Lecture4 - Lecture 4 Semantics for Propositional Logic Four...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 4 Semantics for Propositional Logic Four general steps in analysis of the notion of logical consequence •Specify logical form by formulating an abstract notion of sentence. •Give a semantics for logic: specify how abstract sentences come to be true and false. •Fix a syntactic notion of proof : specify rules by which abstract sentences can be manipulated in proofs. •Prove completeness .
Background image of page 1

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

View Full DocumentRight Arrow Icon
The inferences modeled by PROP The bike is in the garage or the bike is in the basement. p 1 ˮ p 2 The bike is not in the garage. ¬ p 1 Therefore, the bike is in the basement. p 2 If the plant has been infected, then it has splotches on its leaves. p 1 p 2 The plant does not have splotches on its leaves . ¬ p 2 The plant has not been infected. ¬ p 1 Dan speaks Spanish and Stella speaks German. p 1 ˭ p 2 Dan speaks Spanish.
Background image of page 2
Just going by the definition of PROP , however, we have no way of distinguishing correct inferences from incorrect ones. By the definition, an element of PROP is just a sequence of symbols. There is nothing in it to rule out the following as incorrect. p 1 p 2 p 1 ˮ p 2 p 2 p 1 p 1 p 2 These inferences, in contrast to the previous three, are not correct logically, given what we intend and ˮ to represent. Why? One answer : it is possible with both inferences for the premises to be true and the conclusion false.
Background image of page 3

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

View Full DocumentRight Arrow Icon
We thus need a precise way of talking about the elements of PROP as true or false, a way that is compatible with what we intend the connectives ˭ , ˮ , ± ¬ ±  to represent. That is, we need to provide a semantics for PROP . The basic idea : the truth or falsity of a propositional symbol p i is open. Or in other words, the truth values of the propositional symbols can vary arbitrarily.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 14

Lecture4 - Lecture 4 Semantics for Propositional Logic Four...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online