Notes and examples

Lesson 1

The purpose of the lesson is to acquaint you with the fundamental, defining concepts of logic. It is these concepts that logic is about.  The glossary on page 24 defines these fundamental concepts.  Start there, and then read the explanations in the textbook and companion.

From the nine items in the glossary, six will continue to be the subject of our study.  Here they are with examples of each:

    Deductive validity:

    'All men are mortal; all Greeks are men; therefore, all Greeks are mortal' is deductively valid.
    'Some lions are ferocious; Lambert is a lion; therefore, Lambert is ferocious' is deductively invalid.

    Logical consistency:

    The sentences 'John is a man' and 'John is a nurse' form a logically consistent set.
    The sentences 'John is a man' and 'John is a mother' form a logically inconsistent set.

    Logical Equivalence

    The sentences 'if Julie sings everyone is happy' and 'if someone is unhappy, Julie doesn't sing' are
    logically equivalent.
    The sentences 'Tom and Jerry are friends' and 'Tom and Jerry are neighbors' are not logically equivalent.

    Logical truth:

    'All bachelors are unmarried' is logically true.
    'Some bachelors are rascals' is not logically true.

    Logical falsity:

    'I've seen an invisible rainbow' is logically false.
    'I've seen "The Invisible Man"' is not logically false.

    Logical indeterminacy:

    Any true/false sentence at all that is neither logically true nor logically false.
 

Worked Examples:

Page 14

1.4E1

Do these entirely by following what the definitions of the terms tell you.

a.    If an argument is valid, all the premises of that argument are true.

        False.  According to the definition of validity it is not possible for the premises to be true and the
        conclusion to be false.  But it is possible for any other combination of truth and falsity to hold, including
        false premises.

h.    If all the premises of an argument are true and the conclusion is false, then argument is invalid.

        True.  This is by definition, since in a valid argument it is not possible for the premises to be true and the
        conclusion at the same time to be false.
 

Page 20

1.6E1

b.    A consistent set with at least one true member and at least one false member.

        {George W. Bush is the President.  Strom Thurmond is the Vice-President}.  Although the first is true and
        the second is false, they could both have been true.  Consistency is a matter of what is possible not what
        is actually the case.

        Please give a different example when you do your credit exercises.

1.6E2

b.    {Tom, Sue and Robin are all bright.  No one who fails "Poetry for Scientists" is bright.  Tom failed "Poetry
        for Scientists".}

        Inconsistent.  We don't need to know whether these sentences are actually true.  (Who are these people
        anyway?)  We can see that if the last two were true, Tom couldn't be bright.  It's not possible for all these
        sentences to be true at once.
 

Page 23

1.7E1 and 2

b.    No two false sentences are logically equivalent.

        False.  The definition of logical equivalence requires that two sentences be true and false in the same
        circumstances.  A pair of equivalent sentences must both be false at the same time if they are false at all.

Page 43

Focus on exercise sets 1 and 5.  These are the kind you'll meet in the test.

2.1E1

f.    Bob jogs regularly; however, Albert doesn't.

        'Bob jogs regularly' is symbolized with 'B', and 'Albert jogs regularly' is symbolized with 'A'.  So we'll put

                  B      A

        leaving space for other symbols.  Now, 'Albert doesn't' is the negation of 'A'.  So we need to put in '~'.

                B    ~A

        And 'however' has the same logical force as 'and' (see p.29).  So we need to put '&' for 'however':

                B & ~A

2.1E5

n.    Although Albert is healthy he does not jog regularly, but Carol does jog regularly if Bob does.

        We need 'H' for 'Albert is healthy', 'A' for 'Albert jogs regularly ', 'C' for Carol jogs regularly, and 'B' for
        'Bob jogs regularly':

                H    A    C    B

        Then we need a tilde ('~') to negate 'A'
 

               H    ~A    C    B

        The first part of the sentence is a conjunction ('although', see p.29):

               (H & ~A)

        The second part is a conditional of the form 'p if q' which is the same as 'if q then p'

               (B > C)

        The whole thing is a 'but' sentence - another conjunction:

               (H & ~A) & (B > C)

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Last modified 02/10/02
Andrew G. Black
Dept. of Philosophy
UMSL
ablack@umsl.edu