higher-order logic - University of Amsterdam
... EXAMPLE. The notion ‘most A are B ’ is not definable in a first-order logic with identity having, at least, unary predicate constants A B . This time, a refutation involves both compactness and the (downward) Löwenheim-Skolem theorem: Consider any proposed definition (A B ) together with the infin ...
... EXAMPLE. The notion ‘most A are B ’ is not definable in a first-order logic with identity having, at least, unary predicate constants A B . This time, a refutation involves both compactness and the (downward) Löwenheim-Skolem theorem: Consider any proposed definition (A B ) together with the infin ...
Complete Sequent Calculi for Induction and Infinite Descent
... LKID: a sequent calculus for induction in FOLID Extend the usual sequent calculus LKe for classical first-order logic with equality by adding rules for inductive predicates. E.g., right-introduction rules for N are: ...
... LKID: a sequent calculus for induction in FOLID Extend the usual sequent calculus LKe for classical first-order logic with equality by adding rules for inductive predicates. E.g., right-introduction rules for N are: ...
neighborhood semantics for basic and intuitionistic logic
... neighborhood), and a modal formula ϕ is true at a world w, if the set of all states in which ϕ is true is a neighborhood of w. See [2] for more details on neighborhood semantics for modal logic. An interesting question is whether one can define similar neighborhood semantics for Intuitionistic Prop ...
... neighborhood), and a modal formula ϕ is true at a world w, if the set of all states in which ϕ is true is a neighborhood of w. See [2] for more details on neighborhood semantics for modal logic. An interesting question is whether one can define similar neighborhood semantics for Intuitionistic Prop ...
Basic principles of probability theory
... Example 1. Tossing a coin is a random experiment. The outcome space is {H,T} – head and tail. Example 2. Rolling a die. The outcome space is a set - {1,2,3,4,5,6} Example 3. Drawing from an urn with N balls, M of them is red and N-M is white. The outcome space is {R,W} – red and white Example 5. Mea ...
... Example 1. Tossing a coin is a random experiment. The outcome space is {H,T} – head and tail. Example 2. Rolling a die. The outcome space is a set - {1,2,3,4,5,6} Example 3. Drawing from an urn with N balls, M of them is red and N-M is white. The outcome space is {R,W} – red and white Example 5. Mea ...
Propositional Logic
... can be used as a deduction system (or proof system); that is, to construct proofs or refutations. This use of a logical language is called proof theory. In this case, a set of facts called axioms and a set of deduction rules (inference rules) are given, and the object is to determine which facts fol ...
... can be used as a deduction system (or proof system); that is, to construct proofs or refutations. This use of a logical language is called proof theory. In this case, a set of facts called axioms and a set of deduction rules (inference rules) are given, and the object is to determine which facts fol ...
Notes on Classical Propositional Logic
... In fact a stronger result can be proved, which we leave to you as Exercise 5.1. So, we will make sure to choose axiom schemes whose instances are tautologies, and rules of derivation that are sound. Completeness is much harder, however. This will occupy the next several sections. Since it is more co ...
... In fact a stronger result can be proved, which we leave to you as Exercise 5.1. So, we will make sure to choose axiom schemes whose instances are tautologies, and rules of derivation that are sound. Completeness is much harder, however. This will occupy the next several sections. Since it is more co ...
11. Predicate Logic Syntax and Semantics, Normal Forms, Herbrand
... Idea: Elimination of existential quantifiers by applying a function that produces the “right” element. Theorem (Skolem Normal Form): Let ϕ be a closed formula in prenex normal form such that all quantified variables are pair-wise distinct and the function symbols g1 , g2 , . . . do not appear in ϕ. ...
... Idea: Elimination of existential quantifiers by applying a function that produces the “right” element. Theorem (Skolem Normal Form): Let ϕ be a closed formula in prenex normal form such that all quantified variables are pair-wise distinct and the function symbols g1 , g2 , . . . do not appear in ϕ. ...
A Taste of Categorical Logic — Tutorial Notes
... {0, 1}. We take 1 to mean “true” and 0 to mean “false”. If we order 2 by postulating that 0 ≤ 1 then 2 becomes a complete Boolean algebra which in particular means that it is a complete Heyting algebra. Exercise 3.2. Show that given any set X , the set of functions from X to 2, i.e., HomSet (X , 2) ...
... {0, 1}. We take 1 to mean “true” and 0 to mean “false”. If we order 2 by postulating that 0 ≤ 1 then 2 becomes a complete Boolean algebra which in particular means that it is a complete Heyting algebra. Exercise 3.2. Show that given any set X , the set of functions from X to 2, i.e., HomSet (X , 2) ...
A constructive approach to nonstandard analysis*
... axiom of choice (AC). By this procedure it will immediately be clear that the internal theory is constructive, and moreover conservative over HA” + AC. This is in analogy with Nelson’s internal set theory: it is a conservative extension of set theory. 3.1. Arithmetic in all finite types We first pre ...
... axiom of choice (AC). By this procedure it will immediately be clear that the internal theory is constructive, and moreover conservative over HA” + AC. This is in analogy with Nelson’s internal set theory: it is a conservative extension of set theory. 3.1. Arithmetic in all finite types We first pre ...
The Perfect Set Theorem and Definable Wellorderings of the
... THEOREM. Let r be a reasonablepointclass and let M be a perfect set basis for r. If < is a wellorderingof a set of reals and < e r, then the field of < (i.e. the set {a: a < a}) is containedin M. PROOF. Without loss of generality we can assume that the field of < is contained in 20 so that we can wo ...
... THEOREM. Let r be a reasonablepointclass and let M be a perfect set basis for r. If < is a wellorderingof a set of reals and < e r, then the field of < (i.e. the set {a: a < a}) is containedin M. PROOF. Without loss of generality we can assume that the field of < is contained in 20 so that we can wo ...
Clustered states in the fractional quantum Hall effect
... they obey is a radical new one, called non-Abelian statistics. In the past decade it became clear that systems which display such exotic behavior may very well be prime candidates for the building blocks of a topological quantum computer. These are conjectured to remain relatively immune to decohere ...
... they obey is a radical new one, called non-Abelian statistics. In the past decade it became clear that systems which display such exotic behavior may very well be prime candidates for the building blocks of a topological quantum computer. These are conjectured to remain relatively immune to decohere ...
Model theory makes formulas large
... numbers by trees of small height that can be controlled by small first-order formulas. In fact, we show — and use — that full arithmetic on a large initial segment of the positive integers can be simulated by comparably small first-order formulas that operate on the tree encodings of the numbers. It ...
... numbers by trees of small height that can be controlled by small first-order formulas. In fact, we show — and use — that full arithmetic on a large initial segment of the positive integers can be simulated by comparably small first-order formulas that operate on the tree encodings of the numbers. It ...
A review of E infinity theory and the mass spectrum of high energy
... dynamics, namely fractal geometry is reduced to its quintessence, i.e. Cantor sets (see Fig. 5) and employed directly in the geometrical description of the fluctuation of the vacuum. How this is done and how to proceed from there to calculating for instance the mass spectrum of high energy elementary ...
... dynamics, namely fractal geometry is reduced to its quintessence, i.e. Cantor sets (see Fig. 5) and employed directly in the geometrical description of the fluctuation of the vacuum. How this is done and how to proceed from there to calculating for instance the mass spectrum of high energy elementary ...
Propositional Logic
... proposition is a possible “condition'” of the world about which we want to say something. The condition need not be true in order for us to talk about it. In fact, we might want to say that it is false or that it is true if some other proposition is true. In this chapter, we first look at the syntac ...
... proposition is a possible “condition'” of the world about which we want to say something. The condition need not be true in order for us to talk about it. In fact, we might want to say that it is false or that it is true if some other proposition is true. In this chapter, we first look at the syntac ...
why do physicists think that there are extra dimensions
... are extra dimensions of space? Reason #2: mysteries of particle physics all ordinary matter is composed of just three kinds of elementary particles. but in particle accelerators we produce many more! why do these extra particles exist, and why these particles but not others? ...
... are extra dimensions of space? Reason #2: mysteries of particle physics all ordinary matter is composed of just three kinds of elementary particles. but in particle accelerators we produce many more! why do these extra particles exist, and why these particles but not others? ...
Default Reasoning in a Terminological Logic
... the sense that they only allow for monadic predicate symbols, a limited use of negation and no disjunction at all. For our purposes, it is also essential to observe that their monotonic fragment is far less expressive than TLs as, having no term constructors in their syntactic apparatus, they only a ...
... the sense that they only allow for monadic predicate symbols, a limited use of negation and no disjunction at all. For our purposes, it is also essential to observe that their monotonic fragment is far less expressive than TLs as, having no term constructors in their syntactic apparatus, they only a ...
PPT
... A proof of Q from H1, H2, … Hk is finite sequence of propositional forms Q 1, Q 2, … Qn such that Qn is same as Q and every Qj is either one of Hi, (i = 1, 2, … , k) or it follows from the proceedings by the logic rules. Note: In these proofs we will follow the following formats: We begin with by li ...
... A proof of Q from H1, H2, … Hk is finite sequence of propositional forms Q 1, Q 2, … Qn such that Qn is same as Q and every Qj is either one of Hi, (i = 1, 2, … , k) or it follows from the proceedings by the logic rules. Note: In these proofs we will follow the following formats: We begin with by li ...
preference based on reasons
... be raised again in the Discussion section. For now, we develop our theory in the context of utility, with the expectation that most readers will find this setting conceptually familiar. If our agent is presumed to be moral then reasons are meant, very roughly, to be good (at least, not bad ). Morali ...
... be raised again in the Discussion section. For now, we develop our theory in the context of utility, with the expectation that most readers will find this setting conceptually familiar. If our agent is presumed to be moral then reasons are meant, very roughly, to be good (at least, not bad ). Morali ...