2/TRUTH-FUNCTIONS
... ForclassDiscussionsOnly.Teacher.Armand.L.Tan.AssociateProfessor. PhilosophyDepartment.SillimanUniversity s6. S.variable: letter use to symbolize statements such as p, q, r, and s. Statements are either simple such as `Roses are Red’ or compound: `Aristotle is Greek and Russell is English.’ Statement ...
... ForclassDiscussionsOnly.Teacher.Armand.L.Tan.AssociateProfessor. PhilosophyDepartment.SillimanUniversity s6. S.variable: letter use to symbolize statements such as p, q, r, and s. Statements are either simple such as `Roses are Red’ or compound: `Aristotle is Greek and Russell is English.’ Statement ...
Theories.Axioms,Rules of Inference
... has rules of inference that allow us to generate other theorems from those axioms. (Axioms are theorems.) When we start ACL2, it has lots of functions already defined and it correspondingly has axioms for those functions in its theory. Remember from the Lecture 5 that some functions are “primitive” ...
... has rules of inference that allow us to generate other theorems from those axioms. (Axioms are theorems.) When we start ACL2, it has lots of functions already defined and it correspondingly has axioms for those functions in its theory. Remember from the Lecture 5 that some functions are “primitive” ...
Day 29 Presentation - Graphing Linear
... In this presentation, we are interested in summarizing the contents of day 26, 27 and 28 and examine ourselves how best we understood the topics. We are going to discuss the methods of graphing functions, the inverses of functions and the arithmetic sequences. ...
... In this presentation, we are interested in summarizing the contents of day 26, 27 and 28 and examine ourselves how best we understood the topics. We are going to discuss the methods of graphing functions, the inverses of functions and the arithmetic sequences. ...
Chapter 15 Logic Name Date Objective: Students will use
... Chapter 15 Logic Name _______________________________________________ Date __________________ Objective: Students will use propositions to create truth tables and logical equivalences in order to draw logical conclusions Proposition Propositions are statements that may be true or false. Propositions ...
... Chapter 15 Logic Name _______________________________________________ Date __________________ Objective: Students will use propositions to create truth tables and logical equivalences in order to draw logical conclusions Proposition Propositions are statements that may be true or false. Propositions ...
Bound and Free Variables Theorems and Proofs
... How do we decide if a first-order formula is true? Need: • a domain D (what are you quantifying over) • an interpretation I that interprets the constants and predicate symbols: ◦ for each constant symbol c, I(c) ∈ D ∗ Which domain element is Alice? ◦ for each unary predicate P , I(P ) is a predicate ...
... How do we decide if a first-order formula is true? Need: • a domain D (what are you quantifying over) • an interpretation I that interprets the constants and predicate symbols: ◦ for each constant symbol c, I(c) ∈ D ∗ Which domain element is Alice? ◦ for each unary predicate P , I(P ) is a predicate ...
Logic and Proof - Collaboratory for Advanced Computing and
... From K. H. Rosen, Discrete Mathematics and Its Applications (McGraw-Hill) Chapter 1 ...
... From K. H. Rosen, Discrete Mathematics and Its Applications (McGraw-Hill) Chapter 1 ...
Syntax of first order logic.
... Φ ` ϕ if there is a Φ-proof in which ϕ occurs. We call a set Φ of sentences a theory if whenever Φ ` ϕ, then ϕ ∈ Φ (“Φ is deductively closed”). Example. Let L = {≤} be the language of partial orders. Let Φp.o. be the axioms of partial orders, and let Φ be the deductive closure of Φp.o. . Φ is not a ...
... Φ ` ϕ if there is a Φ-proof in which ϕ occurs. We call a set Φ of sentences a theory if whenever Φ ` ϕ, then ϕ ∈ Φ (“Φ is deductively closed”). Example. Let L = {≤} be the language of partial orders. Let Φp.o. be the axioms of partial orders, and let Φ be the deductive closure of Φp.o. . Φ is not a ...
Predicate Calculus pt. 2
... Exercise 1 A set of propositional formulas T is called satisfiable iff there is an assignment of the occuring variables which makes all formulas in T true. The compactness theorem of propositional logic says: T is satisfiable iff every finite subset of T is satisfiable. Proof the compactness theorem ...
... Exercise 1 A set of propositional formulas T is called satisfiable iff there is an assignment of the occuring variables which makes all formulas in T true. The compactness theorem of propositional logic says: T is satisfiable iff every finite subset of T is satisfiable. Proof the compactness theorem ...
Stephen Cook and Phuong Nguyen. Logical foundations of proof
... proofs in P. And a sort of converse to this last statement holds too since the theory T proves the soundness of P. Thus, in the language of the previous paragraph, the proof system P is not only complete, but efficiently so, with respect to the propositional translations of bounded theorems in T . A ...
... proofs in P. And a sort of converse to this last statement holds too since the theory T proves the soundness of P. Thus, in the language of the previous paragraph, the proof system P is not only complete, but efficiently so, with respect to the propositional translations of bounded theorems in T . A ...
Sets and Functions
... How can I decide from a graph whether it is a function? The vertical line test is a way to determine whether or not we have a function. If a vertical line intersects the graph in more than one place, then it is NOT a function. The test is simply a restatement of the definition of a function which st ...
... How can I decide from a graph whether it is a function? The vertical line test is a way to determine whether or not we have a function. If a vertical line intersects the graph in more than one place, then it is NOT a function. The test is simply a restatement of the definition of a function which st ...
Handout 14
... Why would we need a formal system? We are already able to construct wellformed formulas and decide on their truthfulness by means of a truth table. However, imagine we had a set of formulas M and we know that they are true – they represent our knowledge about a certain problem. We would then be inte ...
... Why would we need a formal system? We are already able to construct wellformed formulas and decide on their truthfulness by means of a truth table. However, imagine we had a set of formulas M and we know that they are true – they represent our knowledge about a certain problem. We would then be inte ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.