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

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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
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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
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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
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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
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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
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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 .
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