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The Logic of the Bible

_{.}
When people start to mature,
they find themselves asking
big questions.
Why am I here?
What's the purpose of life? _{.}
Where did it all come from,
and where is it all going?
Why is there anything at all?
What should I do with my life?
And how should I behave in the world?

_{.}
They begin to question everything,
and they want to know
the truth for themselves. _{.}
They become dissatisfied
with simply being told
what to believe.
And they wonder
if there are any real reasons
to believe it.

_{.}
So the questions come:
are there any
principles of reason
that we can rely on
to give us certainty
about what to believe? _{.}
Are there any reliable precepts
that give guidance
on how to behave?
Are there any trustworthy rules
that tell us why the world is
the way it is?

_{.}
I believe there are.
And here I would like
to show you
the general principles of reason
that can be used
to understand reality
and our place in it. _{.}
These are the principles
that underlie many of the
proverbs and precepts of Scripture
such as the Golden Rule.

_{.}
The principles
to which I'm referring
are the basic rules of logic.
Logic is the study
of how true and false statements
relate to each other. _{.}
How can we claim things like
God's Word is true
and fail to study logic
which is the basic math
of true and false? _{.}
And what I present here
is not difficult;
it can be taught
to grade-school children.
I talk about
the basic relationships
of AND's, OR's, and IF's. _{.}
Much of this
you will already know instinctively.
And there are many books and resources
to study logic in depth.
So I will be brief.
If you are already familiar with logic,
the interpretations start
here.

_{.}
Every area of study
such as math, science, biology,
history, politics, and theology
all make statements
about the subject
and show how one fact
in that area leads to another fact.

_{.}
Logic is the study of propositions.
And propositions are sentences
that make statements of fact
that we can consider
to be true or false. _{.}
This differs from other sentences
such as questions, or commands,
or exclamations
which make no claims
that are either true or false. _{.}
Other names
for the concept of a proposition
are "statement", "fact", "claim",
"assertion", "assumption", "supposition", etc. _{.}
Some examples of propositions are:
the light is on;
2+3=4;
the sky is blue;
the grass is green;
no one is perfect; _{.}
all men are mortal;
some men are evil,
and there is only one God.
All these statements
are either true or false.

_{.}
A proposition has a truth-value
that is either true or false.
It cannot be both true and false
at the same time,
and it cannot be
neither true nor false. _{.}
A proposition must be
either true or false.
This is much easier
than a number
which could be any one
of an infinite possibility of values.
So logic is much easier than arithmetic.

_{.}
And a proposition
does not depend
on the language used to say it.
It will mean the same thing
and be either true or false
whether it is written
in English or Spanish
or Japanese or Greek or Hebrew. _{.}
And since we are not focusing
on which language to use,
we might as well make things
as simple as possible
and abbreviate our statements
with as few symbols as possible. _{.}
For example,
we could use the letter, L,
to represent the statement,
"the light is on".
Or we could
represent the statement,
"the sky is blue",
with the letter, B. _{.}
We could use the letter, T,
to represent
that a statement is true.
And we could use F for false. _{.}
Then we can
abbreviate the fact
that the sky is blue
by writing, B=T.
And we can write L=F
when the light is not on. _{.}
As long as the symbols used
are defined somewhere
in the text you're reading,
and as long as the author
uses the symbols to mean
the same thing throughout the book, _{.}
then it should be easy enough
to follow the author's arguments
as he uses those symbols.

_{.}
We can use symbols
to represent propositions
in a more general sense.
If we let the letter, p,
stand for any statement,
then we are not concerned
with what the statement is about. _{.}
We're only concerned
with its essential characteristic
of being either true or _{.}
So we must consider
every possible truth-value
that it might have.
We have to
consider the possibility that
it might be true,
and we have to
consider the possibility
that it might be false. _{.}
And if we use another letter, q,
to stand for any other proposition,
then the only thing
we mean by this
is that it is different
than the proposition, p. _{.}
But q is still either true or false,
and we must consider
every possible truth-value of q.

_{.}
And we can construct
more complicated statements
by connecting simple statements
with words like, "and", "or", "not",
"if", "then", "but" and "when", etc.
These more complicated statements _{.}
are called compound statements,
and they are
just as much propositions
that are also either true or false.
For example, consider the statement,
"the paper is white". _{.}
We can abbreviate this statement
with the symbol, W.
Then an easy compound statement
can be constructed
from this simple statement
by negating it. _{.}
A statement is negated
when the word, "not",
is placed in the statement
to make a true statement false
or a false statement true. _{.}
In this example,
if the paper is indeed white,
then W=T.
The negation of this
is the paper is not white.
Or we can write this
as, "not W". _{.}
It is customary
to use the symbol, "~",
to symbolize the word, "not".
Then the negation
of the statement, W, is ~W
and is read, "not W".
If W=T, then ~W=F,
and if W=F, then ~W=T.

_{.}
If we generalize further,
and use an arbitrary proposition, p,
then we can construct
a truth-table to show the effect of a negation
on every possible truth-value
that p can have.
This is shown in the following table.

_{.}
Below is the truth-table
for the negation of proposition, p.

p | ~p |

T | F_{.}_{.} |

F | T |

1 | 2 | 3 |

p | q | p⋀q |

F | F | F_{.}_{.} |

F | T | F |

T | F | F |

T | T | T |

p | q | p⋁q |

F | F | F |

F | T | T_{.}_{.} |

T | F | T |

T | T | T |

And lastly,

p | q | p→q |

F | F | T |

F | T | T_{.}_{.} |

T | F | F |

T | T | T |

1 | 2 | 3 | 4 | 5 | 6 |

p | q | ~q | p→~q | ~(p→~q) | p⋀q |

F | F | T | T | F | F_{.}_{.} |

F | T | F | T | F | F |

T | F | T | T | F | F |

T | T | F | F | T | T |

There are six columns, labeled 1 through 6. Columns 1 and 2 list every combination of truth-values that p and q can have. Column 3 lists the negation of Column 2.

1 | 2 | 3 | 4 | 5 |

p | q | p⋀q | p→q | (p⋀q) → (p→q) |

F | F | F | T | T_{.}_{.} |

F | T | F | T | T |

T | F | F | F | T |

T | T | T | T | T |

Column 3 in the table above is the standard definition of conjunction, and Column 4 is the definition of implication. What should be noticed here is that there is no combination of p and q for which Column 3 is T when Column 4 is F. This means Column 5 is never false.

1 | 2 | 3 | 4 | 5 | 6 | 7 |

p | q | p⋀q | p→q | q→p | (p→q)→(q→p) | (p⋀q)→[(p→q)→(q→p)] |

F | F | F | T | T | T | T_{.}_{.} |

F | T | F | T | F | F | T |

T | F | F | F | T | T | T |

T | T | T | T | T | T | T |

Column 3 and 4 are conjunction and implication as before. Column 5 is where q implies p and is only false in the second row where q is T but p is F.

(p ⋀ q) → [ (p → q) → (q → p) ] The Golden Rule.

R=p⋀q⋀r⋀s⋀t⋀u⋀v⋀w⋀… ,

(p ⋀ R) → [(p → R) → (R → p)] ,

p = (~p → p) The Diamond Rule

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