3 To expand LK-Quarc B into LK-Quarc 3, we add the rule for Instantiation: ts, Γ Ins* Γ * - where lower sequent does not contain the singular argument t. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 16 / 23
15 LK-Quarc LK-Quarc 3 LK-Quarc 3 - Results Theorem (Cut Elimination) LK-Quarc 3 enjoys the Cut Elimination property. Theorem (Weak Subformula Property) LK-Quarc 3 enjoys the Weak Subformula property. Corollary (Consistency) LK-Quarc 3 is consistent. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 17 / 23
16 LK-Quarc LK-Quarc LK-Quarc Obtained by combining LK-Quarc 2 and LK-Quarc 3, i.e. by adding Identity and Instantiation rules to LK-Quarc B. Consequently, all the previous proofs hold for it. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 18 / 23
17 LK-Quarc LK-Quarc LK-Quarc - Results Theorem (Cut Elimination) LK-Quarc enjoys the Cut Elimination property. Theorem (Weak Subformula Property) LK-Quarc enjoys the Weak Subformula property. Corollary (Consistency) LK-Quarc is consistent. Theorem (Conservativity) LK-Quarc is a conservative expansion of LK-Quarc 3. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 19 / 23
18 Future Work Future Work Work yet to be done/ in progress: 1 Craig s Interpolation Property - done for LK-Quarc with unrestricted quantification, move to restricted is complicated. 2 Modal expansion - using a labeled system, modal expansion is simple. Cut elimination and associated properties hold for a range of quantified modal systems. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 21 / 23
19 Future Work Thank you! Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 22 / 23
20 Future Work Select References Mathias Baaz and Alexander Leitsch (2011). Methods of Cut-Elimination, Trends in Logic vol. 34. Springer. Hanoch Ben-Yami (2014). The Quantified Argument Calculus. The Review of Symbolic Logic 7(1), pp Hanoch Ben-Yami (2004). Logic and Natural Language: On Plural Reference and Its Semantic and Logical Significance. Aldershot: Ashgate. Gerhard Gentzen (1969). The Collected Papers of Gerhard Gentzen, ed. M. Szabo, pp Amsterdam: North-Holland. Norbert Gratzl (2010). A Sequent Calculus for a Negative Free Logic. Studia Logica 96, pp Ran Lanzet and Hanoch Ben-Yami (2006). Logical Inquiries into a New Formal System with Plural Reference. First-order logic revisited, Logische Philosophie series 12, pp Sara Negri and Jan von Plato (2001). Structural Proof Theory. Cambridge: Cambridge University Press. Edi Pavlovic (CEU) Proof-Theoretic Analysis of Quarc PhDs in Logic IX 23 / 23
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Locality for Classical Logic arxiv:math/0301317v1 [mathlo] 27 Jan 2003 Kai Brünnler Technische Universität Dresden Fakultät Informatik - 01062 Dresden - Germany kaibruennler@inftu-dresdende Abstract In
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