On the specification of sequent systems
... elegant since many technical details of the cut-elimination proof were aborbed by the LF. That approach, however, is based on an intuitionistic meta-logic and is not so suitable for handling the dualities of the sequent calculus. In [Mil96,MP04,MP02], classical linear logic was used as a meta-logic ...
... elegant since many technical details of the cut-elimination proof were aborbed by the LF. That approach, however, is based on an intuitionistic meta-logic and is not so suitable for handling the dualities of the sequent calculus. In [Mil96,MP04,MP02], classical linear logic was used as a meta-logic ...
Propositional Logic - faculty.cs.tamu.edu
... The area of logic that deals with propositions is called propositional logic. In addition to propositional variables, we have logical connectives such as not, and, or, conditional, and biconditional. ...
... The area of logic that deals with propositions is called propositional logic. In addition to propositional variables, we have logical connectives such as not, and, or, conditional, and biconditional. ...
A Concise Introduction to Mathematical Logic
... Although the book is primarily designed to accompany lectures on a graduate level, most of the first three chapters are also readable by undergraduates. The first hundred twenty pages cover sufficient material for an undergraduate course on mathematical logic, combined with a due portion of set theo ...
... Although the book is primarily designed to accompany lectures on a graduate level, most of the first three chapters are also readable by undergraduates. The first hundred twenty pages cover sufficient material for an undergraduate course on mathematical logic, combined with a due portion of set theo ...
this PDF file
... form ϕ @ ψ is true iff ϕ is true and false iff ψ is false; ϕ / ψ is related to Blamey’s [7] transplication and can be read as ‘ψ, presupposing ϕ’. This formula has the value of ψ if ϕ is true, but is neither true nor false otherwise. The Π and Σ quantifiers are the duals of ∀ and ∃ and correspond to ...
... form ϕ @ ψ is true iff ϕ is true and false iff ψ is false; ϕ / ψ is related to Blamey’s [7] transplication and can be read as ‘ψ, presupposing ϕ’. This formula has the value of ψ if ϕ is true, but is neither true nor false otherwise. The Π and Σ quantifiers are the duals of ∀ and ∃ and correspond to ...
Sequent calculus for predicate logic
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
Lecture 4 - Michael De
... Since there are n + 1 of them, this must be so under any assignment of values to atoms, since there are only n values. But since I has the disjunction property, it follows that one of the disjuncts is valid; say it is pi ↔ pj . Since i 6= j (given the construction of the disjunction), there is an as ...
... Since there are n + 1 of them, this must be so under any assignment of values to atoms, since there are only n values. But since I has the disjunction property, it follows that one of the disjuncts is valid; say it is pi ↔ pj . Since i 6= j (given the construction of the disjunction), there is an as ...
A Crevice on the Crane Beach: Finite-Degree
... / AC0 , but assert that “contrary to [their] original complexity. Some expressiveness results were also derived hope, [their] Ehrenfeucht-Fraïssé game arguments are not from Crane Beach Properties, for instance Lee [16] shows that simpler than classical lower bounds.” More recent promising FO[+] is ...
... / AC0 , but assert that “contrary to [their] original complexity. Some expressiveness results were also derived hope, [their] Ehrenfeucht-Fraïssé game arguments are not from Crane Beach Properties, for instance Lee [16] shows that simpler than classical lower bounds.” More recent promising FO[+] is ...
Maximal Introspection of Agents
... the modal operator can not be applied directly to the variables—that is, Bi x (or 2x) is not a well-formed modal formula and therefore neither is ¬∃x (Bi x ∨ ¬Bi x). The propositions T1 and T2 of the example above can not be expressed as sentences in a model logic either. Through the use of axioms s ...
... the modal operator can not be applied directly to the variables—that is, Bi x (or 2x) is not a well-formed modal formula and therefore neither is ¬∃x (Bi x ∨ ¬Bi x). The propositions T1 and T2 of the example above can not be expressed as sentences in a model logic either. Through the use of axioms s ...
proceedings version
... This paper is organised as follows. In Section 2 we introduce our modal logic of equilibrium models MEM both semantically and axiomatically. In Section 3 we recall the logic of here-and-there and equilibrium logic. In Section 4 we define the translation tr from the language of the logic of here-and- ...
... This paper is organised as follows. In Section 2 we introduce our modal logic of equilibrium models MEM both semantically and axiomatically. In Section 3 we recall the logic of here-and-there and equilibrium logic. In Section 4 we define the translation tr from the language of the logic of here-and- ...
Certamen 1 de Representación del Conocimiento
... 12 de Octubre, 2012 (a) [1/2 pto] Define a FOL signature S = {Ω, Π} for which formulas in Σ are well-formed. Solution: Ω = {A/0, B/0} and Π = {R/2, P/2} (b) [1/2 pto] Show that Σ is valid (provide an interpretation for S). Solution: Consider the interpretation I = (U, AI , B I , RI , P I ) where U = ...
... 12 de Octubre, 2012 (a) [1/2 pto] Define a FOL signature S = {Ω, Π} for which formulas in Σ are well-formed. Solution: Ω = {A/0, B/0} and Π = {R/2, P/2} (b) [1/2 pto] Show that Σ is valid (provide an interpretation for S). Solution: Consider the interpretation I = (U, AI , B I , RI , P I ) where U = ...
Symbolic Logic II
... that P → Q, Q → R 2LP P → R, we need to show that there is some interpretation in which KVI (P → Q) 6= 0, KVI (Q → R) 6= 0 and KVI (P → R) = 0. If KVI (P → R) = 0, then KVI (P) = 1 and KVI (R) = 0. But if KVI (R) = 0 and KVI (Q → R) 6= 0, then KVI (Q) 6= 1. If KVI (Q) 6= 1 and KVI (P) = 1 and KVI (P ...
... that P → Q, Q → R 2LP P → R, we need to show that there is some interpretation in which KVI (P → Q) 6= 0, KVI (Q → R) 6= 0 and KVI (P → R) = 0. If KVI (P → R) = 0, then KVI (P) = 1 and KVI (R) = 0. But if KVI (R) = 0 and KVI (Q → R) 6= 0, then KVI (Q) 6= 1. If KVI (Q) 6= 1 and KVI (P) = 1 and KVI (P ...
Logical nihilism - University of Notre Dame
... is incomplete with respect to this semantics, and more plausibly one should say that this reading of `IPC φ ⊃ ψ is erroneous. Thus we see a sense in which the phenomenon of structural completeness is related to a sort of semantic completeness: A structurally incomplete logic will be incomplete with ...
... is incomplete with respect to this semantics, and more plausibly one should say that this reading of `IPC φ ⊃ ψ is erroneous. Thus we see a sense in which the phenomenon of structural completeness is related to a sort of semantic completeness: A structurally incomplete logic will be incomplete with ...
First Order Predicate Logic
... A path in a tableaux is contradictory or closed if some atomic formulae α and ~ α appear on the same path. If all the paths of a tableau are closed, then it is called a contradictory tableaux. A tableau proof of a formula α is a contradictory tableau with root as ~ α . Let α be any formula. If table ...
... A path in a tableaux is contradictory or closed if some atomic formulae α and ~ α appear on the same path. If all the paths of a tableau are closed, then it is called a contradictory tableaux. A tableau proof of a formula α is a contradictory tableau with root as ~ α . Let α be any formula. If table ...
Notes on `the contemporary conception of logic`
... In order to make this dependence apparent, let us apply the name statement form to an expression built up from the sentence letters A, B, C, etc., by applications and reapplications of the propositional connectives. . . . So an expression like ‘(((∼ P ) ∨ Q) ⊃ R)’, to use Mendelson’s bracketing conv ...
... In order to make this dependence apparent, let us apply the name statement form to an expression built up from the sentence letters A, B, C, etc., by applications and reapplications of the propositional connectives. . . . So an expression like ‘(((∼ P ) ∨ Q) ⊃ R)’, to use Mendelson’s bracketing conv ...
Structural Multi-type Sequent Calculus for Inquisitive Logic
... We write φ |= ψ if, for any state S , if S |= φ then S |= ψ. If both φ |= ψ and ψ |= φ, then we write φ ≡ ψ. An InqL-formula φ is valid, denoted |= φ, if S |= φ for any state S . The logic InqL is the set of all valid InqL-formulas. An easy inductive proof shows that InqL-formulas have the downward ...
... We write φ |= ψ if, for any state S , if S |= φ then S |= ψ. If both φ |= ψ and ψ |= φ, then we write φ ≡ ψ. An InqL-formula φ is valid, denoted |= φ, if S |= φ for any state S . The logic InqL is the set of all valid InqL-formulas. An easy inductive proof shows that InqL-formulas have the downward ...
A Calculus for Belnap`s Logic in Which Each Proof Consists of Two
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
Chapter 2 - Princeton University Press
... 2.4. A few students — particularly young adults — may begin to study logic in the naive hope of becoming better organized in their personal lives, or of becoming more “logical” people, less prone to making rash decisions based on transient emotions. I must inform those students that, unfortunately, ...
... 2.4. A few students — particularly young adults — may begin to study logic in the naive hope of becoming better organized in their personal lives, or of becoming more “logical” people, less prone to making rash decisions based on transient emotions. I must inform those students that, unfortunately, ...
An Introduction to Prolog Programming
... A Prolog program corresponds to a set of formulas, all of which are assumed to be true. This restricts the range of possible interpretations of the predicate and function symbols appearing in these formulas. The formulas in the translated program may be thought of as the premises in a proof. If Prol ...
... A Prolog program corresponds to a set of formulas, all of which are assumed to be true. This restricts the range of possible interpretations of the predicate and function symbols appearing in these formulas. The formulas in the translated program may be thought of as the premises in a proof. If Prol ...
2. First Order Logic 2.1. Expressions. Definition 2.1. A language L
... First-order logic will involve expressions built from symbols of our language together with additional symbols: • Infinitely many first-order variables, x0 , x1 , . . ., • The logical connectives ∧, ∨, →, ⊥, • Quantifiers ∀ and ∃, • Parentheses ( and ). We usually write x, y, z, . . . for first-orde ...
... First-order logic will involve expressions built from symbols of our language together with additional symbols: • Infinitely many first-order variables, x0 , x1 , . . ., • The logical connectives ∧, ∨, →, ⊥, • Quantifiers ∀ and ∃, • Parentheses ( and ). We usually write x, y, z, . . . for first-orde ...
Fine`s Theorem on First-Order Complete Modal Logics
... arbitrarily large canonical frames can be built for any given logic. The above body of work by Fine can be seen as part of a second wave of research that flowed from the publication by Kripke [41] of his seminal work on the relational semantics of normal propositional modal logics. As is well known, ...
... arbitrarily large canonical frames can be built for any given logic. The above body of work by Fine can be seen as part of a second wave of research that flowed from the publication by Kripke [41] of his seminal work on the relational semantics of normal propositional modal logics. As is well known, ...
higher-order logic - University of Amsterdam
... To this end, various characterizations of first-order definability will be reviewed here in a little more detail than in chapter I.1. 1.4.1 Lindström’s Theorem. Lindström’s result itself gives a definition of first-order logic, in terms of its global properties. Nevertheless, in practice, it is of ...
... To this end, various characterizations of first-order definability will be reviewed here in a little more detail than in chapter I.1. 1.4.1 Lindström’s Theorem. Lindström’s result itself gives a definition of first-order logic, in terms of its global properties. Nevertheless, in practice, it is of ...
Quantified Equilibrium Logic and the First Order Logic of Here
... • M, w |= ∀xϕ(x) iff M, w′ |= ϕ(d) for all w′ ≥ w and d ∈ Dw′ . • M, w |= ∃xϕ(x) iff M, w |= ϕ(d) for some d ∈ Dw . This is one of the standard ways to present Kripke semantics for Int, similar to that of [30]; for alternative but equivalent presentations, see eg. [29]. The semantics can be intuitive ...
... • M, w |= ∀xϕ(x) iff M, w′ |= ϕ(d) for all w′ ≥ w and d ∈ Dw′ . • M, w |= ∃xϕ(x) iff M, w |= ϕ(d) for some d ∈ Dw . This is one of the standard ways to present Kripke semantics for Int, similar to that of [30]; for alternative but equivalent presentations, see eg. [29]. The semantics can be intuitive ...
(A B) |– A
... Hilbert calculus Ad C. The deduction rules are of a form: A1,…,Am |– B1,…,Bm enable us to prove theorems (provable formulas) of the calculus. We say that each Bi is derived (inferred) from the set of assumptions A1,…,Am. ...
... Hilbert calculus Ad C. The deduction rules are of a form: A1,…,Am |– B1,…,Bm enable us to prove theorems (provable formulas) of the calculus. We say that each Bi is derived (inferred) from the set of assumptions A1,…,Am. ...
Lesson 1
... This apple is an agaric. ---------------------------------------------------------------------Hence This apple has a strong toxic effect. The argument is valid. But the conclusion is evidently not true (false). Hence, at least one premise is false (obviously the second). Circumstances according to ...
... This apple is an agaric. ---------------------------------------------------------------------Hence This apple has a strong toxic effect. The argument is valid. But the conclusion is evidently not true (false). Hence, at least one premise is false (obviously the second). Circumstances according to ...
A logical basis for quantum evolution and entanglement
... have left out reflexivity, but concepts like chain (a totally ordered subset) and antichain (a completely unordered subset) work as in partial orders. We use the word “slice” for an antichain in the precedence order. The word is supposed to be evocative of “spacelike slice” as used in relativity, an ...
... have left out reflexivity, but concepts like chain (a totally ordered subset) and antichain (a completely unordered subset) work as in partial orders. We use the word “slice” for an antichain in the precedence order. The word is supposed to be evocative of “spacelike slice” as used in relativity, an ...