The Structure of Sentential Logic

Objectives Distinguish between simple and compound sentences and be able to identify the simple components of compound sentences. Learn the definitions of sentential operator, compound sentence, and simple sentence. Learn five sentential operators and the symbols for them. Learn how compound formulas are constructed out of more elementary parts.

Compound Sentence It contains another complete declarative sentence as a component Compound or not? John loves Mary and Mary loves David. The person who ate the cake has a guilty conscience. John went to New York and Mary went to New York. Mary will be a good student or a good tennis player. (1) A compound sentence (2) Not a compound sentence because the phrase who ate the cake is not a complete declarative sentence. (3) Not a compound sentence. It just has a compound subject. This may be written as John and Mary went to New York. (4) Compound. It has a compound predicate. Mary will be a good student or Mary will be a good tennis player.

Compound Sentences Sentences with compound subjects and/or compound predicates will be considered compound if they can be paraphrased into sentences that are explicitly compound. One sentence contains another if it literally contains the other as a component or it can be paraphrased into an explicitly compound sentence.

Cases when you cannot tell whether the sentence is genuinely compound or just stating a relationship John and Mary are married The art of paraphrasing is not exact

The ff. Sentences are all simple Jose is happy. Dogs like bones. Children fight a lot. Keith likes bananas on his porridge. The man standing by the door is a doctor. The President liked to have complete silence during his many long, tedious speeches about the virtues of democratic government.

The ff. Sentences are all compound Jose and Maria like cats. John likes cats and snakes. Harvey thinks that the earth is flat. If there are flying saucers, then fish live in trees. Jose, Maria and Harvey like lobster and Big Macs.

A Sentential operator Is an expression containing blanks such that, when the blanks are filled with complete sentences, the result is a sentence. There are a lot of sentential operators in English, although we will be using only 5 in sentential logic.

A Sentential operator in Logic Either _____ or _____ Neither _____ nor _____ _____ and _____ If _____, then _____ _____ if and only if _____ _____ unless _____ _____ after _____ _____ only if _____ _____ because _____

5 Sentential operators  or  Negation Not  p Or p , & or  Technical Name Meaning Example  or  Negation Not  p Or p , & or  Conjunction And p  q V Disjunction Or p v q  or V Exclusive OR XOR p  q ,  or  Conditional Implication If ___ then ____ p  q , , iff or  Biconditional If and only if p  q

5 Sentential operators In p  q, p and q are called conjuncts. In p v q, p and q are called disjuncts. In p  q, p is the antecedent, q is the consequent. We will use single capital letters to represent simple English sentences. (4) Will use the first letter of the sentence or anything that reminds use of the meaning of the sentence.

5 Sentential operators and () Let P be P-Noy is president, B be Binay is vice-president. E be Erap is president. M be Mar Roxas is vice-president. Convert these into symbols: Binay is vice-president and P-Noy is president. Erap or P-Noy is president. Binay is vice-president if and only if P-Noy is president. If Binay is vice-president, then P-Noy is president. Erap is not the president. Not both Erap and P-Noy are president. Either Erap is president and Binay is vice-president, or P-Noy is president and Mar Roxas is vice-president. Mar Roxas is vice-president if and only if P-Noy is president. The necessary use of parentheses in arithmetic is as important its use in logic.

Some important terms Sub-formula – a meaningful formula that occurs as a part of another formula. Major operator/connective – the operator that determines the overall form of the sentence, and is the operator introduced last in the process of constructing the formula from its more elementary components.

What is the major operator? ((A  B)  (C v D)) (((A  B)  (C v D)) v ((E F)  G)) (((A  B) v ((C  D) v F))  ((H  E) v C))

Do Worksheet 1