Circuit principles and weak pigeonhole variants
... the context of the propositional proof complexity of the surjective pigeonhole principle (Krajı́ček 2004). ITER(PV , {||id||O(1) }) contains PV and like Θb2 is contained in the class Σb2 . We show that over R22 , mWPHP (ITER(PV , {||id||O(1) })) is equivalent to the k existence of a string S < 22n ...
... the context of the propositional proof complexity of the surjective pigeonhole principle (Krajı́ček 2004). ITER(PV , {||id||O(1) }) contains PV and like Θb2 is contained in the class Σb2 . We show that over R22 , mWPHP (ITER(PV , {||id||O(1) })) is equivalent to the k existence of a string S < 22n ...
Beyond Quantifier-Free Interpolation in Extensions of Presburger
... arithmetic), denoted QPA. Combined with uninterpreted predicates (UP) and uninterpreted functions (UF), this allows us to encode the theory of extensional arrays (AR), using uninterpreted function symbols for read and write operations. Our interpolation procedure extracts an interpolant directly fro ...
... arithmetic), denoted QPA. Combined with uninterpreted predicates (UP) and uninterpreted functions (UF), this allows us to encode the theory of extensional arrays (AR), using uninterpreted function symbols for read and write operations. Our interpolation procedure extracts an interpolant directly fro ...
What is "formal logic"?
... according to the precise definition of this notion related to the theory of recursion: “due to A.M.Turing’s work (Turing 1937) a precise and unquestionably adequate definition of the general notion of formal system can now be given. In my opinion the term “formal system” or “formalism” should never ...
... according to the precise definition of this notion related to the theory of recursion: “due to A.M.Turing’s work (Turing 1937) a precise and unquestionably adequate definition of the general notion of formal system can now be given. In my opinion the term “formal system” or “formalism” should never ...
The Surprise Examination Paradox and the Second Incompleteness
... where PrT,S (pAq) expresses the provability of a formula A from the formula S in the theory T , (formally, PrT,S (pAq) is the formula: there exists w that is the Gödel number of a T -proof for the formula A from the formula S). Note that the formula S is self-referential. Nevertheless, it is well k ...
... where PrT,S (pAq) expresses the provability of a formula A from the formula S in the theory T , (formally, PrT,S (pAq) is the formula: there exists w that is the Gödel number of a T -proof for the formula A from the formula S). Note that the formula S is self-referential. Nevertheless, it is well k ...
Quantified Equilibrium Logic and the First Order Logic of Here
... programs. It also provides a useful logical foundation for answer set programming (ASP), the fast developing paradigm for declarative programming based on the answer set semantics [1]. A costly component of the computation of answer sets is the process of grounding a program containing variables by ...
... programs. It also provides a useful logical foundation for answer set programming (ASP), the fast developing paradigm for declarative programming based on the answer set semantics [1]. A costly component of the computation of answer sets is the process of grounding a program containing variables by ...
page 135 LOGIC IN WHITEHEAD`S UNIVERSAL ALGEBRA
... There are various ways of ‘being together’, that is, various ways of building classes, and this makes ‘togetherness’ an ambiguous notion. The extensionalist Whitehead notes that the particular mode of ‘togetherness’ is already an intension that infects the composite entity. Therefore, in mathematics ...
... There are various ways of ‘being together’, that is, various ways of building classes, and this makes ‘togetherness’ an ambiguous notion. The extensionalist Whitehead notes that the particular mode of ‘togetherness’ is already an intension that infects the composite entity. Therefore, in mathematics ...
pdf
... where · on sets is given by A · B = {u · v | u ∈ A, v ∈ B}, and An is defined inductively as A0 = RM (1) and An+1 = An · A. The image of the interpretation RM together with the operations ∪, ·, ∗ , ∅, {1M } is the algebra of regular sets over M , denoted by Reg M . If M is the free monoid Σ ∗ , the ...
... where · on sets is given by A · B = {u · v | u ∈ A, v ∈ B}, and An is defined inductively as A0 = RM (1) and An+1 = An · A. The image of the interpretation RM together with the operations ∪, ·, ∗ , ∅, {1M } is the algebra of regular sets over M , denoted by Reg M . If M is the free monoid Σ ∗ , the ...
Logic: Introduction - Department of information engineering and
... Modern Logic teaches us that one claim is a logical consequence of another if there is no way the latter could be true without the former also being true. It is also used to disconfirm a theory if a particular claim is a logical consequence of a theory, and we discover that the claim is false, then ...
... Modern Logic teaches us that one claim is a logical consequence of another if there is no way the latter could be true without the former also being true. It is also used to disconfirm a theory if a particular claim is a logical consequence of a theory, and we discover that the claim is false, then ...
Lecture 4 - Michael De
... Assume that instead of interpreting i as a gap, we interpret it as a glut. But then taking the value i means being both true and false, and hence true, and hence designated. So we need to add i to D. The resulting logic is called LP, or the Logic of Paradox, as Priest originally called it. It is the ...
... Assume that instead of interpreting i as a gap, we interpret it as a glut. But then taking the value i means being both true and false, and hence true, and hence designated. So we need to add i to D. The resulting logic is called LP, or the Logic of Paradox, as Priest originally called it. It is the ...
Proof and computation rules
... Some parts of the computation theory are needed here, such as the notion that all the terms used in the rules can be reduced to head normal form. Defining that reduction requires identifying the principal argument places in each term. We give this definition in the next section. The reduction rules ar ...
... Some parts of the computation theory are needed here, such as the notion that all the terms used in the rules can be reduced to head normal form. Defining that reduction requires identifying the principal argument places in each term. We give this definition in the next section. The reduction rules ar ...
full text (.pdf)
... the premises are restricted to commutativity conditions of the form pq = qp for atomic actions p and q, the validity problem is Π01 -complete [13]. However, sometimes the validity of universal Horn formulas with premises of a certain restricted form can be efficiently reduced to the equational theor ...
... the premises are restricted to commutativity conditions of the form pq = qp for atomic actions p and q, the validity problem is Π01 -complete [13]. However, sometimes the validity of universal Horn formulas with premises of a certain restricted form can be efficiently reduced to the equational theor ...
Kripke Models of Transfinite Provability Logic
... Kripke frames for the closed fragment of GLPω , which contains no propositional variables (only ⊥). This fragment, which we denote GLP0ω , is still expressive enough to be used in Beklemishev’s ordinal analysis. Our goal is to extend Ignatiev’s construction for GLP0ω to GLP0Λ , where Λ is an arbitra ...
... Kripke frames for the closed fragment of GLPω , which contains no propositional variables (only ⊥). This fragment, which we denote GLP0ω , is still expressive enough to be used in Beklemishev’s ordinal analysis. Our goal is to extend Ignatiev’s construction for GLP0ω to GLP0Λ , where Λ is an arbitra ...
Logical Argument
... Converse Fallacy of Accident. To argue from a special case to a general rule; a deductive fallacy that can occur when an exception to a generalization is wrongly called for. For example: If we allow people with glaucoma to use medical marijuana then everyone should be allowed to use marijuana. Peopl ...
... Converse Fallacy of Accident. To argue from a special case to a general rule; a deductive fallacy that can occur when an exception to a generalization is wrongly called for. For example: If we allow people with glaucoma to use medical marijuana then everyone should be allowed to use marijuana. Peopl ...
Yablo`s paradox
... regress grounded out in some claim not concerning the sequence, then s could be defined recursively, and it would not require a circular construction to define it. But the sequence is infinite; and it does.7 Given infinitary fantasies, the circularity can be further masked. Consider, for example, th ...
... regress grounded out in some claim not concerning the sequence, then s could be defined recursively, and it would not require a circular construction to define it. But the sequence is infinite; and it does.7 Given infinitary fantasies, the circularity can be further masked. Consider, for example, th ...
Propositions as [Types] - Research Showcase @ CMU
... types of Maietti [Mai98], in a suitable setting. Palmgren [Pal01] formulated a BHK interpretation of intuitionistic logic and used image factorizations, which are used in the semantics of our bracket types, to relate the BHK interpretation to the standard category-theoretic interpretation of proposi ...
... types of Maietti [Mai98], in a suitable setting. Palmgren [Pal01] formulated a BHK interpretation of intuitionistic logic and used image factorizations, which are used in the semantics of our bracket types, to relate the BHK interpretation to the standard category-theoretic interpretation of proposi ...
vmcai - of Philipp Ruemmer
... as integer arithmetic with uninterpreted predicates, often generate interpolants with quantifiers, which makes subsequent calls to decision procedures involving these interpolants expensive. This is not by accident. In fact, in this paper we first show that interpolation of QPA+UP in general require ...
... as integer arithmetic with uninterpreted predicates, often generate interpolants with quantifiers, which makes subsequent calls to decision procedures involving these interpolants expensive. This is not by accident. In fact, in this paper we first show that interpolation of QPA+UP in general require ...
Equality in the Presence of Apartness: An Application of Structural
... The idea of an apartness relation in place of an equality relation appears first in Brouwer’s works on the intuitionistic continuum from the early 1920s. One of the basic insights of intuitionism was that the equality of two real numbers a, b is not decidable: The verification of a = b may require tha ...
... The idea of an apartness relation in place of an equality relation appears first in Brouwer’s works on the intuitionistic continuum from the early 1920s. One of the basic insights of intuitionism was that the equality of two real numbers a, b is not decidable: The verification of a = b may require tha ...
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 : ...
A Brief Introduction to the Intuitionistic Propositional Calculus
... Problem 1 Prove that α ⇒ (β ⇒ γ) `I (α ∧ β) ⇒ γ. Problem 2 Show that α ⇒ β 6`I ¬α ∨ β by demonstrating that there exists a Kripke model K = (W, ≤, |=) and a world w ∈ W such that w |= α ⇒ β, but w 6|= ¬α ∨ β. Problem 3 Show that world w1 in the simple Kripke model in Section 4 does not satisfy Peirc ...
... Problem 1 Prove that α ⇒ (β ⇒ γ) `I (α ∧ β) ⇒ γ. Problem 2 Show that α ⇒ β 6`I ¬α ∨ β by demonstrating that there exists a Kripke model K = (W, ≤, |=) and a world w ∈ W such that w |= α ⇒ β, but w 6|= ¬α ∨ β. Problem 3 Show that world w1 in the simple Kripke model in Section 4 does not satisfy Peirc ...
The Diagonal Lemma Fails in Aristotelian Logic
... A is false, but rather (Ex)Fx “is a necessary precondition not merely of of the truth of what is said, but of its being either true or false.” [Original italics] (Strawson, p. 174) We will, however, do one better and take the entire (∃x)Fx & (Ex)~Gx as the presupposition. Then A is neither true nor ...
... A is false, but rather (Ex)Fx “is a necessary precondition not merely of of the truth of what is said, but of its being either true or false.” [Original italics] (Strawson, p. 174) We will, however, do one better and take the entire (∃x)Fx & (Ex)~Gx as the presupposition. Then A is neither true nor ...
Reducing Propositional Theories in Equilibrium Logic to
... for knowledge representation (see eg [3,8,23]) but proposals for an adequate semantics have differed. Recently however Ferraris [5] has shown how, by modifying somewhat the definition of answer sets for nested programs, a natural extension for arbitrary propositional theories can be obtained. Though ...
... for knowledge representation (see eg [3,8,23]) but proposals for an adequate semantics have differed. Recently however Ferraris [5] has shown how, by modifying somewhat the definition of answer sets for nested programs, a natural extension for arbitrary propositional theories can be obtained. Though ...
Symbolic Logic II
... In the case of Lukasiewicz-validity, we have designated {1} in our definition of validity and have so defined validity as “always true”. If we had wanted a definition of validity to mean never false, then we would have to designate {1, #}. ...
... In the case of Lukasiewicz-validity, we have designated {1} in our definition of validity and have so defined validity as “always true”. If we had wanted a definition of validity to mean never false, then we would have to designate {1, #}. ...
CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction
... Theorem 3.1 (Deduction Theorem for H2 ) For any subset Γ of the set of formulas F of H2 and for any formulas A, B ∈ F , Γ, A `H2 B if and only if Γ `H2 (A ⇒ B). In particular, A `H2 B if and only if `H2 (A ⇒ B). Obviously, the axioms A1, A2, A3 are tautologies, and the Modus Ponens rule leads from t ...
... Theorem 3.1 (Deduction Theorem for H2 ) For any subset Γ of the set of formulas F of H2 and for any formulas A, B ∈ F , Γ, A `H2 B if and only if Γ `H2 (A ⇒ B). In particular, A `H2 B if and only if `H2 (A ⇒ B). Obviously, the axioms A1, A2, A3 are tautologies, and the Modus Ponens rule leads from t ...
Glivenko sequent classes in the light of structural proof theory
... in isolating such classes lies in the fact that it may be easier to prove theorems by the use of classical rather than intuitionistic logic. Further, since a proof in intuitionistic logic can be associated to a lambda term and thus obtain a computational meaning, such results have more recently been ...
... in isolating such classes lies in the fact that it may be easier to prove theorems by the use of classical rather than intuitionistic logic. Further, since a proof in intuitionistic logic can be associated to a lambda term and thus obtain a computational meaning, such results have more recently been ...