Garns Fall 1997
NOTE: Here I must use the " * " for the dot, the " > " for the horseshoe, and the " _ " for the triple bar.
Logic studies the preservation of truth, and propositions or statements are the bearers of truth and falsity.
The truth of a compound statement is systematically dependent upon the truth of the component statements.
Argument forms that reflect this systematic dependence can be shown
to be valid or invalid.
How are the simple pieces of information related to each other?
How can we break down the complex information offered in the premises
to find the simple piece of information in the conclusion that Michael
Jackson is human?
Michael Jackson is a reptile only if he can have reptilean offspring.
If he is not a reptile, then he is either human or an alien.
Jackson can't have reptilean offspring.
And he is not an alien.
So, is Michael Jackson
human?
This argument has four premises, each offering different pieces of information.
No one premise tells us whether Jackson is a human.
1.
Michael Jackson is a reptile only if he can have reptilean offspring.
2.
If he is not a reptile, then he is either human or an alien.
3.
Jackson can't have reptilean offspring.
4.
And he is not an alien.
Premise 2 expresses relationships among three different statements.
Jackson is a reptile.
Jackson is a human.
Jackson is an alien.
If the first statement is not true, then either the second or the third
is true
2.
If he is not a reptile, then he is either human or an alien.
A simple statement does not contain any other statement as a component.
Jackson is a reptile.
A compound statement contains at least one simple statement as a component
and at least one operator or connective.
Jackson is not a reptile.
If Jackson is a reptile, then he has reptilean
offspring.
To display the relationships among statements we abstract the content and use special symbols for the operators and connectives.
In this way we can focus on the form apart from the content.
Use capital letters to stand for particular simple statements.
If Michael Jackson is not a reptile, then he is
either human or an alien.
R = Michael Jackson is a reptile.
H = Michael Jackson is a human.
A = Michael Jackson is an alien.
If not-R, then either H or A.
~ R É (H v A)
| ~ tilda | negation | negates statements | not, it is not the case that |
| * dot | conjunction | conjoins conjuncts | and, also, moreover, but |
| v wedge | disjunction | disjoins disjuncts | (either/)or, unless |
| É > horseshoe | implication, conditional | consequent is conditional on the antecedent | ifÉthen, only if |
| _ triple bar | equivalence | if and only if |
Note: the triple bar looks like "_" in this document and the dot looks like "*"
| ~ A | A negated simple statement |
| ~ (A * B) | A negated conjunction |
| ~ [(A * B) v C] | A negated disjunction; the first disjunct is a conjunction |
| A * B | A conjunction |
| ~ A * B | A conjunction whose first conjunct is a negated statement |
| (A v B) * C | A conjunction whose first conjunct is a disjunction |
| A * [B É > (C v B)] | A conjunction whose second conjunct is a conditional, the consequent of which is a disjunction |
| A v B | A disjunction |
| ~ A v B | A disjunction whose first disjunct is a negated statement |
| (A * B) v C | A disjunction whose first disjunct is a conjunction |
| A É > B | A conditional |
| (A v B) É > ~(B * C) | A conditional whose antecedent is a disjunction and whose consequent is a negated conjunction |
| (A É> B) É > C | A conditional whose antecedent is a conditional |
| A _ B | A biconditional |
| ~ A _ (B É > C) | A biconditional whose left half is a negated statement and whose right half is a conditional |
| [~A * (B v C)] _ B | A biconditional whose left half is a conjunction, the first conjuct is a negated statment and the second conjunct is a disjunction |
| If A, then B | If I pass this course, I will be happy | P É > H |
| A if B | I'll be happy if I pass this course | P É > H |
| A only if B | I'll be happy only if I pass this course | H É > P |
| Only if A, B | Only if I pass this course will I be happy | H É > P |
| A if and only if B | I'll be happy if and only if I pass this course | H _ P |
| Not A, and (/but) B | Gore is not the President, but Clinton is. | ~ G * C |
| Not both A and B | Not both Clinton and Gore are President. | ~ (C * G) |
| Both A and B are not true | Both Clinton and Gore are not blameless. | ~ C * ~ G |
| Neither A nor B | Neither Clinton nor Gore is blameless. | ~ (C v G) |
| Either A is false or B is false | Either Clinton is not the President or Gore is not the President. | ~ C v ~ G |
Upper-case letters (A, B, C, É) are used as names to stand for
particular simple statements.
Lower-case letters (p, q, r, É) are used as variables to stand
for any statement (simple or compound).
The following symbols stand for operators and connectives: * , v, ~,
É, _
Parentheses and brackets are used to avoid confusion by indicated the
scope of particular operators or connectives.