Formal Logic 101 – Part 4: Operators and How to Symbolize Them
In the last post we learned how to symbolize propositions, such as “Clara will come to the party” by simply using the letter “C” as shorthand. We also learned to symbolize “If/then” statements such as “If Clara will come to the party then we’ll have a Good time.” By shortening to “C→G”. Of course, not all statements involve the “if/then” operator, so there are some statements where modus ponens simply doesn’t apply. In this lesson, we’ll cover a few additional operators and how to use and symbolize them.
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There are two propositions here: “compound sentence are Quite common”, and “compound sentences can be symbolized”. This is clearly not an “If/Then” statement, so the → won’t be any use here. As such, we must introduce “And” (symbolized ^). If we let “Q” be our shorthand for “compound sentence are Quite common”, and “S” be our shorthand for “compound sentences can be symbolized”, the full sentence becomes: Q^S
Bearing that in mind, there are many words in the English that serve the same logical function as “and” and are therefore symbolized with ^.
Consider:
John likes Mary; however, she cannot stand him.
It’s raining, nevertheless we will go for a jog.
It’s a nice day, but still, I want to stay home.
Irene is not only a musician, but also an accomplished attorney.
John is invited even though he can’t cook.
In all of these cases, there are two propositions, and the joining words serve to indicate that both are true. As such, all of the above can be symbolized using And.
That said, it is worth remembering that – in Logic – we use the “inclusive” Or. To explain what I mean by that, consider an example:
Let’s say you ask your roommate “did I get a package in the mail?” and for some reason they just won’t give you a straight answer. They’re beating around the bush and just will not answer the question, and eventually you get frustrated and say “Look, either I got a package, or I didn’t. Yes or No.”
That is the “exclusive” Or. You have two options and exactly one of them is true. They can’t both be true, they can’t both be false, you either got a package or you didn’t.
The “inclusive” Or is something more like “do you like the indoors or the outdoors?” As a general rule, everyone likes at least one of them, but also some people like both, and that’s fine too. So with the “inclusive” or, at least one of the options must be true, but it’s totally possible that both are true. So, whenever you see something symbolized as “Or”, work on the basic assumption that both options might be true, even if you know, in the real world, that it’s a “one or the other, but not both” situation.
Once again, start by breaking down our two propositions: 1) Gore was president in 2001. 2) Bush was president in 2001.
Let us now consider a previous example of Modus Tollens:
1. If my kitty likes being pet then he will purr when I pet him. Premise (P)
2. My kitty does not purr when I pet him. P
3. Therefore my kitty does not like being pet. 1,2 Modus tollens.
If we wish to symbolize this, line 1 should be fairly simple. Let “L” be shorthand for “my kitty Likes being pet” and let “P” be shorthand for “my kitty will Purr when I pet him.”
1. L → P
So premise 2 would simply become: ~P, which indicates that P is false in this proof.
So the correctly symbolized argument would be:
1. L→P Premise (P)
2. ~P P – (prove ~L)
3. ~L 1,2 Modus tollens.
“Not” interacts with other operators in several different ways, all of which should be touched upon because it can get a bit tricky if you’re not paying attention. Which is why we’re going to have a whole separate post for “Not” and how it interacts with the other operators. For right now I would like to very briefly turn our attention to the use of Parentheses before closing out the post.
The problem with this symbolization is that it is now ambiguous. It might say “If I Shovel snow, then I will be Cold and Hurt my back.” It also might say “If I Shovel snow then I will be Cold; also, I Hurt my back.” If we only have the current symbolization, we’re left to wonder if “I hurt my back” is something that we’re claiming to be true, or it’s something that we're claiming will result from shoveling.
To help remove the ambiguity we use parentheses to help group the parts of the sentence as such:
By using parentheses, we establish that “I will be Cold” and “I will hurt my back” are both consequences of shoveling.
On the other hand (S→C)^H tells us that I will be cold if I shovel snow, and also that my back will hurt, but that my back pain doesn’t have anything to do with whether I shovel snow or not. My back is just going to hurt no matter what.
In short, if you have multiple operators in a sentence you should always have parentheses grouping your propositions together.
A. Mary hit a home run and a triple.
B. John enjoys a baseball game if he can have popcorn.
C. Dogs don’t like bumblebees.
D. John will have a heart attack if he doesn’t cut back on the hotdogs.
E. Either Mike or David will have to clean up the kitchen after dinner.
F. Humanity will either go extinct or be mutated if there’s nuclear war.
A. H^T (Mary hit a Home run And Mary hit a Triple)
B. P→E (If Popcorn, Then he Enjoys baseball)
C. ~D (Not the case that Dogs like bumblebees)
D. ~C→A (If he does Not cut back, then heart Attack)
E. MvD (Mike has to clean the kitchen, or David has to clean the kitchen)
F. N→(EvM) (If Nuclear war, then Extinction or Mutation)
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And
Consider the sentence:
Compound sentences are Quite common, and they can be Symbolized.
Compound sentences are Quite common, and they can be Symbolized.
There are two propositions here: “compound sentence are Quite common”, and “compound sentences can be symbolized”. This is clearly not an “If/Then” statement, so the → won’t be any use here. As such, we must introduce “And” (symbolized ^). If we let “Q” be our shorthand for “compound sentence are Quite common”, and “S” be our shorthand for “compound sentences can be symbolized”, the full sentence becomes: Q^S
Bearing that in mind, there are many words in the English that serve the same logical function as “and” and are therefore symbolized with ^.
Consider:
John likes Mary; however, she cannot stand him.
It’s raining, nevertheless we will go for a jog.
It’s a nice day, but still, I want to stay home.
Irene is not only a musician, but also an accomplished attorney.
John is invited even though he can’t cook.
In all of these cases, there are two propositions, and the joining words serve to indicate that both are true. As such, all of the above can be symbolized using And.
Or
On the other side of the coin from “And”, we have “Or” (symbolized v). “Or” works pretty much exactly the way you think it would “You can have either a Bowl of soup, or a Salad” becomes BvS.That said, it is worth remembering that – in Logic – we use the “inclusive” Or. To explain what I mean by that, consider an example:
Let’s say you ask your roommate “did I get a package in the mail?” and for some reason they just won’t give you a straight answer. They’re beating around the bush and just will not answer the question, and eventually you get frustrated and say “Look, either I got a package, or I didn’t. Yes or No.”
That is the “exclusive” Or. You have two options and exactly one of them is true. They can’t both be true, they can’t both be false, you either got a package or you didn’t.
The “inclusive” Or is something more like “do you like the indoors or the outdoors?” As a general rule, everyone likes at least one of them, but also some people like both, and that’s fine too. So with the “inclusive” or, at least one of the options must be true, but it’s totally possible that both are true. So, whenever you see something symbolized as “Or”, work on the basic assumption that both options might be true, even if you know, in the real world, that it’s a “one or the other, but not both” situation.
Now that we have that figured out, how do we symbolize something like “Either Gore or Bush was president in 2001”?
Once again, start by breaking down our two propositions: 1) Gore was president in 2001. 2) Bush was president in 2001.
The symbolization thus becomes: GvB
Again, remember, even though – in the real world – we know that only one person can be president at a time, this symbolization does NOT rule out the possibility that they were co-presidents.
Not
The last of our basic operators will be “Not”.
Let us now consider a previous example of Modus Tollens:
1. If my kitty likes being pet then he will purr when I pet him. Premise (P)
2. My kitty does not purr when I pet him. P
3. Therefore my kitty does not like being pet. 1,2 Modus tollens.
If we wish to symbolize this, line 1 should be fairly simple. Let “L” be shorthand for “my kitty Likes being pet” and let “P” be shorthand for “my kitty will Purr when I pet him.”
1. L → P
Premise 2, however, should be a stumbling block. Because while P says “my kitty purrs when I pet him.” Premise 2 is saying “no he doesn’t.” To make this work, we introduce “Not” (symbolized ~).
So premise 2 would simply become: ~P, which indicates that P is false in this proof.
So the correctly symbolized argument would be:
1. L→P Premise (P)
2. ~P P – (prove ~L)
3. ~L 1,2 Modus tollens.
“Not” interacts with other operators in several different ways, all of which should be touched upon because it can get a bit tricky if you’re not paying attention. Which is why we’re going to have a whole separate post for “Not” and how it interacts with the other operators. For right now I would like to very briefly turn our attention to the use of Parentheses before closing out the post.
Parentheses
Consider the phrase: If I Shovel snow, then I will be Cold and Hurt my back.
We could symbolize this as: S→C^H.
We could symbolize this as: S→C^H.
The problem with this symbolization is that it is now ambiguous. It might say “If I Shovel snow, then I will be Cold and Hurt my back.” It also might say “If I Shovel snow then I will be Cold; also, I Hurt my back.” If we only have the current symbolization, we’re left to wonder if “I hurt my back” is something that we’re claiming to be true, or it’s something that we're claiming will result from shoveling.
To help remove the ambiguity we use parentheses to help group the parts of the sentence as such:
S→(C^H)
By using parentheses, we establish that “I will be Cold” and “I will hurt my back” are both consequences of shoveling.
On the other hand (S→C)^H tells us that I will be cold if I shovel snow, and also that my back will hurt, but that my back pain doesn’t have anything to do with whether I shovel snow or not. My back is just going to hurt no matter what.
In short, if you have multiple operators in a sentence you should always have parentheses grouping your propositions together.
Practice
Symbolize the following statements.A. Mary hit a home run and a triple.
B. John enjoys a baseball game if he can have popcorn.
C. Dogs don’t like bumblebees.
D. John will have a heart attack if he doesn’t cut back on the hotdogs.
E. Either Mike or David will have to clean up the kitchen after dinner.
F. Humanity will either go extinct or be mutated if there’s nuclear war.
Answers
(Note: The specific letters you use do not matter, the important part is that your answers have the correct structure)A. H^T (Mary hit a Home run And Mary hit a Triple)
B. P→E (If Popcorn, Then he Enjoys baseball)
C. ~D (Not the case that Dogs like bumblebees)
D. ~C→A (If he does Not cut back, then heart Attack)
E. MvD (Mike has to clean the kitchen, or David has to clean the kitchen)
F. N→(EvM) (If Nuclear war, then Extinction or Mutation)
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