handout
... the left of the turnstile) give propositional tautologies. This is because the typing rules of λ→ correspond exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view o ...
... the left of the turnstile) give propositional tautologies. This is because the typing rules of λ→ correspond exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view o ...
Lesson 1
... If you want to show that your boss is not right, it is not diplomatic to say it in an open way. Instead, you may argue by way of the proof ad absurdum: “Well, you say P – interesting, but P entails Q, and Q entails R, which is obviously false.” (Hence, P must have been false as well.) ...
... If you want to show that your boss is not right, it is not diplomatic to say it in an open way. Instead, you may argue by way of the proof ad absurdum: “Well, you say P – interesting, but P entails Q, and Q entails R, which is obviously false.” (Hence, P must have been false as well.) ...
4 slides/page
... A derivation (or proof ) in an axiom system AX is a sequence of formulas C1 , . . . , C N ; each formula Ck is either an axiom in AX or follows from previous formulas using an inference rule in AX: ...
... A derivation (or proof ) in an axiom system AX is a sequence of formulas C1 , . . . , C N ; each formula Ck is either an axiom in AX or follows from previous formulas using an inference rule in AX: ...
Discrete Computational Structures (CS 225) Definition of Formal Proof
... When we write a proof, we will number each line, and justify it by citing its source. When justifying the application of equivalency rules and argument forms, we will give the name of the logical principle we are using, along with the line number(s) to which the rule or form is applied. For example, ...
... When we write a proof, we will number each line, and justify it by citing its source. When justifying the application of equivalency rules and argument forms, we will give the name of the logical principle we are using, along with the line number(s) to which the rule or form is applied. For example, ...
The Discovery of the Computer
... expressed in a logical system is true or false, without producing all the possible theorems of the system. This so-called “decision problem” was answered by Alan Turing, who showed that it is not possible to decide if any theorem is true or false. In doing this, he discovered the computer. The story ...
... expressed in a logical system is true or false, without producing all the possible theorems of the system. This so-called “decision problem” was answered by Alan Turing, who showed that it is not possible to decide if any theorem is true or false. In doing this, he discovered the computer. The story ...
Lesson 2
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
... • Hence if we prove that the conclusion logically follows from the assumptions, then by virtue of it we do not prove that the conclusion is true • It is true, provided the premises are true • The argument the premises of which are true is called sound. • Truthfulness or Falseness of premises can be ...
Lec 2 Notes
... Unfortunately, the successor law that we have so far doesn’t help that much, since it is only good for showing that 1=2 would hold if 0=1. What we need, is an implication in the other direction which would allow us to prove that 0=1 would hold if 1=2 does. Then, since 06=1 is an axiom, 1=2 cannot ho ...
... Unfortunately, the successor law that we have so far doesn’t help that much, since it is only good for showing that 1=2 would hold if 0=1. What we need, is an implication in the other direction which would allow us to prove that 0=1 would hold if 1=2 does. Then, since 06=1 is an axiom, 1=2 cannot ho ...
A short article for the Encyclopedia of Artificial Intelligence: Second
... standard one. Henkin (1950) developed a notion of general model that included non-standard as well as standard models. In the general setting, it is possible for Dσ→τ to be a proper subset of the set of all functions from Dσ to Dτ as long as there are enough functions to properly interpret all expr ...
... standard one. Henkin (1950) developed a notion of general model that included non-standard as well as standard models. In the general setting, it is possible for Dσ→τ to be a proper subset of the set of all functions from Dσ to Dτ as long as there are enough functions to properly interpret all expr ...
notes
... Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last cent ...
... Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last cent ...
The Role of Mathematical Logic in Computer Science and
... Cantor's Continuum Problem: How many real numbers are there? Cantor's Continuum Hypothesis (CH): Any two uncountable sets of real numbers have the same size. ZFC is not strong enough to answer this question! ...
... Cantor's Continuum Problem: How many real numbers are there? Cantor's Continuum Hypothesis (CH): Any two uncountable sets of real numbers have the same size. ZFC is not strong enough to answer this question! ...
The Diagonal Lemma Fails in Aristotelian Logic
... the term (∃x)Fx as a presupposition. It means that ~(Ex)Fx does not imply that A is false, but rather (Ex)Fx “is a necessary precondition not merely of of the truth of what is said, but of its being either true or false.” [Original italics] (Strawson, p. 174) We will, however, do one better and take ...
... the term (∃x)Fx as a presupposition. It means that ~(Ex)Fx does not imply that A is false, but rather (Ex)Fx “is a necessary precondition not merely of of the truth of what is said, but of its being either true or false.” [Original italics] (Strawson, p. 174) We will, however, do one better and take ...
1. Kripke`s semantics for modal logic
... modality de re, and a mere advocacy of necessity, the belief in modality de dicto. Now, some people say: Let’s give you the concept of necessity. A much worse thing, something creating great additional problems, is whether we can say of any particular that it has necessary or contingent properties…o ...
... modality de re, and a mere advocacy of necessity, the belief in modality de dicto. Now, some people say: Let’s give you the concept of necessity. A much worse thing, something creating great additional problems, is whether we can say of any particular that it has necessary or contingent properties…o ...
Basic Logic - Progetto e
... assume that “All men are mortal” and that “Socratis is a man” and hence conclude “Socratis is mortal”. Here, it is intuitively clear that, if the premises are true, then also the conclusi ...
... assume that “All men are mortal” and that “Socratis is a man” and hence conclude “Socratis is mortal”. Here, it is intuitively clear that, if the premises are true, then also the conclusi ...
Modal_Logics_Eyal_Ariel_151107
... First Order Modal Logic Signature: the unique part of every language - A non-empty set of function symbols. A (maybe empty) set of constants. A (maybe empty) set of predicate symbols. ...
... First Order Modal Logic Signature: the unique part of every language - A non-empty set of function symbols. A (maybe empty) set of constants. A (maybe empty) set of predicate symbols. ...
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 ...
CHAPTER 1 INTRODUCTION 1 Mathematical Paradoxes
... Until recently, till the end of the 19th century, mathematical theories used to be built in an intuitive or axiomatic way. In other words, they were based either intuitive ideas concerning basic notions of the theory - ideas taken from the reality- or on the properties of these notions expressed in ...
... Until recently, till the end of the 19th century, mathematical theories used to be built in an intuitive or axiomatic way. In other words, they were based either intuitive ideas concerning basic notions of the theory - ideas taken from the reality- or on the properties of these notions expressed in ...
Logic Design
... It is common to represent the two states of a binary variable by ‘0’ and ‘1’ Logic circuits are usually implemented using logic gates Circuits in which the output is determined solely by the current inputs are termed combinational logic circuits Logic functions can be described by truth tables ...
... It is common to represent the two states of a binary variable by ‘0’ and ‘1’ Logic circuits are usually implemented using logic gates Circuits in which the output is determined solely by the current inputs are termed combinational logic circuits Logic functions can be described by truth tables ...
Assumption Sets for Extended Logic Programs
... For any models M = hH, T i, M0 = hH 0 , T 0 i, we set M ≤ M0 iff T = T 0 and H ⊆ H 0 . A model M of a program Π is said to be a minimal model of Π, if it is minimal under the ≤-ordering among all models of Π. Definition 3 An N 2-model hH, T i of Π is said to be an equilibrium model of Π iff it is mi ...
... For any models M = hH, T i, M0 = hH 0 , T 0 i, we set M ≤ M0 iff T = T 0 and H ⊆ H 0 . A model M of a program Π is said to be a minimal model of Π, if it is minimal under the ≤-ordering among all models of Π. Definition 3 An N 2-model hH, T i of Π is said to be an equilibrium model of Π iff it is mi ...
x - WordPress.com
... humans. Human beings make decisions based on rules. Although, we may not be aware of it, all the decisions we make are all based on computer like if-then statements. If the weather is fine, then we may decide to go out. If the forecast stays the weather will be bad today, but fine tomorrow, then we ...
... humans. Human beings make decisions based on rules. Although, we may not be aware of it, all the decisions we make are all based on computer like if-then statements. If the weather is fine, then we may decide to go out. If the forecast stays the weather will be bad today, but fine tomorrow, then we ...
Propositions as types
... constructive mathematics, which takes a more conservative view of truth than classical mathematics. Constructive mathematics is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last century. Their views at the time generated ...
... constructive mathematics, which takes a more conservative view of truth than classical mathematics. Constructive mathematics is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last century. Their views at the time generated ...
PDF
... basis of constructive mathematics, which takes a more conservative view of truth than classical mathematics. Constructive mathematics is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last century. Their views at the time g ...
... basis of constructive mathematics, which takes a more conservative view of truth than classical mathematics. Constructive mathematics is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last century. Their views at the time g ...
Second order logic or set theory?
... Evidence for ⊨ ϑM→φ is a proof of φ from ϑM (and comprehension et al. axioms). The proof tells us more than just ⊨ ϑM→φ. If we study a formal system in which the proof is given ...
... Evidence for ⊨ ϑM→φ is a proof of φ from ϑM (and comprehension et al. axioms). The proof tells us more than just ⊨ ϑM→φ. If we study a formal system in which the proof is given ...
INTRODUCTION TO LOGIC Lecture 6 Natural Deduction Proofs in
... Proofs in Natural Deduction Proofs in Natural Deduction are trees of L2 -sentences ...
... Proofs in Natural Deduction Proofs in Natural Deduction are trees of L2 -sentences ...
History and Philosophy of Logic
... that his stature as a philosopher of logic will only increase. O f course, even had he never written a word o f history of logic or o f philosophy of logic, as the founder o f modern modal logic his status as a logician would have been secure. H e has a permanent place in history o f logic. T h i s ...
... that his stature as a philosopher of logic will only increase. O f course, even had he never written a word o f history of logic or o f philosophy of logic, as the founder o f modern modal logic his status as a logician would have been secure. H e has a permanent place in history o f logic. T h i s ...
Chapter 1 Section 2
... true. Then (p ∧ q)∨ ( p ∧ q) would have to be true, but it is not. So, A is not a knight and therefore p must be true. If A is a knave, then B must not be a knight since knaves always lie. So, then both p and q hold since both are knaves. ...
... true. Then (p ∧ q)∨ ( p ∧ q) would have to be true, but it is not. So, A is not a knight and therefore p must be true. If A is a knave, then B must not be a knight since knaves always lie. So, then both p and q hold since both are knaves. ...