COMPLEX STATEMENTS

Conditionals       Order of Parts        Conditionals and Arguments      Diagrams

A complex statement is a statement that has another statement as a part. For instance,  "Today is Tuesday and it is raining" is one statement, but it has two statement-parts: "Today is Tuesday," and "it is raining."   A complex statement as a whole is one statement, even though it has parts that are also statements. It is a complex unity. There are many kinds of complex statements, of which two kinds are of special interest to us now: "or" statements (called disjunctions) and "if-then" statements (called conditionals).

Disjunctions.  A disjunction is just an "or" statement. Sometimes an "either", sometimes an "else" is involved. And sometimes the disjunction is slightly disguised.

All of the following amount to the same thing:

    Joe did it or Sam did it.      Joe did it or else Sam did it.

                  Either Joe did it or Sam did it.

Slightly disguised:     Joe or Sam did it = Joe did it or Sam did it.

                Bill is either tired or sleepy = Bill is tired or Bill is sleepy.

Switching the parts (the disjuncts) of a disjunction has no effect on the meaning of the disjunction as a whole:

    [ 1 ] or [ 2 ] <--- is the same as ---> [ 2 ] or [ 1 ]

It's raining or it's snowing <--is same as--> It's snowing or it's raining.
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Conditionals.  A conditional is an "if - - then - -" statement, or a statement that has an "if-then" meaning. This is the most important type of statement for logic and reasoning. Thus we introduce some very special terminology for dealing with conditionals:

    If-word: a word or phrase that means "if", such as: when, provided that

    Antecedent: the statement right after the "if" in an if-then statement.

    Consequent: the statement right after the "then" in an if-then statement.

    Standard form conditional: a statement that fits the following pattern:

                    if + [antecedent] + then + [consequent]

    Disguised conditional: a statement that has the "if-then" meaning, but
     that is not written as a standard form conditional. Any disguised conditional
     can be re-written as a standard form conditional that means the same thing.

Now for some examples:

    Disguised:     I have logic class if today is Monday.

    Standard Form:    If today is Monday then I have logic class.

[today is Monday] is the antecedent; [I have logic class] is the consequent. Note that "if" is not part of the antecedent, and "then" is not part of the consequent!!!

    Disguised:    Whenever I'm tired I'm sleepy.

    Standard Form:    If I'm tired then I'm sleepy.
 

    Disguised:   Joe sleeps in class every time he gets in late the night before.

    Standard form:   If Joe gets in late the night before then he sleeps in class.
 

    Disguised:    I wouldn't have missed the turn had I been alert.

    Standard Form: If I had been alert then I wouldn't have missed the turn.

There are many if-words and phrases. Here are a few of them:

        when       whenever      provided that        every time          assuming that

       on the condition that            each time             on the assumption that

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The Order of the Parts

Recall that with "or" statements you can switch the order of the parts without changing the meaning of the whole complex statement. Not so with conditionals!  The order of the parts, the antecedent and the consequent, is very important.

Consider:

    If [I get an A] then [I will pass the course] = if [1] then [2].

    If [I pass the course] then [I will get an A] = if [2] then [1].

They have rather different meanings, don't they?  The first is obviously true; the second often isn't!  In general,

    if [ 1 ] then [ 2 ] <-- is not the same as --> if [ 2 ] then [ 1 ]
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Conditionals and Arguments

A conditional statement is NOT an argument!  IF and THEN are not inference indicators!  Why not?  Because in an argument the arguer is asserting the reason(s) to be true, and is also asserting the conclusion to be true.  But in a conditional statement neither the antecedent nor the consequent statement is asserted to be true. In other words, the conditional as a whole can be asserted to be true without making any commitment as to whether the antecedent and consequent by themselves are true. It can be true on Tuesday (when you don't have logic class) that IF it is Monday THEN you have logic class. The whole conditional can true even though both of its parts are false. Other examples:

    A) If [you flunk the final], then [you will flunk the course].

This is not an argument. It does not assert that you will really flunk the final, nor does it assert that you will flunk the course. [1] is not a reason, and [2] is not a conclusion. Now look at the contrast with this one:

    B) Since [you will flunk the final], [you will flunk the course].

Here [1] is asserted to be true, and it is offered as a reason why [2] is true also. A) is a conditional statement, not an argument. B) is an argument, not a conditional statement.

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DIAGRAMMING ARGUMENTS THAT CONTAIN COMPLEX STATEMENTS

Disjunctions ("or" statements) and conditionals ("if-then" statements) are complex statements, because they are statements that have other statements as parts. But a complex statement, as a whole, is still one statement. A complex statement, as a whole, can be a reason or can be a conclusion. But you should never make one part of a complex statement a reason and the other part a
conclusion. Example:

             1                            2                             2                               1
If [Freddy is a dog] then [he can bark]. So, if [Freddy can bark] then [he's a dog].

Diagram:        If [ 1 ] then [ 2]
                               |
                     If [ 2 ] then [ 1 ]

This is an argument with one reason and one conclusion, separated by "So".  The reason is a conditional with an antecedent and a consequent.  The conclusion is also a conditional with an antecedent and a consequent.  The one reason and the one conclusion are both complex statements.  It isn't a very good argument (Freddy might be a barking seal), but it's still an argument.
 

DIAGRAMMING TIPS FOR COMPLEX STATEMENTS

1. Watch out for "or" and if-words. When you see one, hunt around for the TWO component statements which are needed for a disjunction or a conditional!

2. Disjunctions and conditionals must be kept together as wholes (which have smaller statement-parts). NEVER make one part a reason and the other part a conclusion. The whole thing can be a reason, or can be a conclusion.

3. In a finished diagram, conditionals sholud  be put in standard form.

4. Never circle words that are used in disjunctions and conditionals (such as: or, either, else, if, when, whenever, provided that). They are not inference indicators.

5. With disjunctions and conditionals, the same number will often appear more than once in a diagram. (Up until now, that never happened.)

6. Connect disjunctive and conditional reasons with related reasons by a "+", underline the reasons, and just draw one arrow (for one step in reasoning).
 

NUMBERS AND OPPOSITES.  It is very important to notice when there is a pair of opposites in an argument. To show where the opposites are in skeletons and diagrams, use the word "not" in front of a number. Example argument:

    If it's snowing, then the weather is cold. But the weather is very warm. So it's not snowing.

    Skeleton: If [ 1 ] then [ 2 ]. But [ not 2]. So [ not 1 ].

    Diagram:   If [ 1 ] then [ 2 ]  +  [ not 2]
                                        |
                                   [ not 1 ]

What this means is that the statements represented by "[ 2 ]" and by "[ not 2 ]" are a pair of opposites. Similarly, "[ 1 ]" and "[ not 1]" represent a pair of opposites.  The symbol "[ not 1 ]" doesn't mean that the statement it stands for has to be a negative statement. It just means that statements "not 1" and "1" are a pair of opposites. It could be that both of them are completely positive statements.

For example, "This month is June" and "This month is July" are both positive statements, but they are opposites because they can't both be true together.

It doesn't matter which one of a pair of opposites you use the "not" with. What matters is that you recognize where the pairs of opposites are!

In the example above, notice the pair [ 1 ] and [ not 1 ] include a positive statement (it's snowing) and a negative statement (it's not snowing). But the pair [ 2 ] and [ not 2 ] are both positive statements. Neither one has any negative word in it. But they are still a pair of opposites, and you need to pay attention to where the pairs of opposites are.
 

ARGUMENT DIAGRAMMING RULES: REVISED VERSION

Now we make some small changes in the way we do argument diagrams, due to the importance of complex statements in reasoning.  The changes are to allow for disjunctions, conditionals, and pairs of opposites.

1. Put brackets around the statements and circle the inference indicators.

2. Use the same number for repeats of the same statement and use different numbers for different statements EXCEPT: Use the "notting" technique to identify pairs of opposites.

3. Doing a skeleton is not necessary, but it is advisable for complicated arguments.  In complicated arguments, a skeleton can be very helpful for detecting the overall structure of the argument.

4. If no argument is present, write "No argument" and stop. Diagrams are only for passages that contain arguments. If there is no argument, there cannot be a diagram!

5. If an argument is present, do a diagram. Now "notted" numbers will appear in diagrams, and so will disjunctions and conditionals.
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