{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

02-propnd

# 02-propnd - Propositional logic Readings Sections 1.1 and...

This preview shows pages 1–7. Sign up to view the full content.

Propositional logic Readings: Sections 1.1 and 1.2 of Huth and Ryan. In this module, we will consider propositional logic, which will look familiar to you from Math 135 and CS 251. The difference here is that we first define a formal proof system and practice its use before talking about a semantic interpretation (which will also be familiar) and showing that these two notions coincide. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Declarative sentences (1.1) A proposition or declarative sentence is one that can, in principle, be argued as being true or false. Examples: “My car is green” or “Susan was born in Canada”. Many sentences are not declarative, such as “Help!”, “What time is it?”, or “Get me something to eat.” The declarative sentences above are atomic ; they cannot be decomposed further. A sentence like “My car is green AND you do not have a car” is a compound sentence or compositional sentence . 2
To clarify the manipulations we perform in logical proofs, we will represent declarative sentences symbolically by atoms such as p , q , r . (We avoid t , f , T , F for reasons which will become evident.) Compositional sentences will be represented by formulas , which combine atoms with connectives . Formulas are intended to symbolically represent statements in the type of mathematical or logical reasoning we have done in the past. Our standard set of connectives will be ¬ , , , and . (In Math 135, you also used , which we will not use.) Soon, we will describe the set of formulas as a formal language; for the time being, we use an informal description. 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The set of connectives is due to the British mathematician George Boole, who described an algebra using them (now called Boolean algebra) in 1854. We will introduce the connectives in an intuitive fashion, by describing their effect on declarative sentences. In doing so, we anticipate the semantics which we will use to decide if a sentence is true or false. However, it’s important to keep in mind that our proof system is not concerned with true or false; it is concerned with what constitutes a legal proof. Each of the rules makes intuitive sense, and this is not surprising in light of our goal to show that provable equals true. But we maintain a distinction between semantics and syntax at this point. 4
The formal language of propositional logic Let φ range over the set of propositional formulas. Then the following grammar specifies the set of possible values for φ : φ ::= p | φ φ | φ φ | ¬ φ | φ φ The identifier p ranges over atoms; as we noted previously, we may also use q , r , etc. to denote atoms. This language includes four logical connectives: ¬ is a unary connective, and , , and are binary connectives. 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Abstract syntax In CS241 we studied context-free grammars as a tool for parsing programs. There, our concern was in language recognition and parsing, and therefore our grammars were very precise. Such grammars exemplify what is called concrete syntax .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern