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03-propsem

# 03-propsem - The semantics of propositional logic Readings...

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The semantics of propositional logic Readings: Sections 1.3 and 1.4 of Huth and Ryan. In this module, we will nail down the formal definition of a logical formula, and describe the semantics of propositional logic, which will be familiar to you from Math 135 and CS 251. Our final goal is to prove the soundness and completeness of propositional logic, but to do that, we need to be precise about induction, which we review carefully. 1

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The formal language of formulas (1.3) Our formulas can be viewed as strings over an alphabet composed of our atoms ( p, q, r, p 1 , p 2 , . . . ), plus the symbols ¬ , , , , and the open and close parentheses, ( and ). ( is a convenience in our proofs, and shouldn’t appear in formulas.) In the previous module we gave a grammar for the well-formed formulas of propositional logic: φ ::= p | φ φ | φ φ | ¬ φ | φ φ Recall that p ranges over all atoms and φ ranges over all well-formed formulas (wffs) (that is, formulas that conform to the grammar). 2
Also recall the notions of abstract syntax and operator precedence that we discussed in Module 2, and our use of parentheses to explicitly indicate precedence when necessary. 3

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Using notions from CS 135 and CS 241, we can consider the parse tree (a rooted, ordered tree of fanout at most two) for the formula (( ¬ p ) ( q r )) . 4
It’s tempting to try to make this the definition of a formula, especially as you have seen how to prove things about trees in CS 134 and CS 136. But a tree also has a recursive definition, and the drawing is just a visualization of it. The tree is really representing the process of showing that the formula meets our first definition: p , q , r are wffs because they are atoms. ( q r ) is a wff because q and r are wffs. ( ¬ p ) is a wff because p is. (( ¬ p ) ( q r )) is a wff because ( ¬ p ) and ( q r ) are. The formulas occurring in this process are subformulas of (( ¬ p ) ( q r )) . They correspond to complete subtrees rooted at nodes of the parse tree. 5

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Apart from reordering some of the lines in the justification on the previous slide, there is no other way to show that our formula meets the definition. Another way of saying this is that the parse tree is unique. This is true for every formula that satisfies the definition. This will be important for our semantics. The textbook calls this the inversion principle , meaning it is possible to invert the process of building formulas. The textbook asks you to take this on faith, but for those of little faith, we will prove it in a little while, since we’ll have the tools to do so. 6
The ideas of a formal definition of a logical formula and a separate semantics for assigning a truth value to such a formula come from the work of Gottlob Frege in 1879, though the notation he used was quite different. Many mathematicians and philosophers built on Frege’s ideas, including Bertrand Russell, Alfred North Whitehead, Ludwig Wittgenstein, and Emil Post. Throughout their work, the notation changed and evolved.

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03-propsem - The semantics of propositional logic Readings...

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