CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
... set S there is a set S ∗ , such that S ⊂ S ∗ and S ∗ is maximal finitely consistent. Step 4 We use steps 2 and 3 to justify the following reasoning. Given a finitely consistent set S. We extend it, via construction defined in the step 2 to a maximal finitely consistent set S ∗ . By the step 2, S ∗ i ...
... set S there is a set S ∗ , such that S ⊂ S ∗ and S ∗ is maximal finitely consistent. Step 4 We use steps 2 and 3 to justify the following reasoning. Given a finitely consistent set S. We extend it, via construction defined in the step 2 to a maximal finitely consistent set S ∗ . By the step 2, S ∗ i ...
A Resolution-Based Proof Method for Temporal Logics of
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
Formal logic
... But how and why can we conclude that this last sentence follows from the previous two premises? Or, more generally, how can we determine whether a formula ϕ is a valid consequence of a set of formulas {ϕ1 , . . . , ϕn }? Modern logic offers two possible ways, that used to be fused in the time of syl ...
... But how and why can we conclude that this last sentence follows from the previous two premises? Or, more generally, how can we determine whether a formula ϕ is a valid consequence of a set of formulas {ϕ1 , . . . , ϕn }? Modern logic offers two possible ways, that used to be fused in the time of syl ...
Sequent calculus for predicate logic
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
Systems of modal logic - Department of Computing
... One can also have modal operators with arity 2, 3, . . . (even arity 0). Example: ‘since’ and ‘until’, which are binary. I may include some further examples later, for various forms of conditionals, if time allows. ...
... One can also have modal operators with arity 2, 3, . . . (even arity 0). Example: ‘since’ and ‘until’, which are binary. I may include some further examples later, for various forms of conditionals, if time allows. ...
A BRIEF INTRODUCTION TO MODAL LOGIC Introduction Consider
... It is this reading of ‘necessarily true’ as ‘true in all possible worlds’ that lead to the most popular interpretation of modal logic: Kripke’s many-world semantics. Under this interpretation, the truth of a statement is relative to the world in question. For propositional formulae, this is determin ...
... It is this reading of ‘necessarily true’ as ‘true in all possible worlds’ that lead to the most popular interpretation of modal logic: Kripke’s many-world semantics. Under this interpretation, the truth of a statement is relative to the world in question. For propositional formulae, this is determin ...
Normal modal logics (Syntactic characterisations)
... (i) Trivial. We need to show that L contains PL and is closed under MP and US. This is trivial, because L is the set of all formulas. (ii) Easy. PL is a subset of every Σi , so also a subset of the intersection. To show the intersection is closed under MP: suppose A and A → B are formulas in the int ...
... (i) Trivial. We need to show that L contains PL and is closed under MP and US. This is trivial, because L is the set of all formulas. (ii) Easy. PL is a subset of every Σi , so also a subset of the intersection. To show the intersection is closed under MP: suppose A and A → B are formulas in the int ...
PREDICATE LOGIC
... Generally, predicates make statements about individuals. To illustrate this notion, consider the following statements: (a) Mary and Paul are siblings. (b) Jane is the mother of Mary. (c) Tom is a cat. (d) The sum of 2 and 3 is 5. In each of these statements, there is a list of individuals, which is ...
... Generally, predicates make statements about individuals. To illustrate this notion, consider the following statements: (a) Mary and Paul are siblings. (b) Jane is the mother of Mary. (c) Tom is a cat. (d) The sum of 2 and 3 is 5. In each of these statements, there is a list of individuals, which is ...
Logic - United States Naval Academy
... Two (compound) expressions are logically equivalent if and only if they have identical truth values for all possible combinations of truth values for the sub-expressions. If A and B are logically equivalent, we write A B . (Another notation for logical equivalence is ; that is, if A and B are lo ...
... Two (compound) expressions are logically equivalent if and only if they have identical truth values for all possible combinations of truth values for the sub-expressions. If A and B are logically equivalent, we write A B . (Another notation for logical equivalence is ; that is, if A and B are lo ...
Formal deduction in propositional logic
... Intuitive meaning of rules • The elimination (introduction) of a connective means that one occurrence of this connective is eliminated (introduced) in the conclusion of the scheme of formal deducibility generated by the rule. • Remark: In (∨−) it is the ∨ between A and B in A ∨ B that is eliminated ...
... Intuitive meaning of rules • The elimination (introduction) of a connective means that one occurrence of this connective is eliminated (introduced) in the conclusion of the scheme of formal deducibility generated by the rule. • Remark: In (∨−) it is the ∨ between A and B in A ∨ B that is eliminated ...
1 The calculus of “predicates”
... collection of lattice points has both a join and meet, it may still allow for some infinite joins and meets. Thus, in the logic that is built over the lattice there may be quantifiers, but in the lattice there are no distinct points that correspond to quantifiers that do not represent points already ...
... collection of lattice points has both a join and meet, it may still allow for some infinite joins and meets. Thus, in the logic that is built over the lattice there may be quantifiers, but in the lattice there are no distinct points that correspond to quantifiers that do not represent points already ...
Chapter 5 Predicate Logic
... to one of two values: T or F. The logical connectives have their usual function, but we must now include a mechanism for understanding predicates and quantifiers. Consider predicates first. An expression like G(a) is true just in case f (a) is in the subset of D that f assigns G to. For example, if ...
... to one of two values: T or F. The logical connectives have their usual function, but we must now include a mechanism for understanding predicates and quantifiers. Consider predicates first. An expression like G(a) is true just in case f (a) is in the subset of D that f assigns G to. For example, if ...
Logic and Resolution
... then the composition, denoted by θ ◦ σ or θσ , is defined by: {t1 σ/x1 , . . . , tm σ/xm , s1 /y1 , . . . , sn /yn } where each element ti σ/xi is removed for which xi = ti σ and also each element sj /yj for which yj ∈ {x1 , . . . , xm } A substitution σ is called a unifier of E and E ′ if Eσ = E ′ ...
... then the composition, denoted by θ ◦ σ or θσ , is defined by: {t1 σ/x1 , . . . , tm σ/xm , s1 /y1 , . . . , sn /yn } where each element ti σ/xi is removed for which xi = ti σ and also each element sj /yj for which yj ∈ {x1 , . . . , xm } A substitution σ is called a unifier of E and E ′ if Eσ = E ′ ...
Outline of Lecture 2 First Order Logic and Second Order Logic Basic
... A property of τ -structures is a class P of τ -structures which is closed under τ -isomorphisms. • All finite τ -structures. • All {R2,0 }-structures where R2,0 is interpreted as a linear order. • Al finite 3-dimensional matchings 3DM , i.e. all {R3,0 }-structures with universe A where the interpret ...
... A property of τ -structures is a class P of τ -structures which is closed under τ -isomorphisms. • All finite τ -structures. • All {R2,0 }-structures where R2,0 is interpreted as a linear order. • Al finite 3-dimensional matchings 3DM , i.e. all {R3,0 }-structures with universe A where the interpret ...
On Equivalent Transformations of Infinitary Formulas under the
... to check that the equivalence F ↔ G is provable intuitionistically. Some extensions of intuitionistic propositional logic, including the logic of here-and-there, can be used as well. In this note we extend these results to deductive systems of infinitary propositional logic. This goal is closely rel ...
... to check that the equivalence F ↔ G is provable intuitionistically. Some extensions of intuitionistic propositional logic, including the logic of here-and-there, can be used as well. In this note we extend these results to deductive systems of infinitary propositional logic. This goal is closely rel ...
Inquiry
An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.