5.1 Symbolizing Disjunctions and Conjunctions
This module introduces ways to symbolize disjunctive and conjunctive claims. Symbolization helps us abstract away from the content of a sentence to consider its form.
Table of Contents
- 5.1 Symbolizing Disjunctions and Conjunctions
5.1.1 Disjunctive and Conjunctive Claims
“Or” indicates disjunction. “And” indicates conjunction.
Three Types of Claims
Some claims are structured differently from others. We can distinguish types of claims by the structure they have. Many claims make use of a logical operator, a word like “and”, “or”, “some”, or “not”. This logical operator helps determine would make the claim false, or what would serve as a counterexample to the claim. After reviewing atomic sentences and negations (NOT), we will introduce disjunctions (OR), and conjunctions (AND).
Atomic Sentences (Simple Statements)
Recall that atomic sentences, or “simple statements”, do not have an internal structure using propositional connectives like “and”, “not”, “or”, or “if…then”. They can’t be further divided into pieces. For instance, I am home, or Today is Thursday.
Negations (NOT)
The negation of a claim P is true whenever P is false, and the negation of P is false whenever P is true.
e.g., “Jeff is not in the office today” is true on days when Jeff is out of the office, and false on days when Jeff is in the office.
Explanation: The word “not” expresses what is called negation. It changes true claims to false claims, and false claims to true claims.
NOT A is true in every situation where A is false, and false in every situation where A would be true.
For instance “I am not at the store” is true when “I am at the store” is false, and “Sally isn’t wearing a coat” is false when “Sally is wearing a coat” is true.
Double negation — saying NOT NOT A — is just the same thing as saying A.
Remember that philosophical logic tends to be more literal than ordinary English. In ordinary English, “Jim is not happy” means that Jim is unhappy. But in philosophy, “Jim is not happy” is true in any scenario where “Jim is happy” would be false. So, “Jim is not happy” is true if Jim is feeling mediocre or so-so, neither happy nor unhappy, and it is also true if Jim is asleep or dead, or if Jim doesn’t exist.
Disjunctions (OR)
A disjunction consists in two or more claims, P and Q, linked by the word “or”, where “P or Q” is true when P is true, or when Q is true, or when both are true, and false when both P and Q are false.
e.g., “John is happy or Sally is happy” is true if and only if either (i) John is happy, and Sally is not (ii) Sally is happy or John is not, (iii) John is happy and Sally is also happy.
Explanation: Disjunction is one way to put together a new proposition out of two propositions. The word “or” indicates that, although both could be true, so long as one is true, the sentence is true.
This is sometimes called the “inclusive or” in English, and distinguished from the “exclusive or”. The “exclusive or” means that you have to pick one or the other of two options, but not both. For instance, in “Do you want to take the train, or fly?”, it’s implied that doing both is not an option. Most of the time, though, we’ll mean the “inclusive or”. For instance, “Do you want to go out to eat, or go watch a movie?”, “Yes, let’s do both”!
With the inclusive ‘or’, A or B is true whenever A is true, or whenever B is true, including when both A and B are true: either today I’m going to go grocery shopping, or I’m going to go to the post office can be true even if I do both.
By contrast, the “exclusive or” in English is true when A is true, or B is true, but not when both A and B are true: either give me your money or I kill you implies that if you give me your money, I won’t kill you.
A “disjunct” is a statement which appears on either side of the word “or” in a disjunction. For instance, in the disjunction “I am happy or I am sad”, the disjuncts are “I am happy” and “I am sad”.
The word “unless” in English is equivalent to the inclusive “or” in English. When someone says, “I will pay unless the goods aren’t delivered”, the truth value of this sentence is the same as “Either the goods aren’t delivered, or else I will pay.”
Note that the word “OR” in English suggests that both sides of the OR are genuine possibilities, but they need not be. In logic, the sentence “Today is Monday or the moon is made of cheese” is true if in fact today is Monday.
Lastly, note that the words “either” or “else” don’t change the truth value of a sentence with the word “or”. “Either Mr. Garcia or else Mr. Marquez will assign your duties today” means the same thing as “Mr. Garcia or Mr. Marquez will assign your duties today.”
Conjunctions (AND)
A conjunction consists in two or more claims, P and Q, linked by the word “and”, where P and Q is true when both P is true and Q is true, and false otherwise.
e.g., “John is happy and Sally is happy” is true if and only if John is happy and Sally is also happy, not if only one of them is happy.
Explanation: Conjunction is another simple way to put together two propositions, we most often use the English word “and” to do this, also words like “also” and “but” fulfill the same function.
A and B is true only when A is true and B is also true: “Yes, dear, I took out the trash and the recycling!” is false if I only took out the trash and not the recycling.
Note that in Logic the word “and” doesn’t necessarily suggest anything about order in time, unlike in everyday English. “I brushed my teeth and woke up” is true literally speaking even if I woke up before brushing my teeth (as one would hope).
Some words are often used in addition to the word “and” to add emphasis to a conjunction, but they don’t change the truth value of the sentence:
- both: “Both you and I walked” has the same truth value as “You and I walked”
- also: “You walked and I walked also” has the same truth value as “You walked and I walked.”
- in addition to: “Jose yelled in addition to Bill yelling” has the same truth value as “Jose yelled and Bill yelled.”
There are some other words besides “and” in English which represent conjunctions. These words communicate more information than simply conjunction, but they entail the truth of a conjunction, and so they may be symbolized as conjunctions:
- but: “You sang but I stayed quiet” entails the truth of “You sang and I stayed quiet”
- with: “Orson ran with Maria chasing him” entails the truth of “Orson ran and Maria chased him”
- while: “Isaiah tickled while Josiah giggled” entails the truth of “Isaiah tickled and Josiah giggled.”
A “conjunct” is a statement which appears on either side of the word “and” in a conjunction. For instance, in the conjunction “I am happy and I am glad”, the conjuncts are “I am happy” and “I am glad”.
5.1.2 Symbolizing Disjunctions
The ‘wedge’ symbol or lowercase v represents disjunction
Symbolizing Disjunctions
Disjunctions are symbolized using the v symbol, which stands for the truth functional meaning of “or”, on which a disjunction is false if and only if both disjuncts are false, and true otherwise. Technically, the ‘v’ symbol is called the ‘wedge’, and it is supposed to look like ∨, but for simplicity we will use the lowercase letter v on your keyboard. To remember the symbol means “or”, think of two paths diverging in different directions: you can go one way, or you can go another way down the path.
The v symbol is placed between two atomic sentences, with a single space on each side, to represent the disjunction of those sentences. Two examples:
- “Either Joy visited her mother or her mother visited Joy” = J v M
Where
J = “Joy visited her mother”
M = “Joy’s mother visited Joy”.
- “The house will burn down, or else it will be sold” = H v S
Where
H = “The house will burn down”
S = “The house will be sold”
Notice that we ignore punctuation when symbolizing (the ‘,’ is dropped), and we ignore words like ‘either’ or ‘else’ which don’t change the truth value of the sentence.
Multiple Disjunctions
When there are two or more disjunctions, we place parentheses around one of them in order to group them. It doesn’t make any difference in the truth value of the sentence how we choose to group disjunctions, but for simplicity we group them. For instance,
Let
B = “The Bill will pass”
C = “Congress will adjourn”
P = “The President will scream”
Suppose we want to symbolize, “Either the bill will pass, or Congress will adjourn, or else the President will scream.” We have two choices:
B v (C v P)
or (B v C) v P
These sentences are logically equivalent, so it doesn’t matter which we choose, but we must choose one of them. Note that the order doesn’t make a difference to truth value either. That is, this sentence, “Either Congress will adjourn, or else the bill will pass or the President will scream.” is logically equivalent to the sentence above:
C v (B v P)
When symbolizing, however, we will preserve the order in the original English sentence.
Suppose we have a very long disjunction: “Mary or Thad or Greg or Sally or Buck or Jerry will win.”
Let:
M = “Mary will win”
T = “Thad will win”
G = “Greg will win”
S = “Sally will win”
B = “Buck will win”
J = “Jerry will win”
First, we can symbolize without parentheses:
M v T v G v S v B v J
Now, we have many choices about where we choose to put the parentheses, but we must always make sure that where there are two disjunctions, at least one of them has parentheses around it. If needed, we can place parentheses within parentheses. Here are some ways it could be done:
(M v T) v ((G v S) v (B v J))
((M v T) v G) v ((S v B) v J)
M v (T v (G v (S v (B v J))))
((((M v T) v G) v S) v B) v J
Rephrasing More Literally
It is often helpful to rephrase disjunctions more literally. For instance, “Jay or Alyssa stole the car” is better phrased as “Either Jay stole the car or Alyssa stole the car”, we can then symbolize it as
J v A
where:
J = “Jay stole the car”
A = “Alyssa stole the car”
“You can play baseball or basketball” is better phrased as “You can play baseball or you can play basketball”:
S v K
where:
S = “You can play baseball”
K = “You can play basketball”
Unless
Even though the word “unless” sounds very different than “or”, it it is logically equivalent to the word “or”. So, the word “unless” should be symbolized as though it were written “or”.
“The heat is on unless it is broken” is better phrased as “The heat is on or the heat is broken”:
O v B
where:
O = “The heat is on”
B = “The heat is broken”
“Alfred has high hopes unless he oversleeps” is better phrased as “Alfred has high hopes or Alfred oversleeps.”
H v O
where:
H = “Alfred has high hopes”
O = “Alfred oversleeps”
Negation
Disjunctions can contain negated sentences as well as positive sentences. The negation is always placed immediately next to the sentence which it modifies, without a space. For example:
“Either Joy didn’t visit her mother or her mother visited Joy” = ~J v M
Where
J = “Joy visited her mother”
M = “Joy’s mother visited Joy”.
“The house will burn down, or else it won’t be sold” = H v ~S
Where
H = “The house will burn down”
S = “The house will be sold”
“Alfred doesn’t have high hopes or Alfred doesn’t oversleep” = ~H v ~O
where:
H = “Alfred has high hopes”
O = “Alfred oversleeps”
5.1.3 Symbolizing Conjunctions
The ampersand symbol represents conjunction.
Symbolizing Conjunctions
Conjunctions are symbolized using the & symbol, which stands for the truth functional meaning of “and”, on which a conjunction is true if and only if both conjuncts are true, and false otherwise. The & symbol is on most keyboards and is familiar to most people, so we will use it in exercises in this textbook. Alternatively, many logicians use the symbol ∧ or ^, which looks like an upside-down v, which looks like two separate things coming together (or ‘conjoining’), and shows the relationship between conjunction and disjunction. You need to be able to recognize that ∧ or ^ mean the same thing as &, and you can use ∧ when symbolizing by hand if you like, but when typing answers please use &. Some other logicians use the symbol • in place of &, but we will avoid this, because it looks confusingly like the symbol used for multiplication.
The & symbol is placed between two atomic sentences, with a single space on each side, to represent the conjunction of those sentences. Two examples:
- “Jamie visited his mother and he also brought gifts with him” = J & G
where
J = “Jamie visited his mother”
G = “Jamie brought gifts with him”
- “Amos is a bit trigger-happy, but he has a good heart” = A & H
where
A = “Amos is a bit trigger-happy.”
H = “Amos has a good heart.”
Again, notice that we ignore punctuation when symbolizing (the ‘,’ is dropped), we treat the word ‘but’ in the same way as the word ‘and’, and we ignore words like ‘also’ which don’t change the truth value of the sentence.
Multiple Conjunctions
Just as with disjunctions, when there are two or more conjunctions, we place parentheses around one of them in order to group them. It doesn’t make any difference in the truth value of the sentence how we choose to group conjunctions, but for simplicity we group them. For instance,
Let
U = “Ursula made millions”
S = “Sales are up this year”
O = “People buy oranges”
Suppose we want to symbolize, “People buy oranges, sales are up this year, and Ursula made millions.” We have two choices:
O & (S & U)
or (O & S) & U
These sentences are logically equivalent, so it doesn’t matter which we choose, but we must choose one of them. Note that the order doesn’t make a difference to truth value either. That is, this sentence, “Sales are up this year, people buy oranges, and Ursula made millions” is logically equivalent to the sentence above:
S & (O & U)
Nevertheless, as with disjunctions, we will preserve the order in the original English sentence when symbolizing.
Conjunctions with Disjunctions
Grouping with parentheses does make a difference when we have a mixture of conjunctions and disjunctions. The following two sentences are not logically equivalent:
“Either people don’t buy oranges, or else sales are up this year and Ursula made Millions.”
~O v (S & U)
“Either people don’t buy oranges, or sales are up this year; in addition, Ursula made Millions.”
(~O v S) & U
The second sentence requires that it be true that Ursula made millions, even if people didn’t buy oranges, whereas the first sentence would be true even if Ursula didn’t make millions, provided that people didn’t buy oranges. So, using parentheses correctly does matter when dealing with a mixture of conjunctions and disjunctions.
To know how to place parentheses correctly, we will need to rely on clues like punctuation, extra words like ‘else’, ‘also’, and ‘additionally’, and context. For instance, suppose we have the following sentences:
Let:
M = “Marissa will lose”
T = “Toby will lose”
G = “Gabe will lose”
S = “Shania will lose”
Here are different ways these sentences might be grouped, and how the groupings might be indicated in English:
“Marissa will lose and Toby will lose, or else Gabe will lose and Shania will lose”
(M & T) v (G & S)
“Marissa will lose; either Toby or Gabe will lose, and, additionally, Shania will lose.”
M & ((T v G) & S)
“Either Marissa and Toby will both lose, or Gabe will lose; also, Shania will lose.”
((M & T) v G) & S
“Marissa will lose, and either Toby will lose or Gabe and Shania will lose.”
M & (T v (G & S))
Sometimes, the structure of an English sentence is ambiguous and disjuncts and conjuncts could be grouped in two different ways. In this case, the English sentence needs to be made more precise and the ambiguity removed before it can be symbolized.
Conjunctions with Negations
Negated sentences can be conjoined just as positive sentences can be. Here are a few examples.
Let
P = “Peyton has a mansion.”
Q = “Quinton has an apartment.”
R = “Roy has a tent.”
“Peyton doesn’t have a mansion and Quinton doesn’t have an apartment.”
~P & ~Q
“Roy doesn’t have a tent, but either Quiton has an apartment or Peyton has a mansion.”
~R & (Q v P)
“Either Roy doesn’t have a tent and Quinton doesn’t have an apartment, or else Peyton has a mansion.”
(~R & ~Q) v P
“Either Roy has a tent, or else Quinton doesn’t have an apartment and Peyton doesn’t have a mansion.”
R v (~Q & ~P)
5.1.4 Symbolizing Neither-Nor and Not-Both
Either athlete can win the gold medal, but it is not the case that both will win.
Neither Nor
Sometimes we want to negate not just one disjunct, but a pair of disjuncts. The expression used for this in English is “Neither… Nor”. For instance, “Neither the fish nor the chicken is tasty” is the negation of “Either the fish or the chicken is tasty”. If it is true that either is tasty, then it is false that neither is tasty; if it is false that either is tasty, then it is true that neither is tasty. So, we symbolize this by placing the entire disjunction in parentheses, and then placing a negation in front of the parentheses.
Let
F = “The fish is tasty”
C = “The chicken is tasty”
F v C = “Either the fish or the chicken is tasty.”
~(F v C) = “Neither the fish nor the chicken is tasty.”
A very common mistake people tend to make, perhaps because they wrongly think of the ~ sign as something like the ‘minus’ sign in mathematics, is to try to distribute the ~ sign over what is inside of the parentheses. The following inference is invalid:
- ~(F v C)
- ~F v ~C (from 1) !!INVALID!!
Notice that sentence 2 says, “Either the fish is not tasty, or else the chicken is not tasty.” That sentence would be true so long as the fish was not tasty, even if the chicken was still tasty. On the other hand, sentence 1, “Neither the fish nor the chicken is tasty”, would be false if the chicken was tasty, even if the fish was not tasty.
In fact, ~(F v C) is logically equivalent to ~F & ~C. That is, “neither the fish nor the chicken is tasty” is logically equivalent to “the fish is not tasty and the chicken is not tasty.”
Not Both
Sometimes we want to negate a whole conjunction, rather than the individual conjunctions. We want to say that a pair of things will never be the case, as a pair, although one or the other might be the case. For instance, suppose that there can only be one gold medal winner. “France and Sweden won’t both win the gold medal” is true, even though France might win, and Sweden might win. We symbolize this in the following way:
Let
F = “France will win the gold medal.”
S = “Sweden will win the gold medal.”
F & S = “Both France and Sweden will win the gold medal.”
~(F & S) = “France and Sweden won’t both win the gold medal.”
Again, it may be tempting to think that the ~ sign can be distributed within the parentheses, but the following inference is invalid:
- ~(F & C)
- ~F & ~C (from 1) !!INVALID!!
This is invalid, because sentence 2 says that France won’t win the gold and Sweden won’t win the gold. It might be translated, “France and Sweden both won’t win the gold medal”, or, even better, “Neither France nor Sweden will win the gold medal.” This will be false if Sweden wins the gold medal. On the other hand, sentence 1 only says that “It is not the case that both France and Sweden will win the gold medal”, or “France and Sweden won’t both win the gold medal.” This means that either France won’t win, or Sweden won’t win. This would be true even if Sweden wins the gold medal, provided that France doesn’t win the gold medal.
Review
Here is a chart with some common symbolizations so far.
| ~P | Not P |
| P v Q | P or Q |
| P & Q | P and Q |
| ~P & Q | not-P, and also Q |
| ~P v Q | either not-P, or else Q |
| ~(P & Q) | P and Q are not both the case |
| ~(P v Q) | neither P nor Q is the case |
| ~P & ~Q | P and Q are both not the case |
| ~P v ~Q | either P or Q is not the case |
| P v (Q & R) | either P, or else both Q and R |
| P & (Q v R) | P, and also either Q or R |
| (P v Q) & R | P or Q; additionally, R |
| (P & Q) v R | either P and Q, or else R |
Submodule 5.1 Quiz
Licenses and Attributions
Key Sources:
- Watson, Jeffrey (2019). Introduction to Logic. Licensed under: (CC BY-SA).
Next Page: 5.2 Truth Tables for Conjunctions and Disjunctions