In order to define the notion of proof rigorously, we would have to
... In that sense, it is a natural and familiar rule, except that we perhaps never stopped to think about what we are really doing. However, the business about discharging the premise P when we are through with our argument is a bit puzzling. Most people probably never carry out this “discharge step” co ...
... In that sense, it is a natural and familiar rule, except that we perhaps never stopped to think about what we are really doing. However, the business about discharging the premise P when we are through with our argument is a bit puzzling. Most people probably never carry out this “discharge step” co ...
CERES for Propositional Proof Schemata
... schema. The aim is to define a CERES method for proof schemata that will yield a schematic representation of CL(πk ). This can be regarded as a generalization of CERES from proofs π to sequences of proofs (πk )k∈N that are given by such a proof schema. Not only will this close the gap in the applica ...
... schema. The aim is to define a CERES method for proof schemata that will yield a schematic representation of CL(πk ). This can be regarded as a generalization of CERES from proofs π to sequences of proofs (πk )k∈N that are given by such a proof schema. Not only will this close the gap in the applica ...
Chapter 12 - Binus Repository
... a list with the parameters as elements - Predicate Functions: (#T and () are true and false) 1. EQ? takes two symbolic parameters; it returns #T if both parameters are atoms and the two are the same e.g., (EQ? 'A 'A) yields #T (EQ? 'A '(A B)) yields () Note that if EQ? is called with list parameters ...
... a list with the parameters as elements - Predicate Functions: (#T and () are true and false) 1. EQ? takes two symbolic parameters; it returns #T if both parameters are atoms and the two are the same e.g., (EQ? 'A 'A) yields #T (EQ? 'A '(A B)) yields () Note that if EQ? is called with list parameters ...
scheme1 - Computer Science and Electrical Engineering
... • LISP is an acronym for LISt Processing language • Lisp (b. 1958) is an old language with many variants – Fortran is only older language still in wide use – Lisp is alive and well today • Most modern versions are based on Common Lisp • Scheme is one of the major variants – We’ll use Scheme, not Lis ...
... • LISP is an acronym for LISt Processing language • Lisp (b. 1958) is an old language with many variants – Fortran is only older language still in wide use – Lisp is alive and well today • Most modern versions are based on Common Lisp • Scheme is one of the major variants – We’ll use Scheme, not Lis ...
Type Class
... • Referential transparency – symbol always represents the same value – Equational reasoning (equals can be substituted by equals) • easy mathematical manipulation, parallel execution, etc. ...
... • Referential transparency – symbol always represents the same value – Equational reasoning (equals can be substituted by equals) • easy mathematical manipulation, parallel execution, etc. ...
High Level Verification of Control Intensive Systems
... to bound the size of the abstract model by that of the concrete model. We retain most the control variables in the abstract model (the criteria for retaining a control variable is discussed later in this section). The concrete transition relations for these control variables also serve as abstract t ...
... to bound the size of the abstract model by that of the concrete model. We retain most the control variables in the abstract model (the criteria for retaining a control variable is discussed later in this section). The concrete transition relations for these control variables also serve as abstract t ...
Mathematical Logic
... Definition 1.1.5. If A is a formula, the degree of A is the number of occurrences of propositional connectives in A. This is the same as the number of times rules 2 and 3 had to be applied in order to generate A. ...
... Definition 1.1.5. If A is a formula, the degree of A is the number of occurrences of propositional connectives in A. This is the same as the number of times rules 2 and 3 had to be applied in order to generate A. ...
term rewriting.
... Variables in CafeOBJ • Variables in CafeOBJ equations are universally quantified (∀N) equations • Terms with variables, are instantiated by substituting the variables with ground terms. • Here is an example from Jose Meseguer’s CC373 lecture notes. ...
... Variables in CafeOBJ • Variables in CafeOBJ equations are universally quantified (∀N) equations • Terms with variables, are instantiated by substituting the variables with ground terms. • Here is an example from Jose Meseguer’s CC373 lecture notes. ...
Higher Order Logic - Theory and Logic Group
... logic augmented with second order variables, that is, variables ranging over relations and functions (of all arities). Given a vocabulary1 V , V -terms and atomic V -formulas are de ned as in rst order logic (with equality), but using also function-variables and relation-variables in complete analo ...
... logic augmented with second order variables, that is, variables ranging over relations and functions (of all arities). Given a vocabulary1 V , V -terms and atomic V -formulas are de ned as in rst order logic (with equality), but using also function-variables and relation-variables in complete analo ...
Higher Order Logic - Indiana University
... logic augmented with second order variables, that is, variables ranging over relations and functions (of all arities). Given a vocabulary1 V , V -terms and atomic V -formulas are de ned as in rst order logic (with equality), but using also function-variables and relation-variables in complete analo ...
... logic augmented with second order variables, that is, variables ranging over relations and functions (of all arities). Given a vocabulary1 V , V -terms and atomic V -formulas are de ned as in rst order logic (with equality), but using also function-variables and relation-variables in complete analo ...
Non-Classical Logic
... Completeness: If ∆ A then ∆ ` A, for all ∆ and A. that no truth value assignment (interpretation) can make Together these results entail the equivalence of semantic the formula false, so it must be logically valid. and deductive validity. The same process can be used to show that a formula Proofs ...
... Completeness: If ∆ A then ∆ ` A, for all ∆ and A. that no truth value assignment (interpretation) can make Together these results entail the equivalence of semantic the formula false, so it must be logically valid. and deductive validity. The same process can be used to show that a formula Proofs ...
Logic in the Finite - CIS @ UPenn
... It is easy to see that we can make an e ective list A1 ; A2 ; : : : of nite structures for L which contains every such structure up to isomorphism. We may now subject a sentence ' 2 L to the following e ective procedure: successively test whether A1 satis es '; A2 satis es '; : : : ; at the rst st ...
... It is easy to see that we can make an e ective list A1 ; A2 ; : : : of nite structures for L which contains every such structure up to isomorphism. We may now subject a sentence ' 2 L to the following e ective procedure: successively test whether A1 satis es '; A2 satis es '; : : : ; at the rst st ...
Functional Programming and Compiler Design
... Functions and other values begin with small letters … … types begin with capital letters. ...
... Functions and other values begin with small letters … … types begin with capital letters. ...
Thesis Proposal: A Logical Foundation for Session-based
... In Section 3 I develop the interpretation of linear logic as session types that serves as the basis for my work. It has a few variations from that of [5] in that it does not commit to the π-calculus a priori, developing a proof term assignment that can be used as a language for session-typed communi ...
... In Section 3 I develop the interpretation of linear logic as session types that serves as the basis for my work. It has a few variations from that of [5] in that it does not commit to the π-calculus a priori, developing a proof term assignment that can be used as a language for session-typed communi ...
An Introduction to Proof Theory - UCSD Mathematics
... of the truth of theorems. That is to say, a proof is expressed in natural language plus possibly symbols and figures, and is sufficient to convince an expert of the correctness of a theorem. Examples of social proofs include the kinds of proofs that are presented in conversations or published in art ...
... of the truth of theorems. That is to say, a proof is expressed in natural language plus possibly symbols and figures, and is sufficient to convince an expert of the correctness of a theorem. Examples of social proofs include the kinds of proofs that are presented in conversations or published in art ...
Decision procedures in Algebra and Logic
... Algebraic structure • Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix " ' ", and governed by the axioms x'x=1, x(x'y) = xy, x'(yz) = (x'y)(x'z), (xy)'z = (x'z)(y'z). Ringoids: Two binary operations, addition and multiplica ...
... Algebraic structure • Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix " ' ", and governed by the axioms x'x=1, x(x'y) = xy, x'(yz) = (x'y)(x'z), (xy)'z = (x'z)(y'z). Ringoids: Two binary operations, addition and multiplica ...
Boolean Logic - Programming Systems Lab
... expression is always >, and the prime tree normal form of an unsatisfiable expressions is always ⊥. Thus an expression is satisfiable if and only if its prime tree normal form is different from ⊥. We define prime expressions inductively: 1. ⊥ and > are prime expressions. 2. Cxst is a prime expressi ...
... expression is always >, and the prime tree normal form of an unsatisfiable expressions is always ⊥. Thus an expression is satisfiable if and only if its prime tree normal form is different from ⊥. We define prime expressions inductively: 1. ⊥ and > are prime expressions. 2. Cxst is a prime expressi ...
Notes on Modal Logic - Stanford University
... • Alethic Reading: 2ϕ means ‘ϕ is necessary’ and 3ϕ means ‘ϕ is possible’. • Deontic Reading: 2ϕ means ‘ϕ is obligatory’ and 3ϕ means ‘ϕ is permitted’. In this literature, typically ‘O’ is used instead of ‘2’ and ‘P ’ instead of ‘3’. • Epistemic Reading: 2ϕ means ‘ϕ is known’ and 3ϕ means ‘ϕ is cons ...
... • Alethic Reading: 2ϕ means ‘ϕ is necessary’ and 3ϕ means ‘ϕ is possible’. • Deontic Reading: 2ϕ means ‘ϕ is obligatory’ and 3ϕ means ‘ϕ is permitted’. In this literature, typically ‘O’ is used instead of ‘2’ and ‘P ’ instead of ‘3’. • Epistemic Reading: 2ϕ means ‘ϕ is known’ and 3ϕ means ‘ϕ is cons ...
The Development of Mathematical Logic from Russell to Tarski
... and derived notions makes it possible to compare different hypotheticodeductive systems as to logical equivalence. Two systems turn out to be equivalent if for every primitive notion of one we can find an explicit definition in the second one such that all primitive propositions of the first system ...
... and derived notions makes it possible to compare different hypotheticodeductive systems as to logical equivalence. Two systems turn out to be equivalent if for every primitive notion of one we can find an explicit definition in the second one such that all primitive propositions of the first system ...
Preferences and Unrestricted Rebut
... determine whether one argument attacks another. Suppose one considers arguments constructed using reasons, represented as rules. Many rules will be defeasible (for instance, when applying the argument scheme of expert opinion), but some rules will be strict (for instance, Modus ponens).The idea in f ...
... determine whether one argument attacks another. Suppose one considers arguments constructed using reasons, represented as rules. Many rules will be defeasible (for instance, when applying the argument scheme of expert opinion), but some rules will be strict (for instance, Modus ponens).The idea in f ...
First-Order Intuitionistic Logic with Decidable Propositional
... their subsets”. Propositional logic can be considered a part of the mathematics of finite sets because of availability of finite models using truth tables. Thus, LEM for propositional formulas is not really a target of intuitionistic criticism of classical logic. The classical assumption that every ...
... their subsets”. Propositional logic can be considered a part of the mathematics of finite sets because of availability of finite models using truth tables. Thus, LEM for propositional formulas is not really a target of intuitionistic criticism of classical logic. The classical assumption that every ...
Topological aspects of real-valued logic
... Theorem 1.0.3. Let S be a two-sorted metric signature, and let L be a countable fragment of Lω1 ,ω (S). Let T be an L-theory and let M = h M, V, . . . i be a model of T where M has density κ and V has density λ, with κ > λ ≥ ℵ0 . Then there is a model N = h N, W, . . . i ≡L M with N of density ℵ1 an ...
... Theorem 1.0.3. Let S be a two-sorted metric signature, and let L be a countable fragment of Lω1 ,ω (S). Let T be an L-theory and let M = h M, V, . . . i be a model of T where M has density κ and V has density λ, with κ > λ ≥ ℵ0 . Then there is a model N = h N, W, . . . i ≡L M with N of density ℵ1 an ...
Functional Programming
... Extensive polymorphism Structured function returns Constructors for aggregate objects Garbage collection ...
... Extensive polymorphism Structured function returns Constructors for aggregate objects Garbage collection ...
