Logic 100

Logic

Natural Deduction

The method of natural deduction in sentential/propositional logic is a method designed to show how a valid argument can be proven to be a valid argument by proceeding through a finite number of steps, with each step involving some rule or other which can itself be shown to be a valid rule of inference or logical equivalence.

"Natural deduction" is unhelpful when it comes to invalid arguments. The reason for this is that it is designed in such a way that one proceeds from what is true to what is true, and invalid arguments leave open the possibility of going from what is true to what is false. That is, invalid arguments are arguments in which it is possible that the premises are true and the conclusion false. This is not so with valid arguments; that is, a valid argument cannot have true premises and a false conclusion. If the premises are true in a valid argument, then the conclusion must be true as well.

Take, for example, the first rule in the Rules of Inference, that is modus ponens. The rule says that if a conditional exists on some line, such as

(R ® S)

and if the antecedent exists on another line (it has to be on another line by itself), in this case it would be

then one can write the consequent, in this case,

on a further line (again, by itself). Note how these sentences precisely match, in their forms, the sentences of modus ponens, which is

(p ® q), p, q,

where 'p' and 'q' are variables, that is, they can stand for any sentences whatever. The letter 'p', for example, could stand for (represent) a very long sentence, such as

[E ® (D ® H)].

Let 'q' be 'O'. Then

'(p ® q)'

would be

{[E ® (D ® H)] ® O}

Then, if we wanted to use modus ponens using this sentence, it would go like this:

{[E ® (D ® H)] ® O}, [E ® (D ® H)], O

where '[E ® (D ® H)] ® O}' is the conditional referred to above, '[E ® (D ® H)]' is the antecedent, and 'O' is the consequent.

Simple as that. OK, I know it's not that simple. Really, it's just a matter of pattern-matching.

Why does

{[E ® (D ® H)] ® O}

look so complicated? Perhaps because there are three arrows here, only one of which can be the primary/major connective. This sentence is saying that if

[E ® (D ® H)]

is true, then

is true. What

[E ® (D ® H)]

is saying is that if

is true, then

(D ® H)

is true.

Proofs

The method of proof in Natural Deduction is as follows:

Each line of the proof is numbered, beginning with the number '1'.
The conclusion appears twice in the proof.
- First, directly to the right of the last premise (on the same line, but not numbered, following the "conclusion indicator symbol", which is '|-').
- Second, as the last line of the proof itself, numbered.
Each line contains only one sentence.
Each line contains a justification.
Each justification consists of
- First, the number(s) of the line(s) containing the sentence(s) from which the present sentence was derived.
- Second, the rule (only one) by which the peresent sentence was derived.

For example, consider the following proof, using Simplification and Modus ponens:

1. (T ® W)]

2. (T · L)] '|-' W

3. T --2, Si--

4. W --1,3 MP--

Here we see that line #3 comes from line #2, by the rule Simplification. Line #4 comes from lines #1 and #3 using the rule Modus ponens. Note how the justification portion of the proof lines include just this information and nothing more. Also note that the conclusion, 'W', appears twice: first, directly after the conclusion indicator, and then as a sentence, by itself, on the last line (#4) of the proof.

Let's try another proof, this time using Hypothetical Syllogism, Conjunction, Constructive Dilemma, and Modus Tollens:

1. (N ® R)]

2. (R ® O)]

3. (N v R)

4. [(O v O) ® M] '|-' M

5. (N ® O) 1,2 HS

6. (N ® O) · (R ® O) 2,5 Con

7. (O v O) 3,6 CD

8. M 4,7 MT

This is a good proof. The fourth premise looks kind of wierd, but perhaps only because the antecedent to the conditional is a disjunction of the same simple sentence (O). Note that I used premise #2 to help derive both line #5 and line #6 (see the number 2 in the justification lines of #5 and #6). There is no limit to the number of times you may use a line in a proof (whether it be a premise or not).

Make sure you understand how line #7 was derived in accord with Constructive Dilemma. In the book, the rule is stated in terms of the variables 'p', 'q', 'r', and 's'. Line #6 is the conjunction of lines #2 and #5; 'N' in line #6 corresponds to 'p' in the book; the first 'O' corresponds to 'q' in the book; 'R' corresponds to 'r' in the book; and the second 'O' corresponds to 's' in the book. Then, since Constructive Dilemma requires two premises to establish it's conclusion, line #3 is used as the second premise in the use of Constructive Dilemma. So, 'N' in line #3 corresponds to 'p' in the book and 'R' in line #3 corresponds to 'r' in the book. We are then able to derive line #7, where the first 'O' corresponds to 'q' in the book and the second corresponds to 's' in the book.

If you have questions, please contact me at: eMail

Michael F. Goodman
Department of Philosophy
Humboldt State University