Second-order Logic
... Proof. Suppose M |= PA2 Of course, for any n ∈ N, ValM (n) ∈ |M|, so N ⊆ |M|. Let N = {ValM (n) : n ∈ N} and s(X) = N . By assumption, M |= ∀X (X() ∧ ∀x (X(x) → X(x0 ))) → ∀x X(x) and so M, s |= (X() ∧ ∀x (X(x) → X(x0 ))) → ∀x X(x). ValM () ∈ N , and so M, s |= X(). Also, since if x ∈ N then als ...
... Proof. Suppose M |= PA2 Of course, for any n ∈ N, ValM (n) ∈ |M|, so N ⊆ |M|. Let N = {ValM (n) : n ∈ N} and s(X) = N . By assumption, M |= ∀X (X() ∧ ∀x (X(x) → X(x0 ))) → ∀x X(x) and so M, s |= (X() ∧ ∀x (X(x) → X(x0 ))) → ∀x X(x). ValM () ∈ N , and so M, s |= X(). Also, since if x ∈ N then als ...
Symmetric Splitting in the General Theory of Stable Models
... The methods for simplifying SMp [F ] described in [Ferraris et al., 2010] are not directly applicable to (18), but they allow us to simplify (19) and (20). The version of program completion presented in that paper turns the former into p ↔ ¬q ∧ r and the latter into q ↔ ¬p ∧ r. Consequently (18) is ...
... The methods for simplifying SMp [F ] described in [Ferraris et al., 2010] are not directly applicable to (18), but they allow us to simplify (19) and (20). The version of program completion presented in that paper turns the former into p ↔ ¬q ∧ r and the latter into q ↔ ¬p ∧ r. Consequently (18) is ...
Intuitionistic modal logic made explicit
... (i) W 6= ∅ (ii) R is a reflexive and transitive binary relation on W (iii) ≤ is a partial order (reflexive and transitive) on W (iv) V : Prop → P(W ), and for any atomic proposition p, the set V (p) is upwards closed, i.e., : w ≤ v, w ∈ V (p) =⇒ v ∈ V (p) (v) w ≤ v =⇒ R[v] ⊆ R[w] where R[v] := {w ∈ ...
... (i) W 6= ∅ (ii) R is a reflexive and transitive binary relation on W (iii) ≤ is a partial order (reflexive and transitive) on W (iv) V : Prop → P(W ), and for any atomic proposition p, the set V (p) is upwards closed, i.e., : w ≤ v, w ∈ V (p) =⇒ v ∈ V (p) (v) w ≤ v =⇒ R[v] ⊆ R[w] where R[v] := {w ∈ ...
1 The calculus of “predicates”
... adds to the language of the propositional calculus: names of individuals belonging to some domain or universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for p ...
... adds to the language of the propositional calculus: names of individuals belonging to some domain or universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for p ...
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... So our completeness proof cannot rely on an arbitrary attempt to find a tableau proof. After all, completeness only says that it must be possible to prove every valid formula correct with the tableau method but it doesn’t require that any attempt will succeed. And the fact that we weren’t able to f ...
... So our completeness proof cannot rely on an arbitrary attempt to find a tableau proof. After all, completeness only says that it must be possible to prove every valid formula correct with the tableau method but it doesn’t require that any attempt will succeed. And the fact that we weren’t able to f ...
Saturation of Sets of General Clauses
... 2. In the proof, it does not really matter with which negative literal an inference is performed ⇒ choose a negative literal don’t-care-nondeterministically ⇒ selection ...
... 2. In the proof, it does not really matter with which negative literal an inference is performed ⇒ choose a negative literal don’t-care-nondeterministically ⇒ selection ...
CHAPTER 1 The main subject of Mathematical Logic is
... For the human reader such representations are less convenient, so we shall stick to the use of bound variables. In the definition of “substitution of expression E 0 for variable x in expression E”, either one requires that no variable free in E 0 becomes bound by a variable-binding operator in E, wh ...
... For the human reader such representations are less convenient, so we shall stick to the use of bound variables. In the definition of “substitution of expression E 0 for variable x in expression E”, either one requires that no variable free in E 0 becomes bound by a variable-binding operator in E, wh ...
First-Order Loop Formulas for Normal Logic Programs
... compute similar loops and loop formulas every time a program is grounded on a domain. Thus extending loop formulas in logic programming to first-order case is not only theoretically interesting, but may also be of practical relevance. Specifically, in this paper, we propose notions of firstorder loo ...
... compute similar loops and loop formulas every time a program is grounded on a domain. Thus extending loop formulas in logic programming to first-order case is not only theoretically interesting, but may also be of practical relevance. Specifically, in this paper, we propose notions of firstorder loo ...
KnotandTonk 1 Preliminaries
... This raises a further parallel between inferentialist reactions to Knot and semanticist reactions to Tonk. Semanticists sometimes allege that the natural deduction rules for Tonk fail even to define a meaningful connective, on the grounds that Tonk cannot be given semantic conditions. By exactly the ...
... This raises a further parallel between inferentialist reactions to Knot and semanticist reactions to Tonk. Semanticists sometimes allege that the natural deduction rules for Tonk fail even to define a meaningful connective, on the grounds that Tonk cannot be given semantic conditions. By exactly the ...
