Formal Logic 101 – Part 3: Intro to Symbolization
So far, we’ve seen fairly short three-line proofs. For those proofs, writing them out longhand is fine. However, proofs can get much longer. Consider:
Argument 1
1. If Clara will come to the party then we’ll have a good time. premise
2. Clara will come to the party. premise
3. If we have a good time then we’ll come home after midnight. premise
4. If we come home after midnight then we’ll sleep in late. premise
5. We’ll have a good time. from 1 and 2
6. We’ll come home after midnight. from 3 and 5
7. We’ll sleep in late. from 4 and 6
That’s only 7 lines with fairly simple sentences, and 3 applications of modus ponens, and we’re rapidly approaching the point where I think a lot of people – if they had to do it more than once – would find themselves thinking “Isn’t there some way I could do this WITHOUT having to write out every single word every single time?”
That is one of the benefits of symbolizing. We can create a shorthand, do the proof in shorthand, and translate back to longhand if necessary. To begin, consider the shorthand that we used in the previous post to explain the structure of Modus Ponens:
1. If [A] then [B].
2. [A].
3. Therefore, [B].
This will serve as our example for how to symbolize Argument 1. If we look at Premise 1, it exactly mirrors the first premise of the shorthand above. So what we can do is pick a single letter to fill in for individual propositions. Still using Premise 1 as our example, it can be broken down into two propositions: 1) Clara will come to the party, and 2) we’ll have a good time. (Remember from the introduction that a proposition is a statement that can be either true or false.) Instead of having to write out each proposition numerous times, we can assign each proposition a letter to serve as shorthand.
You can pick any letter you like, but I find it most helpful to pick a letter that features prominently in the proposition. So I would make “Clara will come to the party” “C”, and “we’ll have a Good time” “G”.
So Premise 1 becomes: If C then G.
But we’re not quite done. We also will want to symbolize the inference rule. The reason for that is there are several ways to state the same thing in plain English. Consider Premise 1 yet again:
If Clara will come to the party then we’ll have a good time.
Now consider: We’ll have a good time if Clara will come to the party.
Both sentences are equivalent, but they’re phrased very differently. Symbolizing the inference rule lets us cut through the semantic deadwood and simply say “here are the propositions, here is the way that they relate to one another”.
To symbolize an if/then statement like what we see in Premise 1 we will use an arrow (→).
If [A] then [B], thus becomes [A] → [B]. Similarly, the fully symbolized premise 1 is “C→G”. The fully symbolized argument thus becomes:
1. C→G Premise (P)
2. C P
3. G→H P
4. H→S P – (prove “S”)
5. G 1,2 Modus Ponens (MP)
6. H 3,5 MP
7. S 4,6 MP
(Note: There are a few different ways of symbolizing and a few different sets of symbols that can be used. The way that I was first taught involves a less-common set of symbols, but I find them more intuitive, so I’ll be using them for this series.)
(Note 2: Note that all of the Premises are marked as being premises and the final one states “P – (prove ‘S’)”. It is often handy to mark down what your intended conclusion is, as a way of keeping track of it. This is especially useful for long or complicated arguments.)
A. If Max is a poodle then Max is a dog. Max is a poodle. Therefore, Max is a dog.
1.
2.
3.
B. If religions are a good thing then the government should sponsor them. Religion is a good thing. Therefore, the government should sponsor them.
1.
2.
3.
C. Animals can think. If animals can think then they should have rights. Therefore, animals should have rights.
1.
2.
3.
D. If Texas is in the south, then Oregon must be on the coast. Texas is in the south. Oregon is on the coast.
1.
2.
3.
A.
1. P→D P
2. P P
3. D 1,2 MP
B.
1.R→G P
2. R P
3. G 1,2 MP
C.
1. T P
2. T→R P
3. R 1,2 MP
D.
1. T→O P
2. T P
3. O 1,2 MP
Argument 1
1. If Clara will come to the party then we’ll have a good time. premise
2. Clara will come to the party. premise
3. If we have a good time then we’ll come home after midnight. premise
4. If we come home after midnight then we’ll sleep in late. premise
5. We’ll have a good time. from 1 and 2
6. We’ll come home after midnight. from 3 and 5
7. We’ll sleep in late. from 4 and 6
That’s only 7 lines with fairly simple sentences, and 3 applications of modus ponens, and we’re rapidly approaching the point where I think a lot of people – if they had to do it more than once – would find themselves thinking “Isn’t there some way I could do this WITHOUT having to write out every single word every single time?”
That is one of the benefits of symbolizing. We can create a shorthand, do the proof in shorthand, and translate back to longhand if necessary. To begin, consider the shorthand that we used in the previous post to explain the structure of Modus Ponens:
1. If [A] then [B].
2. [A].
3. Therefore, [B].
This will serve as our example for how to symbolize Argument 1. If we look at Premise 1, it exactly mirrors the first premise of the shorthand above. So what we can do is pick a single letter to fill in for individual propositions. Still using Premise 1 as our example, it can be broken down into two propositions: 1) Clara will come to the party, and 2) we’ll have a good time. (Remember from the introduction that a proposition is a statement that can be either true or false.) Instead of having to write out each proposition numerous times, we can assign each proposition a letter to serve as shorthand.
You can pick any letter you like, but I find it most helpful to pick a letter that features prominently in the proposition. So I would make “Clara will come to the party” “C”, and “we’ll have a Good time” “G”.
So Premise 1 becomes: If C then G.
But we’re not quite done. We also will want to symbolize the inference rule. The reason for that is there are several ways to state the same thing in plain English. Consider Premise 1 yet again:
If Clara will come to the party then we’ll have a good time.
Now consider: We’ll have a good time if Clara will come to the party.
Both sentences are equivalent, but they’re phrased very differently. Symbolizing the inference rule lets us cut through the semantic deadwood and simply say “here are the propositions, here is the way that they relate to one another”.
To symbolize an if/then statement like what we see in Premise 1 we will use an arrow (→).
If [A] then [B], thus becomes [A] → [B]. Similarly, the fully symbolized premise 1 is “C→G”. The fully symbolized argument thus becomes:
1. C→G Premise (P)
2. C P
3. G→H P
4. H→S P – (prove “S”)
5. G 1,2 Modus Ponens (MP)
6. H 3,5 MP
7. S 4,6 MP
(Note: There are a few different ways of symbolizing and a few different sets of symbols that can be used. The way that I was first taught involves a less-common set of symbols, but I find them more intuitive, so I’ll be using them for this series.)
(Note 2: Note that all of the Premises are marked as being premises and the final one states “P – (prove ‘S’)”. It is often handy to mark down what your intended conclusion is, as a way of keeping track of it. This is especially useful for long or complicated arguments.)
Part 3 Practice
Symbolize the following valid arguments.A. If Max is a poodle then Max is a dog. Max is a poodle. Therefore, Max is a dog.
1.
2.
3.
B. If religions are a good thing then the government should sponsor them. Religion is a good thing. Therefore, the government should sponsor them.
1.
2.
3.
C. Animals can think. If animals can think then they should have rights. Therefore, animals should have rights.
1.
2.
3.
D. If Texas is in the south, then Oregon must be on the coast. Texas is in the south. Oregon is on the coast.
1.
2.
3.
Answers
(Note: The specific letters you use do not matter, the important part is that your answers have the correct structure)A.
1. P→D P
2. P P
3. D 1,2 MP
B.
1.R→G P
2. R P
3. G 1,2 MP
C.
1. T P
2. T→R P
3. R 1,2 MP
D.
1. T→O P
2. T P
3. O 1,2 MP
Comments
Post a Comment