Introduction to Symbolize Sentential Logic

SYMBOLIC LOGIC TRANSLATING INTO SENTENTIAL LOGIC Translations from natural languages into logical formulas have certain properties. • They are concerned only with truth values • They miss out on subtleties of communication (for example, we translate ‘but’, ‘although,’ and ‘despite the fact that’ as ‘and’). Atomic Claims: Represent each simple claim with a capital letter that corresponds to the sentence (to make it easier to remember). For example, represent the sentence Kara got her hair cut with the letter C. Sentence Connectives: Represent (truth functional) sentence connectives as follows: ~ (NOT), & (AND), V (OR) and  (IF/THEN, IMPLIES) and Ξ (IF AND ONLY IF). In conjunctions, the left conjunct goes to the left of the connective and the right conjunct to the right. Conjunction symbols must have a formula on each side. Example: English: Kara got her hair dyed and cut. Sentential Logic: D & C In disjunctions, the left disjunct goes to the left of the connective and the right disjunct to the right. Conjunction symbols must have a formula on each side. Example: English: Kara got her hair either dyed or cut. Sentential Logic: D V C 1 Parentheses can be used to control the order of processing. For example, if you wanted to translate the sentence “If you either attend enough classes or go to a presentation you will get a good attendance grade,” then you would use parentheses to show that the antecedent is made up of two disjuncts enclosed in parentheses, as follows: (A V P)  G In negations the NOT symbol goes to the left of the formula. Example: English: Kara didn’t get her hair dyed. Sentential Logic: ~D If you want the negation to affect a compound formula (and not just the atomic sentence immediately to its right) you need parentheses. Example: English: It’s not true that Kara got her hair dyed and cut. Sentential Logic: ~(D & C) this means “not both” In conditionals, the antecedent is connected to the consequent with the  symbol, with the antecedent to the left of it and the consequent to its right. Example: English: If Kara got her hair dyed then she got cut. Sentential Logic: D  C NOTE: You must get the antecedent and consequent in the right order around the connective since it says something different to have it the other way. 2 TRANSLATING FROM ENGLISH TO SENTENTIAL LOGIC 1. Identify and circle each sentence connective in the English sentence (and, or, if/then, not, if and only if). 2. Identify and circle each atomic sentence (or shortest complete sentence) in every sentence and choose a capital letter to represent it (one that makes sense is best). 3. Create a dictionary showing what sentence corresponds to each capital letter. 4. Write each sentence connective in its location (and -- &, or – V, if/then , not -- ~, if and only if -- Ξ, leaving blank spaces for parts). 5. Determine where to put each capital letter corresponding to the different atomic sentences (for a conditional it’s important to have the condition or antecedent before the conditional symbol and the result or consequent after it). Write the letters in the appropriate spots. 6. If there is more than one sentence connective, identify the main connective of the English sentence (that affects the biggest pieces). Add parentheses to the formula so that its main connective corresponds to the main connective of the English version of the sentence. Example: If you attend the study skills workshop and bring in the handout you will get attendance credit in the class. 3