Notes
... 3 Natural Deduction (Gentzen, 1943) Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgments of the form ϕ1 , . . . , ϕn ` ϕ, which means that ϕ can be derived from the assumptions ϕ1 , . . . , ϕn . If ` ϕ without assumptions, then ϕ is a theorem of intu ...
... 3 Natural Deduction (Gentzen, 1943) Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgments of the form ϕ1 , . . . , ϕn ` ϕ, which means that ϕ can be derived from the assumptions ϕ1 , . . . , ϕn . If ` ϕ without assumptions, then ϕ is a theorem of intu ...
1 Preliminaries 2 Basic logical and mathematical definitions
... namely Herbrand interpretations. Let assume that the first order language L is defined on a signature Σ which contains at least one 0-ary function symbol. The set τ (Σ) of the ground terms is called the Herbrand universe for the language L, while the Herbrand base BL of L is the set of all the groun ...
... namely Herbrand interpretations. Let assume that the first order language L is defined on a signature Σ which contains at least one 0-ary function symbol. The set τ (Σ) of the ground terms is called the Herbrand universe for the language L, while the Herbrand base BL of L is the set of all the groun ...
FIRST-ORDER QUERY EVALUATION ON STRUCTURES OF
... 2.1. Gaifman locality and first-order logic. A relational signature is a tuple σ = (R1 , . . . , Rl ), each Ri being a relation symbol of arity ri . A relational structure over σ is a tuple A = A, R1A , . . . , RlA , where A = {a1 , . . . , am } is the set of elements of A and RiA is a subset of Ar ...
... 2.1. Gaifman locality and first-order logic. A relational signature is a tuple σ = (R1 , . . . , Rl ), each Ri being a relation symbol of arity ri . A relational structure over σ is a tuple A = A, R1A , . . . , RlA , where A = {a1 , . . . , am } is the set of elements of A and RiA is a subset of Ar ...
And this is just one theorem prover!
... • Learn about ATPs and ATP techniques, with an eye toward understanding how to use them in ...
... • Learn about ATPs and ATP techniques, with an eye toward understanding how to use them in ...
pdf
... itself say, 'explications' Carnap's as a means formal definitions for obtaining desired theorems. Put another ...
... itself say, 'explications' Carnap's as a means formal definitions for obtaining desired theorems. Put another ...
.pdf
... Logic gives a syntactic way to derive or certify truths that can be expressed in the language of the logic. The expressiveness of the language impacts the logic's utility |the more expressive the language, the more useful the logic (at least if the intended use is to prove theorems, as opposed to, s ...
... Logic gives a syntactic way to derive or certify truths that can be expressed in the language of the logic. The expressiveness of the language impacts the logic's utility |the more expressive the language, the more useful the logic (at least if the intended use is to prove theorems, as opposed to, s ...
Topological Completeness of First-Order Modal Logic
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...
Propositional logic
... Definition: an assignment to a set V of variables is a function s: V Æ {T,F}. Each assignment is inductively extended to apply to wffs. For wffs a and b • s(ÿa) = ÿs(a), • s(aŸb) = s(a) Ÿ s(b), • s(a⁄b) = s(a) ⁄ s(b), • s(afib) = s(a) fi s(b), • s(aÛb) = s(a) Û s(b), and • s(T) = T, s(F) = F, Defini ...
... Definition: an assignment to a set V of variables is a function s: V Æ {T,F}. Each assignment is inductively extended to apply to wffs. For wffs a and b • s(ÿa) = ÿs(a), • s(aŸb) = s(a) Ÿ s(b), • s(a⁄b) = s(a) ⁄ s(b), • s(afib) = s(a) fi s(b), • s(aÛb) = s(a) Û s(b), and • s(T) = T, s(F) = F, Defini ...
Adding the Everywhere Operator to Propositional Logic (pdf file)
... as follows. First, extend language C to a language C . The formulas of C will include those of C; the original formulas of C will be called concrete formulas. Then, we give an axiomatization of C —using a finite number of axioms. Finally, we show that the theorems of C that are concrete are preci ...
... as follows. First, extend language C to a language C . The formulas of C will include those of C; the original formulas of C will be called concrete formulas. Then, we give an axiomatization of C —using a finite number of axioms. Finally, we show that the theorems of C that are concrete are preci ...
Comments on predicative logic
... This is a nice alternative, but we discuss another one. Namely, restrict the range of the ∀-elimination rule to atomic formulas. For lack of a better name, let us call this restricted calculus atomic PSOLi . Observe that Theorem 1 still goes through with atomic PSOLi (instead of predicative PSOLi ). ...
... This is a nice alternative, but we discuss another one. Namely, restrict the range of the ∀-elimination rule to atomic formulas. For lack of a better name, let us call this restricted calculus atomic PSOLi . Observe that Theorem 1 still goes through with atomic PSOLi (instead of predicative PSOLi ). ...
Elements of Modal Logic - University of Victoria
... Where L is a logic, let DL be the set of all L-maximal consistent sets. The binary relation RL is defined on DL as follows: RL xy iff ∀α, α ∈ x ⇒ α ∈ y We also define a valuation, VL (n) = {x | pn ∈ x } The model ML = (DL , RL , VL ) is the canonical model of the logic L. Theorem 5 (Fundamental Theore ...
... Where L is a logic, let DL be the set of all L-maximal consistent sets. The binary relation RL is defined on DL as follows: RL xy iff ∀α, α ∈ x ⇒ α ∈ y We also define a valuation, VL (n) = {x | pn ∈ x } The model ML = (DL , RL , VL ) is the canonical model of the logic L. Theorem 5 (Fundamental Theore ...
Completeness through Flatness in Two
... particular, to the flow of time ω of the natural numbers. There are two reasons to do so: first of all, for these structures we can prove a completeness result for flat validity of a system without any non-orthodox derivation rules. An interesting aspect of the proof is that it essentially uses the ...
... particular, to the flow of time ω of the natural numbers. There are two reasons to do so: first of all, for these structures we can prove a completeness result for flat validity of a system without any non-orthodox derivation rules. An interesting aspect of the proof is that it essentially uses the ...
Non-classical metatheory for non-classical logics
... which classical logic is provably sound and complete by its own lights. In order to meet the challenge in a non-classical setting, I propose that we investigate the prospects of a faithful model theory for the non-classical logic. One requirement a faithful model theory must meet is to be able to d ...
... which classical logic is provably sound and complete by its own lights. In order to meet the challenge in a non-classical setting, I propose that we investigate the prospects of a faithful model theory for the non-classical logic. One requirement a faithful model theory must meet is to be able to d ...
And this is just one theorem prover!
... • Learn about ATPs and ATP techniques, with an eye toward understanding how to use them in ...
... • Learn about ATPs and ATP techniques, with an eye toward understanding how to use them in ...
Further Exercises for the tutorials on logic in CS 3511 1. Find a
... c. ∀x∀y(xy ≥ x) Let x be any integer and y any negative integer 2. An advert in an Aberdeen supermarket says: ”All frozen foods reduced by up to 70%”. a) Is the sentence false if all frozen foods have been reduced by 1%? On my readings (see below), it is true. This indicates that there is something ...
... c. ∀x∀y(xy ≥ x) Let x be any integer and y any negative integer 2. An advert in an Aberdeen supermarket says: ”All frozen foods reduced by up to 70%”. a) Is the sentence false if all frozen foods have been reduced by 1%? On my readings (see below), it is true. This indicates that there is something ...
On the paradoxes of set theory
... Theorem B.--The series of ordirials up to and including a given ordinal number, say~, has ordinal number ~ 1. Theorem C.--Tho series of all ordinal numbers is well ordered and hence, has an ordinal number, say.fl. The Burali.-Forti paradox accerbs the incompatibility of the above three theorems. ...
... Theorem B.--The series of ordirials up to and including a given ordinal number, say~, has ordinal number ~ 1. Theorem C.--Tho series of all ordinal numbers is well ordered and hence, has an ordinal number, say.fl. The Burali.-Forti paradox accerbs the incompatibility of the above three theorems. ...
485-291 - Wseas.us
... resulting structure A is called a random structure. For convenience, we assume that the universes of our random structures are of the form n ={ 0,1,2,... } for some ω n. So for fixed n, our probability space is the set of all structures on n (labelled) elements (in other words, elementary events a ...
... resulting structure A is called a random structure. For convenience, we assume that the universes of our random structures are of the form n ={ 0,1,2,... } for some ω n. So for fixed n, our probability space is the set of all structures on n (labelled) elements (in other words, elementary events a ...
Interpolation for McCain
... here. According to Hintikka [1976; 1972], and Harrah [1975] a question can be regarded as denoting its set of possible answers (out of which an appropriate answer selects one). For example, in Harrah’s system our @P would be called the “assertive core” of the question, whereas his indicated replies ...
... here. According to Hintikka [1976; 1972], and Harrah [1975] a question can be regarded as denoting its set of possible answers (out of which an appropriate answer selects one). For example, in Harrah’s system our @P would be called the “assertive core” of the question, whereas his indicated replies ...
Computing Default Extensions by Reductions on OR
... The default α : β / γ is represented by its Konolige translation Bα ∧ Mβ ⊃ γ. If α or β are , i.e. the default is prerequisite-free or justification-free resp., we drop that conjunct. Hence : β / γ translates to Mβ ⊃ γ, while α : / γ translates to Bα ⊃ γ. To translate the whole default theory, ...
... The default α : β / γ is represented by its Konolige translation Bα ∧ Mβ ⊃ γ. If α or β are , i.e. the default is prerequisite-free or justification-free resp., we drop that conjunct. Hence : β / γ translates to Mβ ⊃ γ, while α : / γ translates to Bα ⊃ γ. To translate the whole default theory, ...
Scoring Rubric for Assignment 1
... unclear. Theory is not relevant or only relevant for some aspects; theory is not clearly articulated and/or has incorrect or incomplete components. Relationship between theory and research is unclear or inaccurate, major errors in the logic are present. 0 – 4 pts Conclusion may not be clear and the ...
... unclear. Theory is not relevant or only relevant for some aspects; theory is not clearly articulated and/or has incorrect or incomplete components. Relationship between theory and research is unclear or inaccurate, major errors in the logic are present. 0 – 4 pts Conclusion may not be clear and the ...
The semantics of predicate logic
... statement has the form ∀x∀yφ, refuting it simply amounts to finding specific assignments to x and y such that ¬φ under these ...
... statement has the form ∀x∀yφ, refuting it simply amounts to finding specific assignments to x and y such that ¬φ under these ...
Problem Set 3
... Recall from Problem Set Two that a tournament graph is a directed graph with n ≥ 1 nodes where there is exactly one edge between any pair of distinct nodes and there are no self-loops. Prove that if a tournament graph contains a cycle of any length, then it contains a cycle of length three. ...
... Recall from Problem Set Two that a tournament graph is a directed graph with n ≥ 1 nodes where there is exactly one edge between any pair of distinct nodes and there are no self-loops. Prove that if a tournament graph contains a cycle of any length, then it contains a cycle of length three. ...
Propositional and Predicate Logic - IX
... Soundness - proof (cont.) Otherwise τn+1 is formed from τn by appending an atomic tableau to Vn for some entry P on Vn . By induction we know that An agrees with P. (i) If P is formed by a logical connective, we take An+1 = An and verify that Vn can always be extended to a branch Vn+1 agreeing with ...
... Soundness - proof (cont.) Otherwise τn+1 is formed from τn by appending an atomic tableau to Vn for some entry P on Vn . By induction we know that An agrees with P. (i) If P is formed by a logical connective, we take An+1 = An and verify that Vn can always be extended to a branch Vn+1 agreeing with ...
A course in Mathematical Logic
... Terms and formulas are interpreted in a model. Definition 8. (Definition of a model) Let L be a language. An L-model M is given by a set M of elements (called the universe of the model) and 1. For every function symbol f ∈ L of arity n, a function f M : M n → M ; 2. For every relation symbol R ∈ L o ...
... Terms and formulas are interpreted in a model. Definition 8. (Definition of a model) Let L be a language. An L-model M is given by a set M of elements (called the universe of the model) and 1. For every function symbol f ∈ L of arity n, a function f M : M n → M ; 2. For every relation symbol R ∈ L o ...