Lecture 16 Notes
... results we cited in Lecture 15 that show i FOL to be incomplete with respect to the intuitionistic 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 seman ...
... results we cited in Lecture 15 that show i FOL to be incomplete with respect to the intuitionistic 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 seman ...
Homework #5
... valid. You may use either natural deduction (“Lemmon style”), or tableaux. (We use “⊥” to denote any sentence of the form A ∧ ¬A.) (a) ` ¬A ↔ (A → ⊥) (b) ` ¬A ↔ ¬¬¬A (c) ` (A → B) → (¬¬A → ¬¬B) (d) ` ¬¬(A ∧ B) ↔ (¬¬A ∧ ¬¬B) 4. Suppose that the atomic sentences are given by {p0 , p1 , p2 , . . .}. We ...
... valid. You may use either natural deduction (“Lemmon style”), or tableaux. (We use “⊥” to denote any sentence of the form A ∧ ¬A.) (a) ` ¬A ↔ (A → ⊥) (b) ` ¬A ↔ ¬¬¬A (c) ` (A → B) → (¬¬A → ¬¬B) (d) ` ¬¬(A ∧ B) ↔ (¬¬A ∧ ¬¬B) 4. Suppose that the atomic sentences are given by {p0 , p1 , p2 , . . .}. We ...
Lecture 34 Notes
... gives the “Russell” version (p.17). Next Mike shows that Musser’s attempted fix also fails. That was for the programming language Euclid. He comments that in our book, A Programming Logic, 1978, we use a total correctness logic to avoid these problems. The Nuprl type theory deals with partial correc ...
... gives the “Russell” version (p.17). Next Mike shows that Musser’s attempted fix also fails. That was for the programming language Euclid. He comments that in our book, A Programming Logic, 1978, we use a total correctness logic to avoid these problems. The Nuprl type theory deals with partial correc ...
Homework 5
... During the second half of this course you should work on a self-chosen project related to the topic of applied logic. This could, for instance, be a literature study about an interesting or the implementation (and documentation) of a proof environment. We will discuss a few possibilities in class. P ...
... During the second half of this course you should work on a self-chosen project related to the topic of applied logic. This could, for instance, be a literature study about an interesting or the implementation (and documentation) of a proof environment. We will discuss a few possibilities in class. P ...
PHILOSOPHY 326 / MATHEMATICS 307 SYMBOLIC LOGIC This
... 114, Introduction to Symbolic Logic, is a prerequisite for Philosophy 326 (or Mathematics 307). It is assumed that all students will have a thorough grasp of the fundamentals of the two-valued logic of propositions – including the fundamental vocabulary of formal deductive logic, the basic two-value ...
... 114, Introduction to Symbolic Logic, is a prerequisite for Philosophy 326 (or Mathematics 307). It is assumed that all students will have a thorough grasp of the fundamentals of the two-valued logic of propositions – including the fundamental vocabulary of formal deductive logic, the basic two-value ...
PDF
... theorem A → A, followed by an axiom instance, then C, then the result of modus ponens, then an axiom instance, and finally two applications of modus ponens. Note the second to the last formula is just (A → A) → (A → B). – If C and C → B are axioms or in ∆, then ∆ `i A → B based on the deduction C, C ...
... theorem A → A, followed by an axiom instance, then C, then the result of modus ponens, then an axiom instance, and finally two applications of modus ponens. Note the second to the last formula is just (A → A) → (A → B). – If C and C → B are axioms or in ∆, then ∆ `i A → B based on the deduction C, C ...
pdf
... Russell created a logical theory to express all mathematical concepts based on Frege’s analysis. His logic, called type theory, was also designed to avoid Russell’s paradox, a paradox which arose in a certain application of Frege’s logical foundation for arithmetic and lead to a contradiction. Russe ...
... Russell created a logical theory to express all mathematical concepts based on Frege’s analysis. His logic, called type theory, was also designed to avoid Russell’s paradox, a paradox which arose in a certain application of Frege’s logical foundation for arithmetic and lead to a contradiction. Russe ...
Notes
... SL or construct. We need to use a weaker form of or defined by Gödel and Kolmogorov. They use ∼∼ (α | ∼ α) for α | ∼ α where ∼ α is defined to be α → void. ...
... SL or construct. We need to use a weaker form of or defined by Gödel and Kolmogorov. They use ∼∼ (α | ∼ α) for α | ∼ α where ∼ α is defined to be α → void. ...
handout
... ∀α, β, γ . (α → β) ∗ (β → γ) → (α → γ). If we can construct a term of this type, we will have proved the theorem in intuitionistic logic. The program Λα, β, γ . λp : (α → β) ∗ (β → γ). λx : α. (#2 p) ((#1 p) x) does it. This is a function that takes a pair of functions as its argument and returns th ...
... ∀α, β, γ . (α → β) ∗ (β → γ) → (α → γ). If we can construct a term of this type, we will have proved the theorem in intuitionistic logic. The program Λα, β, γ . λp : (α → β) ∗ (β → γ). λx : α. (#2 p) ((#1 p) x) does it. This is a function that takes a pair of functions as its argument and returns th ...
The Origin of Proof Theory and its Evolution
... First-Order Number Theory - PA (Peano Arithmetic) First-order logic has sufficient expressive power for the formalization of virtually all of mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithm ...
... First-Order Number Theory - PA (Peano Arithmetic) First-order logic has sufficient expressive power for the formalization of virtually all of mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithm ...
Howework 8
... P rov be a provability predicate for the theory Q and X and Y be formulas in the Q. Assume |=Q P rov(dXe) ⊃ Y and |=Q P rov(dY e) ⊃ X ...
... P rov be a provability predicate for the theory Q and X and Y be formulas in the Q. Assume |=Q P rov(dXe) ⊃ Y and |=Q P rov(dY e) ⊃ X ...
notes
... must prove either P or ¬P . It may well be that neither is provable, in which case the intuitionist would not accept that P ∨ ¬P . For intuitionists, the implication P ⇒ Q has a much stronger meaning than merely ¬P ∨ Q, as in classical logic. To prove P → Q, one must show how to construct a proof of ...
... must prove either P or ¬P . It may well be that neither is provable, in which case the intuitionist would not accept that P ∨ ¬P . For intuitionists, the implication P ⇒ Q has a much stronger meaning than merely ¬P ∨ Q, as in classical logic. To prove P → Q, one must show how to construct a proof of ...
ppt
... statements are true, what other statements can you also deduce are true? • If I tell you that all men are mortal, and Socrates is a man, what can you deduce? ...
... statements are true, what other statements can you also deduce are true? • If I tell you that all men are mortal, and Socrates is a man, what can you deduce? ...
Lecture 39 Notes
... ∀[x : D].(P (x) ⇒ P (x)) or ∀[x : D].(P (x) & Q(x) ⇒ P (x)). In Lecture 38 there is a discussion of the close connection between programs with assertions (asserted programs) justified by varieties of programming logics based on Hoare logic and programs that are implicit constructive proofs of assert ...
... ∀[x : D].(P (x) ⇒ P (x)) or ∀[x : D].(P (x) & Q(x) ⇒ P (x)). In Lecture 38 there is a discussion of the close connection between programs with assertions (asserted programs) justified by varieties of programming logics based on Hoare logic and programs that are implicit constructive proofs of assert ...
PDF
... you must prove either P or ¬P . It may well be that neither is provable, in which case the intuitionist would not accept that P ∨ ¬P . For intuitionists, the implication P → Q has a much stronger meaning than merely ¬P ∨Q, as in classical logic. To prove P → Q, one must show how to construct a proof ...
... you must prove either P or ¬P . It may well be that neither is provable, in which case the intuitionist would not accept that P ∨ ¬P . For intuitionists, the implication P → Q has a much stronger meaning than merely ¬P ∨Q, as in classical logic. To prove P → Q, one must show how to construct a proof ...
Propositions as types
... If we are given a well-typed term in System F or λ→ , then its proof tree will look exactly like the proof tree for the corresponding formula in intuitionistic logic. This means that every well-typed program proves something, i.e. is a proof in constructive logic. Conversely, every theorem in constr ...
... If we are given a well-typed term in System F or λ→ , then its proof tree will look exactly like the proof tree for the corresponding formula in intuitionistic logic. This means that every well-typed program proves something, i.e. is a proof in constructive logic. Conversely, every theorem in constr ...
PDF
... you must prove either P or ¬P . It may well be that neither is provable, in which case the intuitionist would not accept that P ∨ ¬P . For intuitionists, the implication P → Q has a much stronger meaning than merely ¬P ∨Q, as in classical logic. To prove P → Q, one must show how to construct a proof ...
... you must prove either P or ¬P . It may well be that neither is provable, in which case the intuitionist would not accept that P ∨ ¬P . For intuitionists, the implication P → Q has a much stronger meaning than merely ¬P ∨Q, as in classical logic. To prove P → Q, one must show how to construct a proof ...
Curry–Howard correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence.