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Logic Unit 2: Symbolization ©2011 Niko Scharer 1 UNIT 2 SENTENTIAL LOGIC: SYMBOLIZATION 2.1 WHAT IS SENTENTIAL LOGIC? Sentential Logic (SL): A branch of logic in which sentences or propositions are used as the basic units. It is also called Propositional Logic or Propositional Calculus. We will use a symbolic language that will let us move from English sentences to symbolic sentences and back again. Each truth-valuable English sentence (statements that can be true or false, rather than questions, orders, exclamations, etc.) will be assigned a symbol and then we can use symbols for logical operators to combine those sentences together. Sentential logic allows us to focus on the logical relations between sentences. By symbolizing truth- functional sentences of natural languages (English, French, Mandarin…), we can focus on the logical structures without being distracted by what the sentences mean. Of course, the disadvantage of this is that it ignores the logical structures within a sentence. Some of those logical relations within sentences will be addressed in the second part of the course (Predicate Logic). The logical connectives that join the simple sentences are ‗truth-functional‘ – they operate on the truth- values of sentences rather than their meanings. For that to work, the atomic sentences need to be simple and (for classical logic), bivalent. The logical operators work on the simple sentences in a systematic way allowing us to calculate the truth-value of complex sentences from the truth-values of simple sentences. Then, we can use the techniques of sentential logic to determine whether sets of complex sentences are consistent and whether arguments are valid or invalid, etc. The truth-functional nature of the logical operators of sentential logic makes it relatively easy to interpret the logical operations electronically or mechanically. ‗Logic gates‘ (AND, OR, NOT…) control the flow of information (truth-values) and are used in logical circuits, calculators and computers. They work by taking the truth-values of one or two sentences as input and outputting truth- values according to the logical function of the ‗gate‘. By assembling and arranging such logic gates, so that the output of one gate is the input for another, one can build more and more complex computing devices. Indeed, you can build simple computing machines out of wooden levers and balls, dominoes, Popsicle sticks, or out of Lego ® or Meccano ® . Logic gates made out of Lego ® . Babbage difference engine made of Meccano ® Constructed by Tim Robinson
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Logic Unit 2: Symbolization ©2011 Niko Scharer 2 2.2 THE SYMBOLS FOR SL In sentential logic we need three types of symbols 1. Symbols for sentences or propositions: capital letters. 2. Symbols for the logical relationships between those sentences: ~ 3. Symbols to keep things clear and organized: brackets and parentheses.
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