Lecture Note
University
Stanford UniversityCourse
Introduction to LogicPages
1
Academic year
2022
Murphyal
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34
p {margin: 0; padding: 0;} .ft00{font-size:24px;font-family:Arial;color:#000000;} .ft01{font-size:18px;font-family:ArialMT;color:#000000;} .ft02{font-size:18px;font-family:Arial;color:#000000;} .ft03{font-size:18px;font-family:Arial;color:#000000;} .ft04{font-size:18px;line-height:23px;font-family:ArialMT;color:#000000;} .ft05{font-size:18px;line-height:23px;font-family:Arial;color:#000000;} Box Logic Box logic, though seemingly simple on the surface, is a sophisticated language usedto create complex logical systems. In this segment we will investigate some of thelatest innovations in logic technology by learning about box logic. Today our goal wasto look at box logic. There are three constants: a large box, a medium box , and a small box. They're all oriented one way or another. It's easy to form compound sentences from ourconsonantsâ€”if we want to negate a sentence,we just invert it (Not medium box; medium box).To form this kind of junction from our constants,we stack them up (small box on top of mediumbox) or place multiple boxes or stacks of boxeson the table (colossal for). Now let's try to apply the rules we've learned tosolve some problems. For example, let'sconsider the following conditional: If it's Monday,Mary loves Pat or Quincy. If Mary loves Pat, then Mary loves Quincy. Let's see howwe can express this statement in our new language. In order to convert a conditional statement of the form "If p, then q" into an equivalentstatement of the form "not p or q," we need to understand that "p" implies "q" asequivalent to "not p or q." In other words, if it is true that "p" is true, then it must alsobe true that "q" is true. Similarly, if it is false that "p" is true, then it must also be falsethat "q" is true. In this case, if Mary loves Pat, then she must love Quincy as well. Sowe write: if it is Monday (a true statement), then Mary loves Pat or Quincy (a falsestatement). However, if it is not Monday (a false statement), then Mary does not loveeither Pat or Quincy (also a false statement). The solution to the problem, which requires only one step, is shown below. First, werepresent each piece of information in the problem with a box. Then, we applyresolution on literals inside boxes. Here, we have two boxes with medium boxesdown; we can combine these two and eliminate their complementary literals to formanswers. Since it's a duplicate, we can throw that one away. That leaves us with justone box on the stack: the conclusion. If it's Monday, Mary loves Quincy. The language of logic is not concerned with particular sets of symbols, and it ispossible to deduce logical truths by manipulating those symbols. Any instantiation ofthe symbols is equally good, and any operations on them are equally good providedthat they respect that basic abstraction.
Box Logic: Complex Logical Systems
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