32.2: Symbolization

When we symbolize a claim, what we are doing is breaking it down into propositions, and the logical connectors that join the propositions. We will represent the propositions with letter variables, and we will represent the logical connectors with symbolic operators. A proposition is a simple sentence. For example: ‘men are mortal’, ‘it is raining’, ‘Tom is lefthanded.,’ etc. When we translate ordinary sentences into logical notation (we call this symbolization), we condense the information down into its most basic parts and represent it with a kind of shorthand. Each proposition is abbreviated with a single capital letter. So, corresponding to the examples of simple sentences from above, we can stipulate the symbolization: ‘M’, ‘R’ and ‘T’. There is no rule that says which letter you should use, so you should just pick one that makes sense, usually based on the letters in the sentence you are symbolizing.

• ‘Men are mortal’ is symbolized as:
  • M
• ‘It is raining’ is symbolized as:
  • R
• ‘Tom is left-handed’ is symbolized as:
  • T

Linking these symbolized propositions are logical operators. Logical operators are the symbolic representation of basic features of our grammar that connect propositions into more complicated claims. We have talked about these features at length in Chapter 2 and Chapter 3. They are features like: ‘and’, ‘or’, ‘if…then’, ‘if and only if’ and ‘not’. The chart of these operators can be found below

Again, using the examples from above:

• ‘Men are mortal, and it is raining’ is symbolized as:
  • M & R
• ‘Men are mortal, or it is raining’ is symbolized as:
  • M v R
• ‘If men are mortal, then it is raining’ is symbolized as:
  • M → R
• ‘Men are mortal, if and only if it is raining’ is symbolized as:
  • M ← → R
• ‘Men are not mortal’ is symbolized as:
  • ~M

When typing your symbolizations, a trick for the conditional and the biconditional is to use the ‘<’ ‘>’ signs and two dashes ‘--’. If you are using Microsoft Word, it will replace what you typed with the correct symbol.

It’s important to remember that every proposition needs to be assigned its own unique letter, but if a proposition repeats, then you need to repeat the letter. So, while:

• ‘Men are mortal or it’s raining’ is symbolized as:
  • M v R
• ‘Men are mortal, or men are not mortal’ is symbolized as:
  • M v ~M

You should also keep in mind the ways that logical connectors can be implied, as we learned in Chapter 3. ‘When it’s raining, the lot is full’ is a conditional, even though it doesn’t explicitly say ‘if’ or ‘then’. Similarly, there are plenty of ways to express a negation that don’t explicitly say ‘not’. Lists often confuse people new to symbolization as well. When you encounter a list, think of the commas in the list as saying either ‘and’ or ‘or,’ depending list you’re looking at. Think carefully about what the is actually meant by the propositions.

• ‘Go to the store and get milk, crackers and apples.’

is actually three propositions connected by ‘&s’:

• ‘Go to the store and get milk,’ and
• ‘Go to the store and get crackers,’ and
• ‘Go to the store and get apples.’

So, we will symbolize it as:

• M & C & A

On the other hand:

• ‘For my birthday, I want to go to the beach, the amusement park or shoot heroin.’

is three propositions connected by ‘v’.

• ‘For my birthday, I want to go to the beach,’ or
• ‘For my birthday, I want to go to the amusement park,’ or
• ‘For my birthday, I want to shoot heroin.’

So, we will symbolize it as:

• B v A v H

The final element used in symbolization is parentheses: ‘( ).’ Parentheses are used to separate independent clauses. When all the logical connectors in the sentence are the same, it doesn’t really matter which pairs of propositions get put in the parentheses. The birthday example above can be symbolized:

• ‘(B v A) v H’ or ‘B v (A v H)’

Outside of lists, though, we will need to be careful about separating the clauses the properly. So:

• ‘Men are mortal and it is raining, or Tom is left-handed.’

is symbolized as:

• (M & R) v T, not M & (R v T)

Grammar can be tricky, so it will sometimes be difficult to quickly parse sentences with a large number of variables and connectors. If you get confused, just take your time and think through carefully what is being expressed (and if you are symbolizing the claims of others and they are available, you can always ask them what they meant). Since formal logic is a tool we use academically, it won’t often be the case that you need to go quickly (unlike informal logic, where you are trying to reason while living your life).

Lastly, when parentheses need to go with another set of parentheses, we use brackets for the outside ones. So, for example:

• [M v (M & R)] v (T v R)

More complicated formal logic systems utilize additional logical operators (for instance ‘all’ and ‘some’). Since we are focused on introducing these concepts, however, we will stop with the five most important operators.

One last thing to keep in mind, if you are looking at materials in other sources, some of the logical operators might be represented differently. There are a number of ways that philosophers have represented ‘not’, ‘and’, ‘or’, ‘if/then’ and ‘if/only/if’ over the years. It doesn’t actually matter which symbols are used, as long as the person writing and the person reading know what is going on. That said, it’s important to consistently use a standard; if you are using this text as your primary resource, we encourage you to use the symbols given to you here.

This really is all there is to symbolizing. While it isn’t actually all that much information, it is confusing for most people at first. We promise you if you stick with it and keep practicing, it will begin to make sense. The only thing you can do to speed up the process is to start working through examples. This is going to be true of everything in formal logic. Things may be hard or confusing at first, but confidence comes from repetition.

Exercises

Symbolize each of the following sentences using the logical notation you have just learned.

1. I hate cats and dogs.
2. I need a fork.
3. The groceries can go in either paper or plastic.
4. John will have a sandwich or a burger, but not fish.
5. Either your mom and dad love you or they don’t.
6. Critical reasoning was interesting, but this isn’t.
7. If we take the bus we can’t take the plane or the train.
8. The Marx Brothers are Harpo, Chico, Groucho and Zeppo.
9. If it rains then the game will be canceled.
10. The dog will bite if and only if you bother it.

Answers to Selected Exercises

1. This sentence actually contains two simple sentences: ‘I hate cats,' and 'I hate dogs'. There is also one connector 'and'. So, we assign any letter we want to each of the simple sentences. Let’s use C for 'I hate cats' and D for 'I hate dogs'. We then just put the connector for 'and' in between them and we are done. So: C & D.
2. This is just one simple sentence, 'I need a fork,' and there are no connectors. So, all we have to do is assign a letter to the sentence (let’s use F), and then we are done. So, the final answer is: F.
3. Here we have three simple sentences: 'John will have a sandwich', 'John will have a burger' and 'John will have fish'. The first thing to do is ascribe each of these sentences a letter. Let’s use S, B, and F, respectively. Now, let’s look at the connectors as they appear in the sentence. The first one we run into is 'or,' and it connects simple sentences S and B. So, now we look at the chart to see that the symbol for 'or' is ‘v’. We are now able to symbolize the first part of the sentence: S v B. There are still two more connectors, however. The next one we see is 'but.' If you recall from earlier in the course, 'but' works just like 'and'. We can now connect another part of the sentence. Remember when you have more than two simple sentences, you need to use parentheses. The way you determine where they go is based on the syntax of the sentence. In this case, the comma is able to indicate which part of the sentence is separate from the rest (the last part). This gives us: (S v B) & F. The last connector to add is the 'not'. Negations can range over any part of a sentence, or over the whole sentence, and where it gets placed will depend on the sentence. In this case, it is easy to figure out the 'not' is in front of 'fish,' so we put the symbol for ‘not’ in front of the letter for the simple sentence about fish. This leaves us with the final answer: (S v B) & ~F.