Chapter 5: Methods of Proof for Boolean Logic
... The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from ...
... The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from ...
A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC–Part
... Metatheorem 4.5. [Inner Generalization] ` 2A ↔ 2(∀x)A. Proof. The ← direction is by 2((∀x)A → A) (boxed axiom (2)), by (M1) and MP. The → direction is (M3). Remark 4.6. The qualifiers “outer” and “inner” are used with respect to the classical logic that M3 extends. Thus, an inner “rule” is not a rul ...
... Metatheorem 4.5. [Inner Generalization] ` 2A ↔ 2(∀x)A. Proof. The ← direction is by 2((∀x)A → A) (boxed axiom (2)), by (M1) and MP. The → direction is (M3). Remark 4.6. The qualifiers “outer” and “inner” are used with respect to the classical logic that M3 extends. Thus, an inner “rule” is not a rul ...
PDF
... In this entry, we will prove the substitution theorem for propositional logic based on the axiom system found here. Besides the deduction theorem, below are some additional results we will need to prove the theorem: 1. If ∆ ` A → B and Γ ` B → C, then ∆, Γ ` A → C. 2. ∆ ` A and ∆ ` B iff ∆ ` A ∧ B. ...
... In this entry, we will prove the substitution theorem for propositional logic based on the axiom system found here. Besides the deduction theorem, below are some additional results we will need to prove the theorem: 1. If ∆ ` A → B and Γ ` B → C, then ∆, Γ ` A → C. 2. ∆ ` A and ∆ ` B iff ∆ ` A ∧ B. ...
Document
... The more fruitful type of definition is a matter of drawing boundary lines that were not previously given at all. What we shall be able to infer from it, cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it. The conclusions we draw from i ...
... The more fruitful type of definition is a matter of drawing boundary lines that were not previously given at all. What we shall be able to infer from it, cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it. The conclusions we draw from i ...
An Overview of Intuitionistic and Linear Logic
... Constructivism is a point of view concerning the concepts and methods used in mathematical proofs, with preference towards constructive concepts and methods. It emerged in the late 19th century, as a response to the increasing use of abstracts concepts and methods in proofs in mathematics. Kronecker ...
... Constructivism is a point of view concerning the concepts and methods used in mathematical proofs, with preference towards constructive concepts and methods. It emerged in the late 19th century, as a response to the increasing use of abstracts concepts and methods in proofs in mathematics. Kronecker ...
Logic - Decision Procedures
... (6) All of them, written by Brown, begin with "Dear Sir"; (7) All of them, written on blue paper, are filed; (8) None of them, written on more than one sheet, are crossed; (9) None of them, that begin with "Dear Sir", are written in the third ...
... (6) All of them, written by Brown, begin with "Dear Sir"; (7) All of them, written on blue paper, are filed; (8) None of them, written on more than one sheet, are crossed; (9) None of them, that begin with "Dear Sir", are written in the third ...
Logic, Sets, and Proofs
... In the discussion that follows, this fixed set will be denoted U . A variable such as x represents some unspecified element from the fixed set U . Example: If Z is the fixed set, then “x is even” is a statement that involves the variable x, and “x > y” involves x and y. When a logical statement cont ...
... In the discussion that follows, this fixed set will be denoted U . A variable such as x represents some unspecified element from the fixed set U . Example: If Z is the fixed set, then “x is even” is a statement that involves the variable x, and “x > y” involves x and y. When a logical statement cont ...
The semantics of propositional logic
... n ≥ 1, if M (n) is true, then M (n + 1) is true (this is the inductive step), then we can conclude that M (n) is true for all natural numbers n. (We don’t actually know how to do this for the specific statement M (n) above, which is called “Goldbach’s conjecture”.) Induction is a way of proving (in ...
... n ≥ 1, if M (n) is true, then M (n + 1) is true (this is the inductive step), then we can conclude that M (n) is true for all natural numbers n. (We don’t actually know how to do this for the specific statement M (n) above, which is called “Goldbach’s conjecture”.) Induction is a way of proving (in ...
MATH 312H–FOUNDATIONS
... This principle has several variants known as recursive definition, inductive definition, finite induction... their meaning is clear whenever they appear in the course. To make it clear at this point by an example: We can say: If n is an even integer, then it can be written in the form 2m where m is ...
... This principle has several variants known as recursive definition, inductive definition, finite induction... their meaning is clear whenever they appear in the course. To make it clear at this point by an example: We can say: If n is an even integer, then it can be written in the form 2m where m is ...
What is a proof? - Computer Science
... Kemp gave a proof that was deemed false 11 years after it was published! His proof, however, contains the essential ideas that were used in subsequent proofs. In our case, we will not learn much from a false proof now, but it will give some insight about the nature of what a proof really is. Conside ...
... Kemp gave a proof that was deemed false 11 years after it was published! His proof, however, contains the essential ideas that were used in subsequent proofs. In our case, we will not learn much from a false proof now, but it will give some insight about the nature of what a proof really is. Conside ...
Propositional Logic
... propositional formulas. Then, for all propositional formulas A and B: M, α |= �A� iff Mα |=PL A, where Mα = {P | M, α |= �P�}. If |=PL A, then |=FOL �A�. If A ≡PL B, then �A� ≡FOL �B�. Some properties of logical equivalence The properties of logical equivalence listed for PL hold for FOL. The followi ...
... propositional formulas. Then, for all propositional formulas A and B: M, α |= �A� iff Mα |=PL A, where Mα = {P | M, α |= �P�}. If |=PL A, then |=FOL �A�. If A ≡PL B, then �A� ≡FOL �B�. Some properties of logical equivalence The properties of logical equivalence listed for PL hold for FOL. The followi ...
Chapter One {Word doc}
... Given a small set of logical equivalences, we can derive new ones in a step by step process. Examples 6→8, pp. 26-27 illustrate how this is done. This process of derivation of new propositions from old is very useful and forms the basis of a lot of work in automated theorem proving. 1.3 - Predicates ...
... Given a small set of logical equivalences, we can derive new ones in a step by step process. Examples 6→8, pp. 26-27 illustrate how this is done. This process of derivation of new propositions from old is very useful and forms the basis of a lot of work in automated theorem proving. 1.3 - Predicates ...
