Chapter 2: The Logic of Compound Statements 2.1 Logical Forms and Equivalence 12.1 Logical Forms and Equivalences Logic is a science of the necessary laws of thought, without which no employment of the understanding and the reason takes place. – Immanuel Kant, 1724 – 1804 Foundation for the Metaphysics of Morals, 1785

Logic is the study of reasoning; specifically whether reasoning is correct. Note: We use p, q, and r to represent propositions. Logic does: – Assess if an argument is valid/invalid Logic does not directly: – Assess the truth of atomic statements 22.1 Logical Forms and Equivalences

Statement or Proposition – any sentence that is true or false but not both 32.1 Logical Forms and Equivalences

Examples of Statements: 1.George Washington was the first president of the United States. 2.Baltimore is the capital of Maryland. 3.Seventeen is an even number. The above statements are either true (1) or false (2, 3) 2.1 Logical Forms and Equivalences4

These are not statements: 1.Earth is the only planet in the universe that contains life. 2.Buy two tickets to the rock concert for Friday 3.Why should we study logic? None of the above can be determined to be true or false so they are not statements. 2.1 Logical Forms and Equivalences5

It is possible that a sentence is a statement yet we can not determine its truth or falsity because of an ambiguity or lack of qualification. Examples: 1.Yesterday it was cold. 2.He thinks Philadelphia is a wonderful city. 3.Lucille is a brunette. For (1) we need to determine what we mean by the word “cold” For (2) we need to know whose opinion is being considered. For (3) it depends on which Lucille we are discussing. 2.1 Logical Forms and Equivalences6

In speech and writing, we combine propositions using connectives such as and or even or. For example, “It is snowing” and “It is cold” can be combined into a single proposition “it is snowing and it is cold.” 2.1 Logical Forms and Equivalences7

Let: p = “It is snowing.” q = “It is cold.” 2.1 Logical Forms and Equivalences8 ConnectiveSymbolNameExample Not ~ Negation ~q: It is not the case that it is cold. And  Conjunction p  q: It is snowing and it is cold. Or (inclusive)  Disjunction p  q: It is snowing or it is cold. Or (exclusive)  Exclusive Or p  q: It is snowing or it is cold but not both. If…then…  Conditional p  q: If it is snowing, then it is cold. If and only if (iff)  Biconditional p  q: It is snowing iff it is cold.

The word but translates the same as and. “Jim is tall but he is not heavy” translates to “Jim is tall AND he is not heavy” 2.1 Logical Forms and Equivalences9

The words neither-nor translates the same as not. “Neither a borrower nor a lender be” translates to “Do NOT be a borrower and do NOT be a lender” 2.1 Logical Forms and Equivalences10

If sentences are statements then the must have well-defined truth values meaning the sentences must be either true or false. 2.1 Logical Forms and Equivalences11

Definition If p is a statement variable, the negation of p is “not p” or “it is not the case that p” and is denoted  p. It has the opposite truth value from p: if p is true,  p is false if p is false,  p is true 2.1 Logical Forms and Equivalences12

p pp T F 2.1 Logical Forms and Equivalences13 The truth value for negation are summarized in the truth table on the right.

Definition If p and q are statement variables, the conjunction of p and q is “p and q”, denoted p  q. It is true when BOTH p and q are true. If either p or q is false, or both are false, p  q is false. 2.1 Logical Forms and Equivalences14

The truth value for conjunction are summarized in the truth table on the right. 2.1 Logical Forms and Equivalences15 pq p  q TT TF FT FF

Definition – If p and q are statement variables, the disjunction of p and q is “p or q”, denoted p  q. It is true when either p is true, q is true, or both p and q are true. If both p and q is false, p  q is false. Example – You may have cream or sugar with your coffee 2.1 Logical Forms and Equivalences16

The truth value for conjunction are summarized in the truth table on the right. 2.1 Logical Forms and Equivalences17 pq p  q TT TF FT FF

Definition – If p and q are statement variables, the exclusive or of p and q is “p or q”, denoted p  q. It is true when either p is true or when q is true, but not both. If both p and q is false, p  q is false. If both p and q is true, p  q is false. Example – Your meal comes with soup or salad. 2.1 Logical Forms and Equivalences18

The truth value for conjunction are summarized in the truth table on the right. 2.1 Logical Forms and Equivalences19 pq p  q TT TF FT FF

Definition – A statement form is an expression made up of statement variables such as p, q, and r and logical connectives that becomes a statement when actual statements are substituted for the component statement variables. 2.1 Logical Forms and Equivalences20

Construct a truth table for the statement form (p  q)  (  p  q). 2.1 Logical Forms and Equivalences21 pq pp qqp  q pqpq(p  q)  (  p  q) TT TF FT FF

Definition – A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables. 2.1 Logical Forms and Equivalences22

Definition – A contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. 2.1 Logical Forms and Equivalences23

Definitions – Two statement forms are logically equivalent iff they have identical truth values for each possible substitution of statements for their statement variables. Logical equivalence of statement forms P and Q is denoted by P  Q. 2.1 Logical Forms and Equivalences24

1.Construct a truth table with one column for the truth values of P and another column for the truth values of Q. 2.Check the combination of truth values of the statement variables. a.If in each row the truth value of P is the same as the truth value of Q, then P  Q. b.If in each row the truth value of P is not the same as the truth value of Q, then P  Q. 2.1 Logical Forms and Equivalences25

Are the statement forms  (  p  q) and p  q logically equivalent?   (  p  q)  p  q 2.1 Logical Forms and Equivalences26 pq pp qq pqpq  (  p  q) (p  q) TT TF FT FF

Given any statement variables p, q, and r, a tautology t, and a contradiction c, the following logical equivalences hold. 2.1 Logical Forms and Equivalences27

The negation of an and statement is logically equivalent to the or statement in which each component is negated.  (p  q)   p   q The negation of the or statement is logically equivalent to the and statement in which each component is negated.  (p  q)   p   q 2.1 Logical Forms and Equivalences28

Use De Morgan’s Laws to write negations for the statement. -10 < x < Logical Forms and Equivalences29

Use Theorem to verify the logical equivalences. Supply a reason for each step. p  (  q  p)  p 2.1 Logical Forms and Equivalences30

Continue with logical equivalence. Discuss valid and invalid arguments. 2.1 Logical Forms and Equivalences31