Lindenbaum lemma for infinitary logics
... Lindenbaum lemma says that for any finitary logic ` (i.e., a finitary substitution-invariant consequence relation over the set of formulas of a given language) each theory (i.e., a set of formulas closed under `) not containing a formula ϕ can be extended into a maximal theory not containing ϕ. The ...
... Lindenbaum lemma says that for any finitary logic ` (i.e., a finitary substitution-invariant consequence relation over the set of formulas of a given language) each theory (i.e., a set of formulas closed under `) not containing a formula ϕ can be extended into a maximal theory not containing ϕ. The ...
completeness theorem for a first order linear
... for the rst order temporal logics with since and until over linear time and rationals were given in [16]. In the case of FOLTL (and similarly when the ow of time is isomorphic to reals or integers) the set of valid formulas is not recursively enumerable, and there is no recursive axiomatization of ...
... for the rst order temporal logics with since and until over linear time and rationals were given in [16]. In the case of FOLTL (and similarly when the ow of time is isomorphic to reals or integers) the set of valid formulas is not recursively enumerable, and there is no recursive axiomatization of ...
The theorem, it`s meaning and the central concepts
... contradiction) via the two true sentences and the rules of deduction – which mean that every sentence would be true in the system. Which is not particularly smart ω-consistency is a slightly stronger version of consistency. The incompleteness theorem says, that if a system contains simple arithmetic ...
... contradiction) via the two true sentences and the rules of deduction – which mean that every sentence would be true in the system. Which is not particularly smart ω-consistency is a slightly stronger version of consistency. The incompleteness theorem says, that if a system contains simple arithmetic ...
Chapter 2
... next states in a given move. Thus several computations are possible on a given input. A word is accepted if there is at least one computation that ends in an accepting state. Nondeterministic fsa (nfsa) accept the same set of languages as fsa. However, the number of states in the equivalent determin ...
... next states in a given move. Thus several computations are possible on a given input. A word is accepted if there is at least one computation that ends in an accepting state. Nondeterministic fsa (nfsa) accept the same set of languages as fsa. However, the number of states in the equivalent determin ...
ppt
... a logical system, can all other facts be derived using the laws of math/logic? Punch line: No! Any formal system breaks down; there are truths that can not be derived ...
... a logical system, can all other facts be derived using the laws of math/logic? Punch line: No! Any formal system breaks down; there are truths that can not be derived ...
PDF
... An important theme in computable model theory is the study of computable models of complete first-order theories. More precisely, given a complete first-order theory T , one would like to know which models of T have computable copies and which do not. A special case of interest is when T is an ℵ1 -c ...
... An important theme in computable model theory is the study of computable models of complete first-order theories. More precisely, given a complete first-order theory T , one would like to know which models of T have computable copies and which do not. A special case of interest is when T is an ℵ1 -c ...
Herbrand Theorem, Equality, and Compactness
... is propositionally unsatisfiable, then by the above lemma, Φ0 is unsatisfiable, and hence Φ is unsatisfiable (because each ground instance is a logical consequence of Φ). Proof (Completeness direction of Herbrand Theorem): We prove the contrapositive: If every finite set of ground instances of Φ is ...
... is propositionally unsatisfiable, then by the above lemma, Φ0 is unsatisfiable, and hence Φ is unsatisfiable (because each ground instance is a logical consequence of Φ). Proof (Completeness direction of Herbrand Theorem): We prove the contrapositive: If every finite set of ground instances of Φ is ...
paper by David Pierce
... (2) to prove that all elements of those sets have certain properties; (3) to define functions on those sets. These three techniques are often confused, but they should not be. Clarity here can prevent mathematical mistakes; it can also highlight important concepts and results such as Fermat’s (Little ...
... (2) to prove that all elements of those sets have certain properties; (3) to define functions on those sets. These three techniques are often confused, but they should not be. Clarity here can prevent mathematical mistakes; it can also highlight important concepts and results such as Fermat’s (Little ...
notes
... Let P be a propositions containing the (distinct) atomic formulas A 1 , . . . , An and v1 , . . . v2n its interpretations. We denote with v P the boolean function associated with P , i.e. vP : {0, 1}n → {0, 1} is defined as follows: for each (a 1 , . . . , an ), ai ∈ {0, 1}, there exists i ∈ {1, . ...
... Let P be a propositions containing the (distinct) atomic formulas A 1 , . . . , An and v1 , . . . v2n its interpretations. We denote with v P the boolean function associated with P , i.e. vP : {0, 1}n → {0, 1} is defined as follows: for each (a 1 , . . . , an ), ai ∈ {0, 1}, there exists i ∈ {1, . ...
Autoepistemic Logic and Introspective Circumscription
... By the theorem, I U S b i8 a model of A*(P, BP). Consequently, S b is a model of 3pA*(p, BP). Now take a model U of 3pA*(p, BP), and let I be a subset of P such that I U U is a model of A*(P, BP). By the theorem, there exist an autoepistemic model S of A(P, BP) and an interpretation J E S such that ...
... By the theorem, I U S b i8 a model of A*(P, BP). Consequently, S b is a model of 3pA*(p, BP). Now take a model U of 3pA*(p, BP), and let I be a subset of P such that I U U is a model of A*(P, BP). By the theorem, there exist an autoepistemic model S of A(P, BP) and an interpretation J E S such that ...
Using model theory for grammatical inference
... 3. Motivation for the Present Work Why model theoretic treatments of formal languages? There are two reasons. The first is that it provides a bridge to a range of techniques that have been used to learn concepts expressed as logical sentences. The second is that there are many more kinds of models o ...
... 3. Motivation for the Present Work Why model theoretic treatments of formal languages? There are two reasons. The first is that it provides a bridge to a range of techniques that have been used to learn concepts expressed as logical sentences. The second is that there are many more kinds of models o ...
Jacques Herbrand (1908 - 1931) Principal writings in logic
... ES(A,p) that makes the expansion true and assigns the numerical value q to the constant c. œxœy∑zı(x,y,z) expresses the existence, for any p and q, of interpretations that make ES(A,p) true, and give the constant c the value q. If the theory is a true theory of arithmetic, then œxœy∑zı(x,y,z) is tru ...
... ES(A,p) that makes the expansion true and assigns the numerical value q to the constant c. œxœy∑zı(x,y,z) expresses the existence, for any p and q, of interpretations that make ES(A,p) true, and give the constant c the value q. If the theory is a true theory of arithmetic, then œxœy∑zı(x,y,z) is tru ...
an interpretation of aristotle`s syllogistic and a certain fragment of set
... is aj ). This is not a syllogistic thesis. If, now, in some interpretation would be allowed using only at most k objects (i.e. k natural numbers in Leibnitz’s interpretation and k sets in Slupecki’s interpretation), then formula A would not be rejected. This can be seen especially well in Slupecki’s ...
... is aj ). This is not a syllogistic thesis. If, now, in some interpretation would be allowed using only at most k objects (i.e. k natural numbers in Leibnitz’s interpretation and k sets in Slupecki’s interpretation), then formula A would not be rejected. This can be seen especially well in Slupecki’s ...
full text (.pdf)
... (i) is Σ1α -complete and problem (ii) is Π2α -complete. Both problems remain hard for their respective complexity classes even if Σ is restricted to contain only a single constant, a single unary function symbol, and a single monadic predicate. It follows from (ii) that there can exist no relatively ...
... (i) is Σ1α -complete and problem (ii) is Π2α -complete. Both problems remain hard for their respective complexity classes even if Σ is restricted to contain only a single constant, a single unary function symbol, and a single monadic predicate. It follows from (ii) that there can exist no relatively ...
DOC - John Woods
... 4. If A, B are formulas, so too are AB A B A B A B. Further Parts of the Grammar a. Scope. If αA is a formula, then A is the scope of . b. Freedom and bondage of occurrences. An occurrence of a variable α in a formula is bound in a formula iff either it is the variable of a quantifier or it o ...
... 4. If A, B are formulas, so too are AB A B A B A B. Further Parts of the Grammar a. Scope. If αA is a formula, then A is the scope of . b. Freedom and bondage of occurrences. An occurrence of a variable α in a formula is bound in a formula iff either it is the variable of a quantifier or it o ...
Morley`s number of countable models
... awkward in Lω1 ,ω . We could define it explicitly but instead we simply assume there is a function Subst(i, j, ϕ) defined for all i, j ∈ ω and all formulas ϕ in Lω1 ,ω such that SB1 Subst(i, i, ϕ) = ϕ, SB2 i 6= j ⇒ vi does not occur free in Subst(i, j, ϕ), SB3 |= vi = vj → (ϕ ↔ Subst(i, j, ϕ)). The ...
... awkward in Lω1 ,ω . We could define it explicitly but instead we simply assume there is a function Subst(i, j, ϕ) defined for all i, j ∈ ω and all formulas ϕ in Lω1 ,ω such that SB1 Subst(i, i, ϕ) = ϕ, SB2 i 6= j ⇒ vi does not occur free in Subst(i, j, ϕ), SB3 |= vi = vj → (ϕ ↔ Subst(i, j, ϕ)). The ...
PDF
... propositional case. However, I is now a first-order valuation over U instead of a boolean valuation and the definition of |=, in contrast to S-value, is based on the interpretation of quantifiers as well. The proof of this fact is very similar to the one we had before. It just needs to consider γ an ...
... propositional case. However, I is now a first-order valuation over U instead of a boolean valuation and the definition of |=, in contrast to S-value, is based on the interpretation of quantifiers as well. The proof of this fact is very similar to the one we had before. It just needs to consider γ an ...
The Anti-Foundation Axiom in Constructive Set Theories
... subsection, it is pivotal to observe that, unlike Aczel’s type of iterative sets, a strong system type is not required to have an inductive structure, i.e. there need not be an elimination rule for it. The strength of a type theory ML1 with the type constructors Π, Σ, +, I, N, N0 , N1 and one univer ...
... subsection, it is pivotal to observe that, unlike Aczel’s type of iterative sets, a strong system type is not required to have an inductive structure, i.e. there need not be an elimination rule for it. The strength of a type theory ML1 with the type constructors Π, Σ, +, I, N, N0 , N1 and one univer ...
An Independence Result For Intuitionistic Bounded Arithmetic
... theory for all intuitionistic theories we will mention. Let M and N be two models of BASIC. Let Log(M ) = {a ∈ M : ∃b ∈ M a ≤ |b|}. N is called a weak end extension of M and it is written that M ⊆w.e. N , if N extends M and Log(N ) is an end extension of Log(M ). This means that, for all a ∈ Log(M ) ...
... theory for all intuitionistic theories we will mention. Let M and N be two models of BASIC. Let Log(M ) = {a ∈ M : ∃b ∈ M a ≤ |b|}. N is called a weak end extension of M and it is written that M ⊆w.e. N , if N extends M and Log(N ) is an end extension of Log(M ). This means that, for all a ∈ Log(M ) ...
Set Theory (MATH 6730) HOMEWORK 1 (Due on February 6, 2017
... Let Γ ∪ {ϕ} be a set of LC -formulas, and let d be a constant symbol not in C. If Γ ∪ {Subf xd (ϕ)} is an inconsistent set of formulas in the language LC∪{d} , then Γ ∪ {∃x ϕ} is an inconsistent set of formulas in the language LC . 2. Let Γ be a set of LC -formulas. (i) Show that the following condi ...
... Let Γ ∪ {ϕ} be a set of LC -formulas, and let d be a constant symbol not in C. If Γ ∪ {Subf xd (ϕ)} is an inconsistent set of formulas in the language LC∪{d} , then Γ ∪ {∃x ϕ} is an inconsistent set of formulas in the language LC . 2. Let Γ be a set of LC -formulas. (i) Show that the following condi ...
LOGIC AND PSYCHOTHERAPY
... 1. In 1956 Bateson and his colleagues proposed the “double-bind theory”: “...an important factor in the development of schizophrenic thought disorder is the constant subjection of an individual to a so-called double-bind situation which includes the following elements. 1) The individual has an inten ...
... 1. In 1956 Bateson and his colleagues proposed the “double-bind theory”: “...an important factor in the development of schizophrenic thought disorder is the constant subjection of an individual to a so-called double-bind situation which includes the following elements. 1) The individual has an inten ...
Bound and Free Variables Theorems and Proofs
... by plugging in arbitrary formulas for A, B, C A proof is a sequence of formulas A1, A2, A3, . . . such that each Ai is either 1. An instance of Ax1 and Ax2 2. Follows from previous formulas by applying MP • that is, there exist Aj , Ak with j, k < i such that Aj has the form A ⇒ B, Ak is A and Ai is ...
... by plugging in arbitrary formulas for A, B, C A proof is a sequence of formulas A1, A2, A3, . . . such that each Ai is either 1. An instance of Ax1 and Ax2 2. Follows from previous formulas by applying MP • that is, there exist Aj , Ak with j, k < i such that Aj has the form A ⇒ B, Ak is A and Ai is ...
full text (.pdf)
... COROLLARY 1.6 (Zero-one law). For anyfirst-order sentence 40, limn ~ F R A C T I O N (40, n) exists and equals zero or one. Notice that this theorem and corollary would be false if we had permitted zero-place relation symbols in a, for such a symbol would be a 40 with F R A C T I O N (40, n) = ½ for ...
... COROLLARY 1.6 (Zero-one law). For anyfirst-order sentence 40, limn ~ F R A C T I O N (40, n) exists and equals zero or one. Notice that this theorem and corollary would be false if we had permitted zero-place relation symbols in a, for such a symbol would be a 40 with F R A C T I O N (40, n) = ½ for ...
PDF
... An immediate consequence of Theorem 1 is Church’s undecidability theorem, which states that first-order logic is undecidable. We can easily reduce the decidability problem for first-order logic to the decidability problem for theories that can represent the computable functions, provided we find a ...
... An immediate consequence of Theorem 1 is Church’s undecidability theorem, which states that first-order logic is undecidable. We can easily reduce the decidability problem for first-order logic to the decidability problem for theories that can represent the computable functions, provided we find a ...
FOR HIGHER-ORDER RELEVANT LOGIC
... and theories. Thus far, γ has at most been proved, in [2], for first-order relevant logics. (Related methods are applied, in [1], to yield a new proof of elementary logic, the classical adaptation of the γ-techniques as refined in [3] having been carried out by Dunn.) It is time to move up; at the h ...
... and theories. Thus far, γ has at most been proved, in [2], for first-order relevant logics. (Related methods are applied, in [1], to yield a new proof of elementary logic, the classical adaptation of the γ-techniques as refined in [3] having been carried out by Dunn.) It is time to move up; at the h ...