# 18 Another example of usage (Ad)

## Introduction

In logic, there is a logical connective which is usually written as a long, single-lined arrow. It is used to connect two statements, e.g., p and q, like this:

p --> q

It means "If p is true, then q is true". According to the definition of this logical connective, this statement is true unless p is true but q is false. This also means that if p is false, then the whole statement is true regardless of whether q is true or false.

The statement can defined in terms of the boolean operators "and" and "not" as follows:

p --> q is equivalent to not(p and not(q))

This definition allows us to use Peirce's rules to form such a statement. It can be formulated as follows:

```
<!--
var agt=navigator.userAgent.toLowerCase();
var is_win = ( (agt.indexOf("win")!=-1) || (agt.indexOf("16bit")!=-1) );
var is_gecko = (agt.indexOf('gecko') != -1);
if (is_win && !is_gecko) {
document.write("<FONT FACE=\"Symbol\">&#216;</FONT>");
} else {
document.write("&not;");
}
-->
[Proposition:
P

<!--
var agt=navigator.userAgent.toLowerCase();
var is_win = ( (agt.indexOf("win")!=-1) || (agt.indexOf("16bit")!=-1) );
var is_gecko = (agt.indexOf('gecko') != -1);
if (is_win && !is_gecko) {
document.write("<FONT FACE=\"Symbol\">&#216;</FONT>");
} else {
document.write("&not;");
}
-->
[Proposition:
Q
]
]
```

where P and Q stand for any conceptual graphs. Verify for yourself that this closely matches the definition.

Thus we have a way to say "if p is true, then q is true".

## Application to Peirce's rules

Consider the statement, "If I were rich, I would be happy". This is not generally true, of course, but this is just an example. What we are now going to do is to prove that, given that I am happy, then one reason for this could be that I am rich.

Thus we are going to start with the statement "I am happy" and apply Peirce's rules to say "If I am rich, then I am happy."

## Step 1

```[Person: #I]<-(Expr)<-[State: Happy]
"I am the experiencer of a State which is Happy."
```

We will assume that this is true.

## Step 2

Then we apply the "Double negation" rule:

Double negation:
A double negation may be drawn around or removed from any graph or set of graphs in any context.
```
<!--
var agt=navigator.userAgent.toLowerCase();
var is_win = ( (agt.indexOf("win")!=-1) || (agt.indexOf("16bit")!=-1) );
var is_gecko = (agt.indexOf('gecko') != -1);
if (is_win && !is_gecko) {
document.write("<FONT FACE=\"Symbol\">&#216;</FONT>");
} else {
document.write("&not;");
}
-->
[Proposition:

<!--
var agt=navigator.userAgent.toLowerCase();
var is_win = ( (agt.indexOf("win")!=-1) || (agt.indexOf("16bit")!=-1) );
var is_gecko = (agt.indexOf('gecko') != -1);
if (is_win && !is_gecko) {
document.write("<FONT FACE=\"Symbol\">&#216;</FONT>");
} else {
document.write("&not;");
}
-->
[Proposition:
[Person: #I]<-(Expr)<-[State: Happy]
]
]
```

The statement is still true:

"It is not the case that it is not the case that I am happy," which is equivalent to "I am happy."

## Step 3

Then we apply the "Insertion" rule:

Insertion:
Any graph may be inserted in any oddly enclosed context.
```
<!--
var agt=navigator.userAgent.toLowerCase();
var is_win = ( (agt.indexOf("win")!=-1) || (agt.indexOf("16bit")!=-1) );
var is_gecko = (agt.indexOf('gecko') != -1);
if (is_win && !is_gecko) {
document.write("<FONT FACE=\"Symbol\">&#216;</FONT>");
} else {
document.write("&not;");
}
-->
[Proposition:
[Person: #I]<-(Expr)<-[State: Rich]

<!--
var agt=navigator.userAgent.toLowerCase();
var is_win = ( (agt.indexOf("win")!=-1) || (agt.indexOf("16bit")!=-1) );
var is_gecko = (agt.indexOf('gecko') != -1);
if (is_win && !is_gecko) {
document.write("<FONT FACE=\"Symbol\">&#216;</FONT>");
} else {
document.write("&not;");
}
-->
[Proposition:
[Person: #I]<-(Expr)<-[State: Happy]
]
]
"It is not the case that: (I am rich AND it is not the case that: (I
am happy))"
```

This perfectly fits our schema for the "if...then" statement. So what we have said, in effect, is "If I am rich, I am happy."

## Summary

We started by assuming that I was happy. Therefore, there is nothing magical in this derivation. The statement is bound to be true, and we have just proved the statement "If I am rich, I am happy" from the true statement "I am happy". An implication can only be false if the premise is true and the conclusion is false. Since the conclusion is true, i.e., since it is true that I am happy, there is no way the implication can be false.

## Next

Next, we have a summary.

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Up: Part IV: Peirce's rules (Ad)