Lecture 16 Notes
... version of classical Tarski semantics. We briefly touched on this semantics in Lecture 14, citing Troelstra and van Dalen for the result that i FOL is incomplete for this “standard intuitionistic semantics.” We will look briefly at the incompleteness result since that has received a good deal of att ...
... version of classical Tarski semantics. We briefly touched on this semantics in Lecture 14, citing Troelstra and van Dalen for the result that i FOL is incomplete for this “standard intuitionistic semantics.” We will look briefly at the incompleteness result since that has received a good deal of att ...
Chapter 1: The Foundations: Logic and Proofs
... Tautology: A compound proposition that is always true. Contradiction: A compound proposition that is always false. Contingency: A compound proposition that is neither a tautology nor a contradiction. ...
... Tautology: A compound proposition that is always true. Contradiction: A compound proposition that is always false. Contingency: A compound proposition that is neither a tautology nor a contradiction. ...
A Simple Exposition of Gödel`s Theorem
... In October 1997 I was asked to join in a discussion of the Gödelian argument at an undergraduate philosophy club in King's College, London; and I was asked to preface it with a very simple exposition of Gödel's (first) Theorem at a level at which first-year students could understand. Although there ...
... In October 1997 I was asked to join in a discussion of the Gödelian argument at an undergraduate philosophy club in King's College, London; and I was asked to preface it with a very simple exposition of Gödel's (first) Theorem at a level at which first-year students could understand. Although there ...
Sub-Birkhoff
... name with its rule is called an axiom. Subequational logics generate subequational theories. Definition 2 For a subequational logic L = hS,Ii its theory L is generated by the following inference rules, where an inference rule (i) only applies if i ∈ I. s, t and r range over terms. `sLs ...
... name with its rule is called an axiom. Subequational logics generate subequational theories. Definition 2 For a subequational logic L = hS,Ii its theory L is generated by the following inference rules, where an inference rule (i) only applies if i ∈ I. s, t and r range over terms. `sLs ...
8 predicate logic
... can be represented as As ⊃ Ap; the proposition “Socrates is altruistic but Plato is not” can be represented as As · ~Ap, and so on. Representing quantified propositions in predicate logic requires a little more symbolic apparatus. First, we require the idea of an individual variable. We shall alloca ...
... can be represented as As ⊃ Ap; the proposition “Socrates is altruistic but Plato is not” can be represented as As · ~Ap, and so on. Representing quantified propositions in predicate logic requires a little more symbolic apparatus. First, we require the idea of an individual variable. We shall alloca ...
Lecture 39 Notes
... subtyping, becomes “immediate implication”, e.g. the evidence a for A is also evidence for B, i.e. A ⇒ B. A ∩ B intersection, becomes a strong &, e.g. the evidence for A and B can be the same, hence A ∩ B ⇒ A&B. T (A ⇒ A) intersection of an indexed family. The element λ(y.y) ∈ (A ⇒ A) shows A:Ui tha ...
... subtyping, becomes “immediate implication”, e.g. the evidence a for A is also evidence for B, i.e. A ⇒ B. A ∩ B intersection, becomes a strong &, e.g. the evidence for A and B can be the same, hence A ∩ B ⇒ A&B. T (A ⇒ A) intersection of an indexed family. The element λ(y.y) ∈ (A ⇒ A) shows A:Ui tha ...
The semantics of predicate logic
... In this module, we will precisely define the semantic interpretation of formulas in our predicate logic. In propositional logic, every formula had a fixed, finite number of models (interpretations); this is not the case in predicate logic. As a consequence, we must take more care in defining notions ...
... In this module, we will precisely define the semantic interpretation of formulas in our predicate logic. In propositional logic, every formula had a fixed, finite number of models (interpretations); this is not the case in predicate logic. As a consequence, we must take more care in defining notions ...
Chapter 2
... + fire + retire). One of the nicest features of regular languages is that they have a dual characterization using fsa and regular expressions. Indeed, Kleene’s theorem says that a language L is regular iff it can be specified by a regular expression. There are several important variations of fsa tha ...
... + fire + retire). One of the nicest features of regular languages is that they have a dual characterization using fsa and regular expressions. Indeed, Kleene’s theorem says that a language L is regular iff it can be specified by a regular expression. There are several important variations of fsa tha ...
Introduction to formal logic - University of San Diego Home Pages
... is equivalent to “She was old and ugly” - NOT “She was old or ugly.” “You can’t both have your cake and eat it” is equivalent to “You either don’t have your cake or you don’t ...
... is equivalent to “She was old and ugly” - NOT “She was old or ugly.” “You can’t both have your cake and eat it” is equivalent to “You either don’t have your cake or you don’t ...
this PDF file
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
A program for the countable choice axiom
... Let us look at the execution of a process γ f.π : γ f.π χ Yξ.π (where ξ = λxλn(cc)λk f στ depends only on f ) Yξ nπ .π ξ η.nπ .π (with η = Yξ ). Therefore γ f.π f σf π .τf π .π with σf π = σ[η/x, nπ /n, kπ /k]. Now σf π is simply the triple <η, nπ , kπ >, in other words σf π stores the current state ...
... Let us look at the execution of a process γ f.π : γ f.π χ Yξ.π (where ξ = λxλn(cc)λk f στ depends only on f ) Yξ nπ .π ξ η.nπ .π (with η = Yξ ). Therefore γ f.π f σf π .τf π .π with σf π = σ[η/x, nπ /n, kπ /k]. Now σf π is simply the triple <η, nπ , kπ >, in other words σf π stores the current state ...
