Symbolic Logic: Introduction to Sentential Logic

SYMBOLIC LOGIC INTRODUCTION TO SENTENTIAL LOGIC All arguments studied by logic are made up of sentences that state claims. For example, consider the following argument. Premise: If the butler committed the crime, he could not have locked the door. Premise: The screen door was locked. Conclusion: The butler didn’t commit the crime. By translating arguments into logic, it is easier to judge whether the argument is VALID or not, that is, whether it is possible for the conclusion to be FALSE even when all the premises are TRUE. There are many different types of logic. Sentential logic is the study of logic which takes the smallest complete sentence (that expresses a complete thought) as the basic unit, or atomic sentence. In translating sentences into sentential logic you will create simple (atomic) formulas and compound formulas -- that are made up of simple claims (the bricks) and sentential connectives (the mortar). TYPES OF CLAIMS A simple or atomic claim is the shortest possible unit of expression that expresses a complete thought (something capable of being true or false). Kara got her hair cut. A compound claim is any claim made up of simple claims and logical sentence connectives. Compound Claim It’s not true that Kara got her hair cut. Component Atomic Claims Kara got her hair cut. Kara got her hair dyed and cut. Kara got her hair dyed. Kara got her hair cut. In these compound claims the sentential connectives are “It’s not true that” and “and.” TRUTH VALUES Logic only studies sentences that make statements that are true or false (or partially true or partially false). For example, logic does not study sentences that are expressives (Hurray!) or imperatives (Go home!) In classical logic (conceived by Aristotle) every sentence in logic has one of two possible truth values: T (TRUE) or F (FALSE). We cannot evaluate the truth value of any given atomic sentence (say R) unless we know what it says and whether the facts in the world make the sentence true or false, if we know the truth This is because each truth functional sentence connective affects the truth value of the sentence in a deterministic way. For example, if two atomic sentences are connected by an “AND” and both of the atomic sentences are true, then the compound sentence formed by combining the two with the “AND” connective is also true. On the other hand, if one of the atomic sentences is false (i.e. if Kara did not get her hair cut), then the resulting compound sentence is also false. SENTENCE CONNECTIVES Sentence connectives are the glue or mortar that allow you to translate compound claims by building compound formulas from atomic formulas (the bricks). For example, to represent the claim above, you must use the logical or logical sentence connective “AND” to get Kara got her hair dyed AND Kara got her hair cut. TYPES OF SENTENCES There are 4 sentence connectives in classical logic, and any given formula can be identified by the type of sentence connective that dominates it. Connective And Type of Sentence Conjunction Sentence Parts Conjuncts (2) Or Disjunction Disjuncts (2) Not Negation If/then Conditional If and only if Bi-conditional Antecedent & Consequent MAIN CONNECTIVE We identify the sentence type by the main connective. The main connective of any formula with just one sentence connective is that sentence connective. However, compound sentences (and their formulas) often have more than one connective. For example, someone could say “It’s not true that I went to the store and the library.” This sentence contains both a “NOT” and an “AND.” When there is more than one connective we need to figure out which is the main connective. To do so, look for the connective that connects the biggest parts. For example, in the sentence above, the “AND” connects two atomic sentences (creating a conjunction of S and L) and the NOT modifies the whole consequent (the whole conjunction), which is bigger than each atomic sentence since it contains more connectives. Thus the NOT is the main connective in that claim.