Lecture Notes
... Problem: Given a structure M , a world w of M and a formula φ, decide if M, w |= φ. Theorem: For finite M , and φ ∈ L{K1 ,...,Kn ,CG } there exists an algorithm that solves the problem in time linear in |M | · |φ|, where |M | and |φ| are the amount of space needed to write down M and φ, ...
... Problem: Given a structure M , a world w of M and a formula φ, decide if M, w |= φ. Theorem: For finite M , and φ ∈ L{K1 ,...,Kn ,CG } there exists an algorithm that solves the problem in time linear in |M | · |φ|, where |M | and |φ| are the amount of space needed to write down M and φ, ...
1 LOGICAL CONSEQUENCE: A TURN IN STYLE KOSTA DO SEN
... This talk summarizes some of the things that contemporary logic and, in particular, proof theory stemming from Gentzen have to say about the notion of consequence. It starts from very elementary facts, the understanding of which doesn’t require any technical knowledge, to reach the more specialized ...
... This talk summarizes some of the things that contemporary logic and, in particular, proof theory stemming from Gentzen have to say about the notion of consequence. It starts from very elementary facts, the understanding of which doesn’t require any technical knowledge, to reach the more specialized ...
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... now on), if awareness is generated by primitive propositions, then it is impossible for an agent to explicitly know that he is unaware of a specific fact. Nevertheless, an agent may well be aware that there are relevant facts that he is unaware of. For example, primary-care physicians know that spec ...
... now on), if awareness is generated by primitive propositions, then it is impossible for an agent to explicitly know that he is unaware of a specific fact. Nevertheless, an agent may well be aware that there are relevant facts that he is unaware of. For example, primary-care physicians know that spec ...
Simple multiplicative proof nets with units
... conclusion and no premise; however, ⊥-links are treated like 0-ary ?-links, i.e., they must be given a default jump. Sequentialisation is immediate. At first sight, cut elimination is unproblematic: replace a cut between conclusions 1 and ⊥ of zero-ary links with. . . nothing. But we notice a new pr ...
... conclusion and no premise; however, ⊥-links are treated like 0-ary ?-links, i.e., they must be given a default jump. Sequentialisation is immediate. At first sight, cut elimination is unproblematic: replace a cut between conclusions 1 and ⊥ of zero-ary links with. . . nothing. But we notice a new pr ...
Structural Multi-type Sequent Calculus for Inquisitive Logic
... We write φ |= ψ if, for any state S , if S |= φ then S |= ψ. If both φ |= ψ and ψ |= φ, then we write φ ≡ ψ. An InqL-formula φ is valid, denoted |= φ, if S |= φ for any state S . The logic InqL is the set of all valid InqL-formulas. An easy inductive proof shows that InqL-formulas have the downward ...
... We write φ |= ψ if, for any state S , if S |= φ then S |= ψ. If both φ |= ψ and ψ |= φ, then we write φ ≡ ψ. An InqL-formula φ is valid, denoted |= φ, if S |= φ for any state S . The logic InqL is the set of all valid InqL-formulas. An easy inductive proof shows that InqL-formulas have the downward ...
Back to Basics: Revisiting the Incompleteness
... We said we wanted formal features of the theory being mechanically checkable. Here’s a quick and dirty argument that this holds in the right kind of way for the relevant properties of PA, making the two components of (1) true. First, to evaluate diag(m) we need to check whether m is the Gödel code ...
... We said we wanted formal features of the theory being mechanically checkable. Here’s a quick and dirty argument that this holds in the right kind of way for the relevant properties of PA, making the two components of (1) true. First, to evaluate diag(m) we need to check whether m is the Gödel code ...
Structural Logical Relations
... We shall prove that for every derivation of eτ , there exists a v τ , s.t. e −→∗ v and v ⇑ τ via a unary structural logical relation. The challenge, however, is the choice of a predicate P . It may come as a surprise that it is sufficient to characterize the fact that a term has a normal form withou ...
... We shall prove that for every derivation of eτ , there exists a v τ , s.t. e −→∗ v and v ⇑ τ via a unary structural logical relation. The challenge, however, is the choice of a predicate P . It may come as a surprise that it is sufficient to characterize the fact that a term has a normal form withou ...
Argument construction and reinstatement in logics for
... Arguments based on ordered theories, such as those depicted in the Tweety and Nixon examples, can conflict with one another; and it is at first tempting to think of two arguments as conflicting just in case they contain complementary literals as conclusions. This notion of conflict fails to account, ...
... Arguments based on ordered theories, such as those depicted in the Tweety and Nixon examples, can conflict with one another; and it is at first tempting to think of two arguments as conflicting just in case they contain complementary literals as conclusions. This notion of conflict fails to account, ...
11. Predicate Logic Syntax and Semantics, Normal Forms, Herbrand
... us to determine satisfiability or general validity (by transformation into DNF or CNF). But: we can reduce the satisfiability problem in predicate logic to the satisfiability problem in propositional logic. In general, however, this produces a very large number of propositional formulae (perhaps inf ...
... us to determine satisfiability or general validity (by transformation into DNF or CNF). But: we can reduce the satisfiability problem in predicate logic to the satisfiability problem in propositional logic. In general, however, this produces a very large number of propositional formulae (perhaps inf ...
pdf file
... 2. d1 ‹i d2 iff d2 is of the form β ∝ α and d1 « α; 3. d1 ‹ d2 iff (a) d1 ‹i d2 or (b) there is a default d3 such that d1 ‹ d3 and d3 ‹ d2. Intuitively d « α implies that the conclusion of d may be used to prove α; d1 ‹i d2 means that the conclusion of d1 may be used to prove the exception of d2; an ...
... 2. d1 ‹i d2 iff d2 is of the form β ∝ α and d1 « α; 3. d1 ‹ d2 iff (a) d1 ‹i d2 or (b) there is a default d3 such that d1 ‹ d3 and d3 ‹ d2. Intuitively d « α implies that the conclusion of d may be used to prove α; d1 ‹i d2 means that the conclusion of d1 may be used to prove the exception of d2; an ...
CDM Recursive Functions Klaus Sutner Carnegie Mellon University
... S of arity 1. Compound terms are constructed as follows: Given n terms τi of arity m and one term σ of arity n, Comp[σ, τ1 , . . . , τn ] is a new term of arity m. Given terms σ of arity n + 2 and τ of arity n, PRec[σ, τ ] is a new term of arity n + 1. Our language of terms is the inductive closure. ...
... S of arity 1. Compound terms are constructed as follows: Given n terms τi of arity m and one term σ of arity n, Comp[σ, τ1 , . . . , τn ] is a new term of arity m. Given terms σ of arity n + 2 and τ of arity n, PRec[σ, τ ] is a new term of arity n + 1. Our language of terms is the inductive closure. ...
full text (.pdf)
... although with a little extra work this result can be shown to follow from a 1979 result of Berstel 2] (see also 9]). The proof in 17] only needed extra commutativity conditions of the form bp = pb, where b is a test. But as shown in that paper, this equation is equivalent to bpb + bpb = 0. Thus i ...
... although with a little extra work this result can be shown to follow from a 1979 result of Berstel 2] (see also 9]). The proof in 17] only needed extra commutativity conditions of the form bp = pb, where b is a test. But as shown in that paper, this equation is equivalent to bpb + bpb = 0. Thus i ...
Combining Paraconsistent Logic with Argumentation
... but plausible argument orderings are possible in which C 6≺ D2 and so C defeats D2 . This is problematic, since s can be any formula, so any defeasible argument unrelated to A2 or B2 , such as D2 , can, depending on the argument ordering, be defeated by C. Clearly, this is extremely harmful, since t ...
... but plausible argument orderings are possible in which C 6≺ D2 and so C defeats D2 . This is problematic, since s can be any formula, so any defeasible argument unrelated to A2 or B2 , such as D2 , can, depending on the argument ordering, be defeated by C. Clearly, this is extremely harmful, since t ...
An introduction to ampleness
... We show that tp(A/B) does not divide over C. Suppose (Bi : i < ω) is a sequence of translates of B over C. Let X be the ≤-closure of the union of these and let Y be the free amalgam of X and A over C. As Bi ≤ X we have that A and Bi are freely amalgamated over C and A ∪ Bi ≤ Y . We may assume that Y ...
... We show that tp(A/B) does not divide over C. Suppose (Bi : i < ω) is a sequence of translates of B over C. Let X be the ≤-closure of the union of these and let Y be the free amalgam of X and A over C. As Bi ≤ X we have that A and Bi are freely amalgamated over C and A ∪ Bi ≤ Y . We may assume that Y ...
Algebraic Laws for Nondeterminism and Concurrency
... The denotational approach to the semantics of programming languageshas been well developed in recent years [ 1, 111and applied successfullyto many nontrivial languages.Even languageswith parallel constructs have been treated in this way, using the power-domain constructions of [3], [7], and [lo]. In ...
... The denotational approach to the semantics of programming languageshas been well developed in recent years [ 1, 111and applied successfullyto many nontrivial languages.Even languageswith parallel constructs have been treated in this way, using the power-domain constructions of [3], [7], and [lo]. In ...
Computability and Incompleteness
... that is not entirely fair: the two topics we have discussed share a common emphasis on philosophical and conceptual clarification, of “computation” in the first case, and “proof” in the second. But we will see that the relationship is much deeper than that. Computability is needed to define the noti ...
... that is not entirely fair: the two topics we have discussed share a common emphasis on philosophical and conceptual clarification, of “computation” in the first case, and “proof” in the second. But we will see that the relationship is much deeper than that. Computability is needed to define the noti ...
Chapter 2 Propositional Logic
... then we can put an → between them, enclose the whole thing in parentheses, and we get another wff. (The resulting wff is often called a “conditional”, whose “antecedent” is the wff before the → and whose “consequent” is the wff after the →.) For example, since we know that Q is a wff (clause i)), an ...
... then we can put an → between them, enclose the whole thing in parentheses, and we get another wff. (The resulting wff is often called a “conditional”, whose “antecedent” is the wff before the → and whose “consequent” is the wff after the →.) For example, since we know that Q is a wff (clause i)), an ...
Logical Omniscience As Infeasibility - boris
... normal modal logics yields the possibility of using the semantics of Kripke models, which have proved to be a convenient and intuitively clear tool for reasoning about knowledge, based on the Leibnizian supposition of multiple possible worlds. These postulates, however, have an unrealistic consequen ...
... normal modal logics yields the possibility of using the semantics of Kripke models, which have proved to be a convenient and intuitively clear tool for reasoning about knowledge, based on the Leibnizian supposition of multiple possible worlds. These postulates, however, have an unrealistic consequen ...
Basic Concepts of Formal Logic
... is false. The negation of a proposition, in turn, can be shown to be false if, from it, another proposition that is known to be false can be validly derived. If the conclusion of a valid argument is a false proposition, then at least one of the premises from which that conclusion has been derived mu ...
... is false. The negation of a proposition, in turn, can be shown to be false if, from it, another proposition that is known to be false can be validly derived. If the conclusion of a valid argument is a false proposition, then at least one of the premises from which that conclusion has been derived mu ...
Sketch-as-proof - Norbert Preining
... properties, which are those concerned with measurements of distances, angles, and areas, and descriptive properties, which are those concerned with the positional relations of geometric figures to one another. For example, the length of a line segment and the congruence of three lines are metric pro ...
... properties, which are those concerned with measurements of distances, angles, and areas, and descriptive properties, which are those concerned with the positional relations of geometric figures to one another. For example, the length of a line segment and the congruence of three lines are metric pro ...
F - Teaching-WIKI
... “evaluated” to determine its truth value (True or False) • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sentence that is True under all interpretations, no matter what the wo ...
... “evaluated” to determine its truth value (True or False) • A model for a KB is a “possible world” (assignment of truth values to propositional symbols) in which each sentence in the KB is True • A valid sentence or tautology is a sentence that is True under all interpretations, no matter what the wo ...
this PDF file
... Anderson and Belnap prove with a set of eight-element matrices (cf. [3], §22.1.3) that R has the vsp (so, that E has the vsp) and with a set of ten-element ones that E has the A.P (cf. [3], §22.1.1). Now, as the reader can readily check both sets of matrices satisfy γ. Consequently, TWcrγ , as well ...
... Anderson and Belnap prove with a set of eight-element matrices (cf. [3], §22.1.3) that R has the vsp (so, that E has the vsp) and with a set of ten-element ones that E has the A.P (cf. [3], §22.1.1). Now, as the reader can readily check both sets of matrices satisfy γ. Consequently, TWcrγ , as well ...
Modal Logic - Web Services Overview
... Syntax of Modal Logic (□ and ◊) Formulae in (propositional) Modal Logic ML: • The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML. • If and are formulae in ML, then ...
... Syntax of Modal Logic (□ and ◊) Formulae in (propositional) Modal Logic ML: • The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML. • If and are formulae in ML, then ...
Formal systems of fuzzy logic and their fragments∗
... The logic BCK plus this axiom of prelinearity will be the starting point for us in this paper—we call this logic Fuzzy BCK logic (FBCK for short). This logic is obviously complete with respect to the BCK-chains and this is the rationale for the name “Fuzzy BCK”, as the authors believe that completen ...
... The logic BCK plus this axiom of prelinearity will be the starting point for us in this paper—we call this logic Fuzzy BCK logic (FBCK for short). This logic is obviously complete with respect to the BCK-chains and this is the rationale for the name “Fuzzy BCK”, as the authors believe that completen ...
LPF and MPLω — A Logical Comparison of VDM SL and COLD-K
... This allows a large class of recursive and inductive definitions of functions and predicates to be expressed as formulae of MPLω . This was first sketched in [KR89, Section 4] and later worked out in detail by Renardel de Lavalette in [Ren89]. If A is a formula, then the term ιx : S (A) can be forme ...
... This allows a large class of recursive and inductive definitions of functions and predicates to be expressed as formulae of MPLω . This was first sketched in [KR89, Section 4] and later worked out in detail by Renardel de Lavalette in [Ren89]. If A is a formula, then the term ιx : S (A) can be forme ...