Logic

I.  Some basic concepts:

    A. Logical and View video

        Written notes to summarize the video:  
        "Logical and" is a little different from ordinary English "and" because "logical and" can only join sentences. So        
        you get something like: <Short sentence A> and <short sentence B> and ... <short sentence Z>.  For the whole
        long sentence to be true, each part of it must be true.  We would not accept as true a statement such as,
        "George Washington was a Russian, and he was the first president of the United States," even though the second
        part of the long sentence is true

    B. Logical or (non-exclusive or)  View video

        Written notes to summarize the video:
        "Logical or" is sometimes called "non-exclusive or" because one part of it being correct does not exclude
        the second part of it from being correct.  We  often use "or" sentences when we have made a partial determination
        but can't settle on a single answer, e.g., "She is a medical doctor, or she is an attorney at law." Possibly she is both.

    C. The symbols used to indicate "and" and "or" sentences  View video

        Written notes to summarize the video:
        Logic books can use different symbols for "and," "or," and other logical connectives.  We will use the
        ampersand (&)  because it is a way of writing the Latin word for "and" (et) as the letters "e" and "t" joined together.  
        (Some books use "∧.")

        Most logic books use "∨" for "or."  That is because the Latin word for "non-exclusive or" is "vel."  

        People ordinarily chose the first letter of one key word in a sentence to represent that sentence in logical form. So for
        "Minnows are small and whales are large," we might write:  M & W.  For "She is a singer or she is a pianist," we
        might write: "S ∨ P."
       

    D. The truth table for and sentences. View video



        Written notes to summarize the truth table for and sentences.  
        This truth table is very straightforward. It shows that "logical and" sentences are only true when both shorter, component,
        sentences are true.

    E. The truth table for or sentences.View video




        Written notes to summarize the truth table for or sentences:
        This truth tables is also pretty straightforward.  They just look at two simple sentences at a time, noting whether
        each simple sentence is true or false, and then set down a determination as to whether we would call their logical
       "or"  combination true or false.  
       
   
    F. If...Then... sentences and their truth tables View video



       
        Written notes to summarize the truth table for "if... then..."  sentences:
        This truth table often causes trouble if ridiculous examples are provided students. Professor Maylon Hepp gives a
        very useful example in his book Thinking things through: An introduction to logic:  "If I give you a penny, then I
        will give you a dollar."  Under what circumstances would we call Professor Hepp a liar? Only if he gives us a
        penny and then refuses to give us a dollar.  If he just gives us a dollar, we have no right to complain. He never said
        he would give us a dollar only if he first gave us a penny. He's just being generous.  If he gives us nothing, he's not
        been demonstrated to be a liar either. He never promised to give us anything. He just promised or predicted what
        he would do providing he first gave us a penny.  

G. .....If and only if..... sentences and their truth tables View First Video View Second Video     



        Written notes to summarize the truth table for "if... then..."  sentences:
        If I say, "I will give you a dollar if and only if I give you a penny," then I am not telling the truth if I only give you
        a penny or if I only give you a dollar. If I give you both, it is clear that I was telling the truth. If I give you nothing
        then I have not been demonstrated to be a liar.

    H. Overview of all truth tables examined so far.View video



                Written notes to summarize the truth table for all  sentences covered so far:
                Note that the crucial thing in the "and" table is to have all the short sentences (A, B,...) be true.
                Note that the crucial thing in the "or" table is to have at least one of the short sentences be true.
                Note that the crucial thing in the "if... then..." table is the case where the "if" part is true, but the "then"  part
                is not true.  Also note that if neither R or D is true, then nothing relevant has happened, so the "if-then"
                statement has not been tested. In that case it is counted as being true.  We would never know for sure
                whether the logical implication was true unless a case occurred in which R was true.  
                Note that the crucial thing for the "if and only if" table are the cases where one of the shorter sentences is
                true and its counterpart is false. When both are false, the whole thing is counted as being true because its
                truth has not been challenged.