Proof theory of witnessed G¨odel logic: a
... supported by the FWF Austrian Science Fund projects START Y544-N23 and P22416. ...
... supported by the FWF Austrian Science Fund projects START Y544-N23 and P22416. ...
Geometric Modal Logic
... claims incomparably more than saying that this proposition is simply necessary. Speaking of something as ‘possibly possible’, we implicitly let the variation system itself vary, we shift from a given system of possibility into a frame inside which this system is only one among others, and we say tha ...
... claims incomparably more than saying that this proposition is simply necessary. Speaking of something as ‘possibly possible’, we implicitly let the variation system itself vary, we shift from a given system of possibility into a frame inside which this system is only one among others, and we say tha ...
The Relative Efficiency of Propositional Proof
... theorem" to be used as a rule. According to the deduction theorem, from a derivation n in a Frege system 9 showing Al, . . ., A, t- B we can construct a derivation n' in 9 showing Al, ..., AmP1t- A, 3 B. The trouble is that n' may be twice as long as n, so that if a natural deduction derivation has ...
... theorem" to be used as a rule. According to the deduction theorem, from a derivation n in a Frege system 9 showing Al, . . ., A, t- B we can construct a derivation n' in 9 showing Al, ..., AmP1t- A, 3 B. The trouble is that n' may be twice as long as n, so that if a natural deduction derivation has ...
Proof Theory of Finite-valued Logics
... for A is decidable. The unification algorithm (see Chang and Lee [1973]) calculates the most general unifier if it exists. ...
... for A is decidable. The unification algorithm (see Chang and Lee [1973]) calculates the most general unifier if it exists. ...
Strong Completeness and Limited Canonicity for PDL
... i.e. when | ϕ implies that there is a finite ⊆ with | ϕ, hence | → ϕ. This is, for example, the case in propositional and predicate logic, and in many modal logics such as K and S5. Segerberg’s axiomatization of PDL is only weakly complete, since PDL is not compact: we have that {[a ...
... i.e. when | ϕ implies that there is a finite ⊆ with | ϕ, hence | → ϕ. This is, for example, the case in propositional and predicate logic, and in many modal logics such as K and S5. Segerberg’s axiomatization of PDL is only weakly complete, since PDL is not compact: we have that {[a ...
Argumentative Approaches to Reasoning with Maximal Consistency Ofer Arieli Christian Straßer
... Dung’s semantics for abstract argumentation frameworks. Given a framework AF (Definition 1), a key issue in its understanding is the question what combinations of arguments (called extensions) can collectively be accepted from AF. According to Dung (1995), this is determined as follows: Definition 6 L ...
... Dung’s semantics for abstract argumentation frameworks. Given a framework AF (Definition 1), a key issue in its understanding is the question what combinations of arguments (called extensions) can collectively be accepted from AF. According to Dung (1995), this is determined as follows: Definition 6 L ...
Modalities in the Realm of Questions: Axiomatizing Inquisitive
... (vi) If ϕ ∈ L! ∪ L? and a ∈ A, then Ka ϕ ∈ L! (vii) If ϕ ∈ L! ∪ L? and a ∈ A, then Ea ϕ ∈ L! Importantly, conjunction, implication, and the modalities are allowed to apply to interrogatives as well as declaratives. Also, notice how the two syntactic categories are intertwined: from a sequence of dec ...
... (vi) If ϕ ∈ L! ∪ L? and a ∈ A, then Ka ϕ ∈ L! (vii) If ϕ ∈ L! ∪ L? and a ∈ A, then Ea ϕ ∈ L! Importantly, conjunction, implication, and the modalities are allowed to apply to interrogatives as well as declaratives. Also, notice how the two syntactic categories are intertwined: from a sequence of dec ...
GLukG logic and its application for non-monotonic reasoning
... Hence, We came up with a basic mathematical question. Can we use any other type of monotonic logic different to the constructive intermediate logics such that the following properties hold? 1. The logic should have the following desirable properties. It should satisfy the replacement and deduction ...
... Hence, We came up with a basic mathematical question. Can we use any other type of monotonic logic different to the constructive intermediate logics such that the following properties hold? 1. The logic should have the following desirable properties. It should satisfy the replacement and deduction ...
First-Order Intuitionistic Logic with Decidable Propositional
... Apparently, every formula derivable in LJ+ is also derivable in LK. LK is Gentzen’s formulation of classical logic. Gentzen [Ge] alternatively defines LK as LJ (i.e. all sequents have no more than one formula in the succedent) plus axiom schema |-- A\/¬A that is applicable to any formula A. LJ+ is c ...
... Apparently, every formula derivable in LJ+ is also derivable in LK. LK is Gentzen’s formulation of classical logic. Gentzen [Ge] alternatively defines LK as LJ (i.e. all sequents have no more than one formula in the succedent) plus axiom schema |-- A\/¬A that is applicable to any formula A. LJ+ is c ...
Logic 1 Lecture Notes Part I: Propositional Logic
... John is happy only if Yoko is away John is happy ...
... John is happy only if Yoko is away John is happy ...
Logic3
... The sequence of wffs (w1, w2, …, wn) is called a proof (or deduction) of wn from a set of wffs Δ iff each wi in the sequence is either in Δ or can be inferred from a wff (or wffs) earlier in the sequence by using a valid rule of inference. If there is a proof of wn from Δ, we say that wn is a theore ...
... The sequence of wffs (w1, w2, …, wn) is called a proof (or deduction) of wn from a set of wffs Δ iff each wi in the sequence is either in Δ or can be inferred from a wff (or wffs) earlier in the sequence by using a valid rule of inference. If there is a proof of wn from Δ, we say that wn is a theore ...
06_chapter 2
... Aristotle presented this argument in the form of a syllogism. Hence this argument is sometimes known as "Inductive Syllogism". In this example, we proceed by the enumeration of all the particular cases; we have given a list of long-lived animals ; we have also stated that all these long-lived animal ...
... Aristotle presented this argument in the form of a syllogism. Hence this argument is sometimes known as "Inductive Syllogism". In this example, we proceed by the enumeration of all the particular cases; we have given a list of long-lived animals ; we have also stated that all these long-lived animal ...
The Taming of the (X)OR
... a part of the overall formula. This is indeed the case for DES encodings whereas xor clauses are just the hard core part (4% of whole) [Mas99,Li00]. This is true for many other problems such as model checking [BCC+99,Li00]. So we want to have affine-logic reasoning in our calculus and, at the same t ...
... a part of the overall formula. This is indeed the case for DES encodings whereas xor clauses are just the hard core part (4% of whole) [Mas99,Li00]. This is true for many other problems such as model checking [BCC+99,Li00]. So we want to have affine-logic reasoning in our calculus and, at the same t ...
Loop Formulas for Circumscription - Joohyung Lee
... 1997; Giunchiglia et al., 2004a], the problem of determining whether a theory is consistent can be reduced to the satisfiability problem for propositional logic by the process of “literal completion”—a translation similar to Clark’s completion. This idea has led to the creation of the Causal Calcula ...
... 1997; Giunchiglia et al., 2004a], the problem of determining whether a theory is consistent can be reduced to the satisfiability problem for propositional logic by the process of “literal completion”—a translation similar to Clark’s completion. This idea has led to the creation of the Causal Calcula ...
Modular Sequent Systems for Modal Logic
... permute down below the rules cut, 2 and ∧, so we generalise these rules as in Figure 3. We define a contraction-free system K− as K− = K − ctr + {med, m2, m∧} and will show cut elimination for that system, but first we develop the machinery to show that cut elimination for K− leads to cut-eliminatio ...
... permute down below the rules cut, 2 and ∧, so we generalise these rules as in Figure 3. We define a contraction-free system K− as K− = K − ctr + {med, m2, m∧} and will show cut elimination for that system, but first we develop the machinery to show that cut elimination for K− leads to cut-eliminatio ...
Master Thesis - Yoichi Hirai
... Middle, which states either a proposition or the negation of it is always valid. The Law of Excluded Middle asserts that either a message has reached the intended receiver or it has not reached the intended receiver. We point out that this reasoning assumes the existence of a current state of the wo ...
... Middle, which states either a proposition or the negation of it is always valid. The Law of Excluded Middle asserts that either a message has reached the intended receiver or it has not reached the intended receiver. We point out that this reasoning assumes the existence of a current state of the wo ...
Knowledge Representation: Logic
... The elements of the map could be associated with object classes divided into point-like, line-like and so on. Addition of a new component may be then achieved by adding a new subclass, but it can be impossible, for example for street names. We may as well express the content of the map using logic a ...
... The elements of the map could be associated with object classes divided into point-like, line-like and so on. Addition of a new component may be then achieved by adding a new subclass, but it can be impossible, for example for street names. We may as well express the content of the map using logic a ...
pdf
... computing. Most of that work has used standard Kripke structures to model knowledge, where an agent knows a fact ϕ if ϕ is true in all the worlds that the agent considers possible. While this approach has proved useful for many applications, it suffers from a serious shortcoming, known as the logica ...
... computing. Most of that work has used standard Kripke structures to model knowledge, where an agent knows a fact ϕ if ϕ is true in all the worlds that the agent considers possible. While this approach has proved useful for many applications, it suffers from a serious shortcoming, known as the logica ...
A Logical Expression of Reasoning
... and serious concern from the tradition. That is why for so long, at least until Hume [12], but even after him (deductive) logic and correct reasoning, or even “rationality”, has been taken as synonymous. However, real life reasoning, meaning a wide range of forms of inference, covering from common s ...
... and serious concern from the tradition. That is why for so long, at least until Hume [12], but even after him (deductive) logic and correct reasoning, or even “rationality”, has been taken as synonymous. However, real life reasoning, meaning a wide range of forms of inference, covering from common s ...
The Science of Proof - University of Arizona Math
... If these three sentences have an interpretation in some domain of objects (events), then it follows that there are infinitely many objects. In other words, there is no first cause. This is an example where simple syntax (three sentences) has profound semantic implications. It also shows how convenie ...
... If these three sentences have an interpretation in some domain of objects (events), then it follows that there are infinitely many objects. In other words, there is no first cause. This is an example where simple syntax (three sentences) has profound semantic implications. It also shows how convenie ...
Interactive Theorem Proving with Temporal Logic
... Each of the rules of this sequent calculus can be given an intuitive reading. These readings will be re ected directly in the generation of text from proofs. Most of the propositional rules are straightforward. The ^ right rule for example states that if A and B each hold from the assumptions , then ...
... Each of the rules of this sequent calculus can be given an intuitive reading. These readings will be re ected directly in the generation of text from proofs. Most of the propositional rules are straightforward. The ^ right rule for example states that if A and B each hold from the assumptions , then ...
A SHORT AND READABLE PROOF OF CUT ELIMINATION FOR
... Theorem 2.6 (cf. [10, 12, 9]). If (Γ ⊢ ∆) [a] is provable with order m and b is some other free variable, then (Γ ⊢ ∆) [b] is provable with order ≤ m. Proof. By induction on the order of derivation, m, of (Γ ⊢ ∆)[a]. For m = 0, (Γ ⊢ ∆)[a] is an axiom. Then so is (Γ ⊢ ∆)[b]. For the induction step we ...
... Theorem 2.6 (cf. [10, 12, 9]). If (Γ ⊢ ∆) [a] is provable with order m and b is some other free variable, then (Γ ⊢ ∆) [b] is provable with order ≤ m. Proof. By induction on the order of derivation, m, of (Γ ⊢ ∆)[a]. For m = 0, (Γ ⊢ ∆)[a] is an axiom. Then so is (Γ ⊢ ∆)[b]. For the induction step we ...
A Proof Theory for Generic Judgments
... an assumption (that is, on the left of the sequent arrow) is essentially equated to having instead all instances Bt for terms t of type τ . There are cases (one is considered in more detail in Section 6) where we would like to make inferences from an assumption of the form ∀τ x.Bx that holds indepen ...
... an assumption (that is, on the left of the sequent arrow) is essentially equated to having instead all instances Bt for terms t of type τ . There are cases (one is considered in more detail in Section 6) where we would like to make inferences from an assumption of the form ∀τ x.Bx that holds indepen ...
(pdf)
... may take on any value between and including 0 and 1 in a fuzzy system. From this basic fact follow many rules and rule changes which will comprise the bulk of this paper. For now, though, we shall adapt Hájek’s definition to say: A fuzzy system is a logical system whose elements may take truth valu ...
... may take on any value between and including 0 and 1 in a fuzzy system. From this basic fact follow many rules and rule changes which will comprise the bulk of this paper. For now, though, we shall adapt Hájek’s definition to say: A fuzzy system is a logical system whose elements may take truth valu ...
Propositional logic - Computing Science
... Need to know how to represent information, knowledge, facts and beliefs, and how to apply logic, … ...
... Need to know how to represent information, knowledge, facts and beliefs, and how to apply logic, … ...