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Lecture 10 Notes
... We see both philosophical and technical reasons for exploring this new semantics. On the philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important factors in the technology of knowled ...
... We see both philosophical and technical reasons for exploring this new semantics. On the philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important factors in the technology of knowled ...
Propositional/First
... • A valid sentence is true in all worlds under all interpretations • If an implication sentence can be shown to be valid, then—given its premise—its consequent can be derived • Different logics make different commitments about what the world is made of and what kind of beliefs we can have regarding ...
... • A valid sentence is true in all worlds under all interpretations • If an implication sentence can be shown to be valid, then—given its premise—its consequent can be derived • Different logics make different commitments about what the world is made of and what kind of beliefs we can have regarding ...
Tautologies Arguments Logical Implication
... 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: ...
handout
... the type ∀α .α also corresponds to logical falsity and is uninhabited as well. Note that we can produce terms with these types if we have recursive functions, as in the following term with type 0: ...
... the type ∀α .α also corresponds to logical falsity and is uninhabited as well. Note that we can produce terms with these types if we have recursive functions, as in the following term with type 0: ...
General Proof Theory - Matematički institut SANU
... essentially by Gödel in his famous incompleteness theorem, carried on further by Gerhard Gentzen with his cut elimination theorem. In 1957, at a famous conference in Ithaca, proof theory was recognized as one of the four pillars of mathematical logic (along with model theory, recursion theory and s ...
... essentially by Gödel in his famous incompleteness theorem, carried on further by Gerhard Gentzen with his cut elimination theorem. In 1957, at a famous conference in Ithaca, proof theory was recognized as one of the four pillars of mathematical logic (along with model theory, recursion theory and s ...
AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic
... (ii) NEGATION, if Φ is a wff, then the expression denoted by ¬Φ is also a wff (iii) CONJUNCTION, if Φ and Ψ are both wffs, then the expression denoted by ( Φ ∧ Ψ) is a wff (iv) DISJUNCTION if Φ and Ψ are both wffs, then the expression denoted by (Φ ∨ Ψ) is a wff (v) CONDITIONAL (with ANTECEDENT and ...
... (ii) NEGATION, if Φ is a wff, then the expression denoted by ¬Φ is also a wff (iii) CONJUNCTION, if Φ and Ψ are both wffs, then the expression denoted by ( Φ ∧ Ψ) is a wff (iv) DISJUNCTION if Φ and Ψ are both wffs, then the expression denoted by (Φ ∨ Ψ) is a wff (v) CONDITIONAL (with ANTECEDENT and ...
Jacques Herbrand (1908 - 1931) Principal writings in logic
... ES(A,p) that makes the expansion true and assigns the numerical value q to the constant c. œxœy∑zı(x,y,z) expresses the existence, for any p and q, of interpretations that make ES(A,p) true, and give the constant c the value q. If the theory is a true theory of arithmetic, then œxœy∑zı(x,y,z) is tru ...
... ES(A,p) that makes the expansion true and assigns the numerical value q to the constant c. œxœy∑zı(x,y,z) expresses the existence, for any p and q, of interpretations that make ES(A,p) true, and give the constant c the value q. If the theory is a true theory of arithmetic, then œxœy∑zı(x,y,z) is tru ...
IntroToLogic - Department of Computer Science
... Russell (1872-1970) and Whitehead Goals was to derive all of mathematics through formal operations on a collection of axioms. Theorem-proving would be mechanical. No intuition would be involved. Strict syntax and formal rules of inference. ...
... Russell (1872-1970) and Whitehead Goals was to derive all of mathematics through formal operations on a collection of axioms. Theorem-proving would be mechanical. No intuition would be involved. Strict syntax and formal rules of inference. ...
Translating Propositions into Categorical Form
... Affirmo and Nego • AffIrmo affirms the statement, and nEgO denies the statement • A: all is/are true • I: some is/are true • E: none is/ are true • O: some is/are not true ...
... Affirmo and Nego • AffIrmo affirms the statement, and nEgO denies the statement • A: all is/are true • I: some is/are true • E: none is/ are true • O: some is/are not true ...
notes
... the type ∀α .α also corresponds to logical falsity and is uninhabited as well. Note that we can produce terms with these types if we have recursive functions, as in the following term with type 0: ...
... the type ∀α .α also corresponds to logical falsity and is uninhabited as well. Note that we can produce terms with these types if we have recursive functions, as in the following term with type 0: ...
1 The Principle of Mathematical Induction
... Exercise 3. Prove the Principle of Mathematical Induction (PMI), assuming the Well-Ordering Principle. A proof by induction involves setting up a statement that we want to prove for all natural numbers1 , and then proving that the two statements (1) and (2) hold for that statement. Exercise 4. Let D ...
... Exercise 3. Prove the Principle of Mathematical Induction (PMI), assuming the Well-Ordering Principle. A proof by induction involves setting up a statement that we want to prove for all natural numbers1 , and then proving that the two statements (1) and (2) hold for that statement. Exercise 4. Let D ...
Language of Logic 1-2B - Winterrowd-math
... What counterexamples to Alice’s logic were given by the Mad Hatter, the March Hare, and the Dormouse? What is the difference between “I mean what I say” and “I say what I mean”? If you had attended the tea party, what counterexample to Alice’s logic could you have added to the conversation? ...
... What counterexamples to Alice’s logic were given by the Mad Hatter, the March Hare, and the Dormouse? What is the difference between “I mean what I say” and “I say what I mean”? If you had attended the tea party, what counterexample to Alice’s logic could you have added to the conversation? ...
Philosophy 120 Symbolic Logic I H. Hamner Hill
... if, but only if, it is possible to construct a proof of the conclusion from the premises. • Semantic definition- An argument is valid if, but only if, there is no row in the truth table for the argument in which all the premises are true and the conclusion false. ...
... if, but only if, it is possible to construct a proof of the conclusion from the premises. • Semantic definition- An argument is valid if, but only if, there is no row in the truth table for the argument in which all the premises are true and the conclusion false. ...
Propositional logic, I
... » An interpretation is a set of associations of atoms to propositions in the world. – In an interpretation, the proposition associated to an atom is called the denotation of that atom. » Under a given interpretation atoms have truth values (True or False) that are determined by the truth or falsity ...
... » An interpretation is a set of associations of atoms to propositions in the world. – In an interpretation, the proposition associated to an atom is called the denotation of that atom. » Under a given interpretation atoms have truth values (True or False) that are determined by the truth or falsity ...