![]() | 4. Tautological Implications and Tautological Equivalences | ![]() | Section 5 Exercises | ![]() | 6. Arguments and Proofs | ![]() | Main Logic Page | ![]() | "Real World" Page |
In the last section, we wrote out all our tautologies in what we called "argument form." For instance, Modus Ponens [(pq)
p]
q was represented as
In words, this says the following:
In other words, if the first two lines of the argument are known to be true statements, then Modus Ponens assures us that the last line is also a true statement.
According to our convention, small letters stand for atomic statements. Now there is no real reason to restrict modus ponens to apply only to such statements. It seems perfectly reasonable to have Modus Ponens apply to all statements, whether or not they are atomic. As an example, we would like to say the following:
In symbols, this is:
In other words, we would like to replace Modus Ponens with the more general and hence more usable form:
where A and B are any statements we care to choose-atomic or not.
In this form, Modus Ponens becomes our first rule of inference:
Apply Modus Ponens to statements 1 and 3 in the following list of premises (that is, statements that we take to be true).
Solution
Notice that all the statements are compound statements, and that they have the following patterns:
Statement A appears twice; in lines (1) and (3). Looking at Modus Ponens, we see that we can deduce B = r~s from these lines. (Line (2) is not going to be used at all; it just goes along for the ride.) Thus, we can enlarge our list as follows:
1. (p![]() ![]() ![]() |
Premise |
2. ~r![]() |
Premise |
3. p![]() |
Premise |
4. r![]() |
1,3 Modus Ponens |
On the right we have given the justification for each line: lines (1) through (3) were given as premises, and line (4) follows by an application of Modus Ponens to lines (1) and (3); hence the justification "1,3 Modus Ponens."
Write the following in symbolic form, and justify it: If Socrates is a man, then Socrates is mortal. Socrates is a man. Therefore, Socrates is mortal.
Solution
Let p: "Socrates is a man" and let q: "Socrates is mortal." Then the given statements have the form:
1. p![]() |
Premise |
2. p | Premise |
3. q | 1,2 Modus Ponens |
We have put the justifications on the right.
Before We Go On ...
We shall talk more about arguments and proofs in the next section. What this list of statements and justifications says is that, if you accept the premises, then you are logically compelled to accept the conclusion, "Socrates is mortal," as well.
Rule of Inference T1
Any tautology that appears on the list at the end of the last section can be used as a rule of inference. |
Apply Modus Tollens to lines (1) and (2) in the following list of premises:
Solution
Looking at the given premises, we see the pattern:
As a rule of inference, Modus Tollens has the following form:
(In words, if AB appears on the list, and if ~B also appears on the list, we can add ~A to the list of true statements.)
This allows us to write:
1. (p![]() ![]() ![]() |
Premise |
2. ~(r![]() |
Premise |
3. (p![]() ![]() |
Premise |
4. ~(p![]() |
1,2 Modus Tollens |
Before We Go On ...
We used C to represent the statement (pq)
p, although we could just as well have represented it by A
D. Since we're not using this statement at all, it doesn't matter how we represent it. On the other hand, in order to be able to use Modus Tollens on lines (1) and (2), it was imperative that we represented line (1) by A
B, and not by the single letter A. If you look at the argument form of Modus Tollens, you will see that it requires a statement of the form A
B (as well as ~B, of course).
Caution
Modus Ponens tells us that, if AB appears on the list, and if A also appears on the list, then we can add B to the list of true statements. If A
B appears on the list, but if A does not appear on the list, then we cannot add B to the list. Put another way, if A implies B is true, then we cannot conclude that B is true until we know that A is true.
Extend the list in Example 3 by first applying De Morgan's Law to line (4), and then applying Simplification, to obtain ~p.
Solution
Here we must apply two laws in succession. The first is De Morgan's Law. The form we need for line (4) is this:
To use this on line (4), we see that A corresponds to p and B to q. Thus we get:
Next, we must apply Simplification to line (5). The argument form of Simplification is:
Here, A = (~p) and B = (~q). Thus we can write
Thus the whole list becomes:
1. (p![]() ![]() ![]() |
Premise |
2. ~(r![]() |
Premise |
3. (p![]() ![]() |
Premise |
4. ~(p![]() |
1,2 Modus Tollens |
5. (~p)![]() |
4, De Morgan |
6. ~p | 5, Simplification |
Before We Go On ...
Note that the sequence of steps above allowed us to conclude ~p from the three listed premises. (Actually, we needed only the first two.) This sequence of steps is called a proof of the argument
(p![]() ![]() ![]() |
~(r![]() |
(p![]() ![]() |
![]() |
![]() |
We'll say this again in the next section.
Rule of Inference T2
We can add any tautology that appears in the list of tautologies at the end of the last section as a new line in our list of true statements. |
Consider the following argument.
- |
![]() |
![]() ![]() |
What this means is that we are given no premises, and are asked to arrive at the conclusion pp. In the following proof, lines (1) and (2) are obtained using a T2 rule of inference from the Double Negative tautology in the general form A
~(~A), while line (3) is an application of a T1 rule obtained from transitivity.
1. p![]() |
Double Negative |
2. ~(~p)![]() |
Double Negative |
3. p![]() |
1,2 Transitivity |
Note that we are permitting ourselves to break up a tautological implication of the form AB into two statements: A
B and B
A. In other words, each tautological equivalence is really giving us two tautological implications for the price of one. That is why we listed two argument forms for most of the equivalences. As for line (3), we used the Transitivity rule:
A![]() |
B![]() |
![]() |
![]() ![]() |
Rule of Inference S (Substitution)
We can use a tautological equivalence to replace any part of a compound statement with an equivalent statement. For instance, we can replace the statement p |
Rule of Inference C (Conjunction)
If A and B are any two lines on our list, then we can add the new line A
All this is saying is that, if A and B are both true statements, then so is A |
Question Are we done yet?
Answer Not quite. What we are doing is giving rules for what lines can be written in the proof of a given argument. We have already been using one rule with gay abandon, and we should write it down:
Rule of Inference P (Premises)
We can write down any premise at any time. |
Of course, this does not entitle us to make up premises as we go along; we will always be told what the premises are before we start, and Rule P applies only to those. Also, it is traditional to write down all the premises at once at the beginning, although some people like to write them down only as they are needed.
To summarize, Here are all the rules of inference we shall be using.
Rules of Inference
Any tautology that appears on the list at the end of the last section can be used as a rule of inference. | |
We can add any tautology that appears in the list at the end of the last section as a new line in our list of true statements. | |
We can use a tautological equivalence to replace any part of a compound statement with an equivalent statement. | |
If A and B are any two lines on our list, then we can add the new line A![]() | |
If We can write down any premise at any time. |
In the following rather tricky proof, we start with two premises, and shall manage to use every single law of inference except for T2:
What we have proved is the validity of the following argument:
a![]() |
b![]() |
![]() |
![]() ![]() ![]() |
In words, if a and b each imply q, then either a or b implies q. Although this seems intuitively obvious, its proof is not!
![]() | 4. Tautological Implications and Tautological Equivalences | ![]() | Section 5 Exercises | ![]() | 6. Arguments and Proofs | ![]() | Main Logic Page | ![]() | "Real World" Page |
![]() | Stefan Waner (matszw@hofstra.edu) | ![]() | Steven R. Costenoble (matsrc@hofstra.edu) |