Sequentiality by Linear Implication and Universal Quantification
... Sequentialization is achieved in linear logic by a controlled form of backchaining, whose non-determinism is eliminated by the linearity of the calculus (linear implication) and a declarative way of producing unique identifiers (universal quantification). In our case study these two mechanisms, toge ...
... Sequentialization is achieved in linear logic by a controlled form of backchaining, whose non-determinism is eliminated by the linearity of the calculus (linear implication) and a declarative way of producing unique identifiers (universal quantification). In our case study these two mechanisms, toge ...
Aristotle`s work on logic.
... The first letter indicates to which one of the four perfect moods the mood is to be reduced: ‘B’ to Barbara, ‘C’ to Celarent, ‘D’ to Darii, and ‘F’ to Ferio. The letter ‘s’ after the ith vowel indicates that the corresponding proposition has to be simply converted, i.e., a use of si . The letter ‘p’ ...
... The first letter indicates to which one of the four perfect moods the mood is to be reduced: ‘B’ to Barbara, ‘C’ to Celarent, ‘D’ to Darii, and ‘F’ to Ferio. The letter ‘s’ after the ith vowel indicates that the corresponding proposition has to be simply converted, i.e., a use of si . The letter ‘p’ ...
Dissolving the Scandal of Propositional Logic?
... implication was never introduced in propositional logic to capture merely some connection of dependence between two statements. It was introduced to capture the notion of implication or logical consequence in natural language. So having to rely on an adjunctive interpretation of material implication ...
... implication was never introduced in propositional logic to capture merely some connection of dependence between two statements. It was introduced to capture the notion of implication or logical consequence in natural language. So having to rely on an adjunctive interpretation of material implication ...
A Proof Theory for Generic Judgments
... Γ0 , ∀xB −→ C is proved using the introduction of ∀ on the left from the premise Γ0 , B[t/x] −→ C, where t is some term. To reduce the rank of the cut formula ∀x.B between the sequents Γ −→ ∀x.B and Γ0 , ∀xB −→ C, the eigenvariable c in the sequent calculus proof Π(c) must be substituted by t to yie ...
... Γ0 , ∀xB −→ C is proved using the introduction of ∀ on the left from the premise Γ0 , B[t/x] −→ C, where t is some term. To reduce the rank of the cut formula ∀x.B between the sequents Γ −→ ∀x.B and Γ0 , ∀xB −→ C, the eigenvariable c in the sequent calculus proof Π(c) must be substituted by t to yie ...
Temporal Here and There - Computational Cognition Lab
... inference rules for induction. In this setting, traditional proofs of completeness (see [11, Chap. 9]) are based on canonical model and filtration. In our HT setting, however, the usual filtration method does not allow to transform, as it is the case in ordinary temporal logic, the canonical model int ...
... inference rules for induction. In this setting, traditional proofs of completeness (see [11, Chap. 9]) are based on canonical model and filtration. In our HT setting, however, the usual filtration method does not allow to transform, as it is the case in ordinary temporal logic, the canonical model int ...
A modal perspective on monadic second
... (found in [18]) is of no particular importance for the present paper, as we give a virtually self-contained exposition of all our results. As a by-product of our investigations we obtain a simple, effective procedure (inspired by the approach of ten Cate [8]) that translates MSOsentences to equivale ...
... (found in [18]) is of no particular importance for the present paper, as we give a virtually self-contained exposition of all our results. As a by-product of our investigations we obtain a simple, effective procedure (inspired by the approach of ten Cate [8]) that translates MSOsentences to equivale ...
Nelson`s Strong Negation, Safe Beliefs and the - CEUR
... Since it was introduced in [3], strong negation has been well accepted in the answer set programming community2 . However, this connective has not received a fair treatment. While the answer set semantics has been extended to always more flexible classes of logic programs where conjunctions, disjunc ...
... Since it was introduced in [3], strong negation has been well accepted in the answer set programming community2 . However, this connective has not received a fair treatment. While the answer set semantics has been extended to always more flexible classes of logic programs where conjunctions, disjunc ...
SORT LOGIC AND FOUNDATIONS OF MATHEMATICS 1
... The concept of a free occurrence of a variable in a formula is defined as in first order logic. As a new concept we have the concept of a free occurrence of a sort in a formula. We define it as follows, following the intuition that if a sort occurs “free” in a formula, either as the sort of an indiv ...
... The concept of a free occurrence of a variable in a formula is defined as in first order logic. As a new concept we have the concept of a free occurrence of a sort in a formula. We define it as follows, following the intuition that if a sort occurs “free” in a formula, either as the sort of an indiv ...
Discrete Mathematics - Lecture 4: Propositional Logic and Predicate
... Using Propositional Logic for designing proofs A mathematical statement comprises of a premise (or assumptions). And when the assumptions are satisfied the statement deduces something. If A is the set of assumptions and B is the deduction then a mathematical statement is of the form ...
... Using Propositional Logic for designing proofs A mathematical statement comprises of a premise (or assumptions). And when the assumptions are satisfied the statement deduces something. If A is the set of assumptions and B is the deduction then a mathematical statement is of the form ...
Document
... two-valued logic – every sentence is either true or false some sentences are minimal – no proper part which is also a sentence others – can be taken apart into smaller parts we can build larger sentences from smaller ones by using connectives ...
... two-valued logic – every sentence is either true or false some sentences are minimal – no proper part which is also a sentence others – can be taken apart into smaller parts we can build larger sentences from smaller ones by using connectives ...
propositions and connectives propositions and connectives
... propositions names: p, q, r, …, p0, p1, p2, … a name for false : ...
... propositions names: p, q, r, …, p0, p1, p2, … a name for false : ...
lecture notes in Mathematical Logic
... In turned out that some parts of logic are of a special nature: they can be entirely carried out by a mechanical procedure; for example, to verify that one formula is an instance of another, or that a given sequence of formulas constitutes a formal proof. Finding a proof, on the other hand, is usual ...
... In turned out that some parts of logic are of a special nature: they can be entirely carried out by a mechanical procedure; for example, to verify that one formula is an instance of another, or that a given sequence of formulas constitutes a formal proof. Finding a proof, on the other hand, is usual ...
Annals of Pure and Applied Logic Automata and logics
... In this section we introduce a variant of classical word automata called ST-NFA’s which are a convenient formalism for generating signals. We recall that a non-deterministic finite state automaton (NFA) over an alphabet A is a structure A = (Q , S , δ, F ) where Q is a finite set of states, S is the ...
... In this section we introduce a variant of classical word automata called ST-NFA’s which are a convenient formalism for generating signals. We recall that a non-deterministic finite state automaton (NFA) over an alphabet A is a structure A = (Q , S , δ, F ) where Q is a finite set of states, S is the ...
A SHORT AND READABLE PROOF OF CUT ELIMINATION FOR
... step we prove the case for m > 0. Cases (2)–(8) are numbered by the rule number (Definition 2.1) of the last rule applied in deriving (Γ ⊢ ∆)[a]. (2) (Γ ⊢ ∆)[a] = Γ[a], A[a] → B[a] ⊢ ∆[a]. Thus the premises of the rule,8 Γ[a], A[a] → ⊥ ⊢ ∆[a] and Γ[a], B[a] ⊢ ∆[a], are each derived with orders < m. ...
... step we prove the case for m > 0. Cases (2)–(8) are numbered by the rule number (Definition 2.1) of the last rule applied in deriving (Γ ⊢ ∆)[a]. (2) (Γ ⊢ ∆)[a] = Γ[a], A[a] → B[a] ⊢ ∆[a]. Thus the premises of the rule,8 Γ[a], A[a] → ⊥ ⊢ ∆[a] and Γ[a], B[a] ⊢ ∆[a], are each derived with orders < m. ...
Foundations of Logic Programmin:
... language, as well as deeper proof-theoretic issues. The semantics is concerned with the meanings attached to the well-formed formulas and the symbols they contain. ...
... language, as well as deeper proof-theoretic issues. The semantics is concerned with the meanings attached to the well-formed formulas and the symbols they contain. ...
Bounded Functional Interpretation
... given f , compactness reasons (that can be put in intuitionistic clothes) yield a bound for the n’s. Notwithstanding, it is not continuity that is responsible for the elimination of the FAN theorem by b.f.i.: It is majorizability! This was first observed in [15] in connection with the FAN rule (see ...
... given f , compactness reasons (that can be put in intuitionistic clothes) yield a bound for the n’s. Notwithstanding, it is not continuity that is responsible for the elimination of the FAN theorem by b.f.i.: It is majorizability! This was first observed in [15] in connection with the FAN rule (see ...
• Propositional definite clauses ctd • Monotone functions and power
... We can think of using our inference system to do forward inference. • Any unit clauses (atoms pi ∈ S) are given as true, and are provable as axioms. • We can deduce that a new atom r is true whenever we have deduced that p1, . . . , pn true, and p1 ∧ · · · ∧ pn → r is in S. A derivation here uses an ...
... We can think of using our inference system to do forward inference. • Any unit clauses (atoms pi ∈ S) are given as true, and are provable as axioms. • We can deduce that a new atom r is true whenever we have deduced that p1, . . . , pn true, and p1 ∧ · · · ∧ pn → r is in S. A derivation here uses an ...
pdf - at www.arxiv.org.
... been argued in [6, 18, 22]. In the classical approach [18], the semantic view was taken: if a nonterminating SLD-resolution derivation for Φ and A accumulates computed substitutions σ0 , σ2 , . . . in such a way that . . . (σ2 (σ0 (A))) is an infinite ground formula, then . . . (σ2 (σ0 (A))) is said ...
... been argued in [6, 18, 22]. In the classical approach [18], the semantic view was taken: if a nonterminating SLD-resolution derivation for Φ and A accumulates computed substitutions σ0 , σ2 , . . . in such a way that . . . (σ2 (σ0 (A))) is an infinite ground formula, then . . . (σ2 (σ0 (A))) is said ...
On Natural Deduction in Classical First-Order Logic: Curry
... The informal idea expressed by the associated reductions is to assume ∀α P and try to produce some complete proof of C out of u by reducing inside u. Whenever u needs the truth of an instance P[n/α] of the assumption ∀α P, it checks it, and if it is true, it replaces it by its canonical proof which ...
... The informal idea expressed by the associated reductions is to assume ∀α P and try to produce some complete proof of C out of u by reducing inside u. Whenever u needs the truth of an instance P[n/α] of the assumption ∀α P, it checks it, and if it is true, it replaces it by its canonical proof which ...
A really temporal logic
... sequences) depends on the interpretation of the variables .x and y. The explicit quantification of variables, however, is typically omitted from first-order temporal specifications [Ostroff, 1990; Pnueli and Harel, 1988]. Moreover, specification languages quantifier prefwes ...
... sequences) depends on the interpretation of the variables .x and y. The explicit quantification of variables, however, is typically omitted from first-order temporal specifications [Ostroff, 1990; Pnueli and Harel, 1988]. Moreover, specification languages quantifier prefwes ...
General Dynamic Dynamic Logic
... logic [7,11,12,20]. A significant difference from the epistemic setting is the need to describe dynamic operators that change the relational structure of the underlying model, not just the size of its domain (announcement) or the propositional valuations (real-world change). For example, if one mode ...
... logic [7,11,12,20]. A significant difference from the epistemic setting is the need to describe dynamic operators that change the relational structure of the underlying model, not just the size of its domain (announcement) or the propositional valuations (real-world change). For example, if one mode ...
One-dimensional Fragment of First-order Logic
... Two-variable logic and guarded-fragment are examples of decidable fragments of first-order logic that are not based on restricting the quantifier alternation patterns of formulae, unlike the prefix classes studied in the context of the classical decision problem. Surprisingly, not many such framewor ...
... Two-variable logic and guarded-fragment are examples of decidable fragments of first-order logic that are not based on restricting the quantifier alternation patterns of formulae, unlike the prefix classes studied in the context of the classical decision problem. Surprisingly, not many such framewor ...
Document
... x,y,z (x1 = x, x2 = [y|z], member(x,z)) x1 x2 disjoint(x1,x2) x (x1 = [], x2 = x) x,y,z (x1 = [x|y], x2 = z, member(x,z), disjoint(y,z)) ( plus the standard axioms for equality and inequality) Then, e.g. comp(p) |= member(a,[a,b]), member(a,[ ]), member(a,[b,c]) ...
... x,y,z (x1 = x, x2 = [y|z], member(x,z)) x1 x2 disjoint(x1,x2) x (x1 = [], x2 = x) x,y,z (x1 = [x|y], x2 = z, member(x,z), disjoint(y,z)) ( plus the standard axioms for equality and inequality) Then, e.g. comp(p) |= member(a,[a,b]), member(a,[ ]), member(a,[b,c]) ...
FIRST DEGREE ENTAILMENT, SYMMETRY AND PARADOX
... assigned only one value, ρ′ might assign both. The only requirement on quotation names for this fixed point construction to succeed is that quotation names for different sentences are different. This means that the construction will work whatever we take the denotation of other constants to be. So, ...
... assigned only one value, ρ′ might assign both. The only requirement on quotation names for this fixed point construction to succeed is that quotation names for different sentences are different. This means that the construction will work whatever we take the denotation of other constants to be. So, ...
Outline of Lecture 2 First Order Logic and Second Order Logic Basic
... • MSOL has no complete provability system: The Peano axioms are expressible in MSOL and characterize the structure h IN, +, ×, 0, 1i up to isomorphims. If there were a complete provability system, the set of MSOL(τarith )sentences true in h IN, +, ×, 0, 1i would be computable. But this contradicts G ...
... • MSOL has no complete provability system: The Peano axioms are expressible in MSOL and characterize the structure h IN, +, ×, 0, 1i up to isomorphims. If there were a complete provability system, the set of MSOL(τarith )sentences true in h IN, +, ×, 0, 1i would be computable. But this contradicts G ...