1. Chapter 1 Section 1.5 Analyzing Arguments

2. Syllogisms A syllogism is a type of deductive reasoning that draws a logical conclusion from two statements. We will show examples of three types of syllogisms in this section: Direct Reasoning, Indirect Reasoning and Transitive Reasoning. Direct Reasoning This uses an implication and a specific instance of the hypothesis to draw a conclusion which is marked with the symbol . Look at the following example. If Dr. Daquila voted then Dr. Daquila is over 18. Dr. Daquila voted. Therefore, Dr. Daquila is over 18. If we let the variables p and q represent the following statements: p : Dr. Daquila voted. q : Dr. Daquila is over 18. In symbolic form (i.e. using letters) the statements consist of the following: (( p  q ) Λ p )  q p  q p  q p  q  q

3. Indirect Reasoning This uses an implication and a specific instance of the negation of the conclusion to draw a specific conclusion that negates the hypothesis. Look at the following example. If you have retired then you collect social security. You do not collect social security. Therefore, you have not retired. If we let the variables p and q represent the following statements: p : You have retired. q : You collect social security. In symbolic form (i.e. using letters) the statements consist of the following: (( p  q ) Λ ~ q )  ~ p p  q p  q ~ q  ~ p pq ~p~p ~q~q p  qp  q ( p  q ) Λ ~ q (( p  q ) Λ ~ q )  ~ p TTFFTFT TFFTFFT FTTFTFT FFTTTTT

4. Transitive Reasoning This uses two implication statements with the conclusion of the first being the hypothesis of the next. Look at the following example. If Hillary wins Iowa then Hillary wins the nomination. If Hillary wins the nomination then Hillary wins the presidency. Therefore, if Hillary wins Iowa then Hillary wins the presidency. If we let the variables p, q and r represent the following statements: p : Hillary wins Iowa. q : Hillary wins the nomination. r : Hillary wins the presidency. In symbolic form (i.e. using letters) the statements consist of the following: (( p  q ) Λ ( q  r ))  ( p  r) The truth table to show this statement is a tautology is very long it has 8 columns and 8 rows. p  q p  q q  r q  r  p  r

5. pqr p→qp→qq→rq→r (p→q)Λ(q→r)(p→q)Λ(q→r) p→rp→r (( p → q )Λ( q → r ))→( p → r ) TTTTTTTT TTFTFFFT TFTFTFTT TFFFTFFT FTTTTTTT FTFTFFTT FFTTTTTT FFFTTTTT The truth table below shows all 8 rows for analyzing the argument that symbolically is given by: (( p → q )Λ( q → r ))→( p → r ). We can see from the last column this represents a tautology.