Truth Table Definitions of Logical Connectives
1. Truth Functions: Logicians DEFINE logical operators in terms of their relation to the truth or falsehood of the statement(s) that they are operating on.
A. Negation: Let’s consider negation first. If we have “¬P”, what happens when “P” is TRUE? What happens when “P” is FALSE?
To answer these questions, let’s take an example:
“It is not the case that it is raining.” Translation: ¬P
Here, “P” = “it is raining”. First, imagine that it IS raining (i.e., “it is raining” is true). In that case, “It is not the case that it is raining” would be a LIE. It would be FALSE. Now, imagine that it is NOT raining (i.e., “it is raining” is false). In that case, “It is not the case that it is raining would be telling the TRUTH. It would be TRUE.
In short, when “P” is true, “¬P” is false. But, when “P” is false, “¬P” is true. Basically, negation just takes the truth value of the thing being negated and makes it the OPPOSITE of whatever it is. For instance, since “The sky is blue” is TRUE, then “It is NOT the case that the sky is blue” is false.
This relation of the dash operator (“¬”) to the truth of the propositions being operated on (“P”) is called the truth function of that operator. And it can be expressed in a truth table, like this one (“T” means “True” and “F” means “False”):
Negation P ¬P T F F T
Basically, this says that, when “P” is true, “¬P” is false. And when “P” is false, “¬P” is true.
Examples:
“It is not the case that the moon is made of cheese” is true because “the moon is made of cheese” is false.
“It is not the case that grass is green” is false because “grass is green” is true.
1 B. Conjunction: All of the other operators require TWO statements, “P” AND “Q”. What is the relation between “” and the truth or falsehood of “P” and “Q”? Well, it turns out that “P Q” is ONLY true when BOTH “P” AND “Q” are true. It will not do to have only one or the other be true. They must BOTH be true. Consider some examples:
P and Q are true: “The sky is blue and grass is green.” (this sentence is true)
P is true, but Q is false: “Grass is green and dogs are birds.” (this sentence is false)
P is false, but Q is true: “Snakes are insects and the sky is blue.” (this sentence is false)
P and Q are both false: “Snakes are insects and dogs are birds.” (this sentence is false)
We can write the truth table for “” as follows:
Conjunction P Q P Q T T T T F F F T F F F F
C. Disjunction: What is the relation between the “” operator and the truth or falsehood of “P” and “Q”? Well, it turns out “P Q” is ONLY false when BOTH “P” AND “Q” are false. Consider some examples:
P and Q are true: “The sky is blue or grass is green.” (this sentence is true)
P is true, but Q is false: “Grass is green or dogs are birds.” (this sentence is true because grass is green)
P is false, but Q is true: “Snakes are insects or the sky is blue.” (this sentence is true because the sky is blue)
P and Q are both false: “Snakes are insects or dogs are birds.” (this sentence is false)
2 We can write the truth table for “” as follows:
Disjunction P Q P Q T T T T F T F T T F F F
Note: If the first line seems counter-intuitive, that is because “or” is INCLUSIVE. Read on.
“Or” is inclusive: If line-1 seems counter-intuitive, note that this is because we have interpreted “or” as being “inclusive”. This makes sense in many contexts. For instance, when I say that “I will eat strawberry OR chocolate ice cream” and you hand me a bowl with BOTH strawberry and chocolate ice cream, what I have said does not automatically mean that I would refuse the dish.
What “or” means here, then, is that AT LEAST one of the two disjuncts must be true. But, maybe they could BOTH be true as well. So, we say that “or” is inclusive.
But, sometimes, we DO use the word “or” to be “exclusive”. For instance, I might say that, “Timmy is either five or six years old.” Surely I do not mean to allow that Timmy might also be five AND six years old. That is impossible. I mean “or” to be EXCLUSIVE here. That is, if Timmy is five, then this excludes the possibility of him being six, and vice versa. Now, there IS a way to account for an EXCLUSIVE “or”. We would do it like this:
(F S) ¬(F S)
What this says is that “Timmy is either five or six, AND he is NOT both five AND six.” This is how we capture the idea behind the word “or” when it is meant to be exclusive. (We will discuss these more complicated sorts of statements soon.)
D. Conditional: What is the relation between the “” operator and the truth or falsehood of “P” and “Q”? Well, it turns out that “P Q” is ONLY false when “P” (the antecedent) is true, and “Q” (the consequent) is false.
Note: Students sometimes find this counter-intuitive, because it seems like “P Q” should turn out to be FALSE when both “P” and “Q” are false. For instance, “If the moon is made of cheese, then unicorns are invading the capitol” seems totally false, but it turns out to be TRUE according to the definition of “” that I have just given.
3 To understand why “” is defined this way, it may help to consider a scenario, and ask yourself: In which of these four scenarios did I tell a lie?
I say to you, “If you come over and help me move my couch on Saturday, then I will buy you some pizza.” Translation: P Q
Scenario 1: You DO help me, and I DO buy you pizza (P and Q are both true). Scenario 2: You DO help me, but I do NOT buy you pizza (P is true, Q is false). Scenario 3: You do NOT help me, but I DO buy you pizza anyway (P is false, Q is true). Scenario 4: You do NOT help me, and I do NOT buy you pizza (P and Q are both false).
Now, in which of these four scenarios did I tell a lie, or break my promise to you? It seems that I ONLY told a lie in the scenario where you DID come over to help me, but I did NOT buy you pizza. So, “P Q” is only false when “P” is true and “Q” is false.
Alternatively, think of it this way: Which of the 4 people are breaking this law?
Law: “If someone is consuming alcohol, then they are at least 21 years of age.”
Scenario 1: Peggy IS consuming alcohol, and IS over 21 (P and Q are both true). Scenario 2: Sue IS consuming alcohol, but is NOT over 21 (P is true, Q is false). Scenario 3: Billy is NOT consuming alcohol, but IS over 21 (P is false, Q is true). Scenario 4: Jean is NOT consuming alcohol, and is NOT over 21 (P and Q are both false).
ONLY SUE is breaking the law. Again, “P Q” is only violated when P is true and Q is false. We can write the truth table for “” as follows:
Conditional P Q P Q T T T T F F F T T F F T
E. Bi-Conditional: Imagine that I say:
“Peggy will go to the party if and only if Quinn goes to the party.”
There are four possible scenarios. Ask yourself: In which of these four scenarios is the statement above a lie?
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Scenario 1: Peggy goes to the party, AND Quinn goes too. (P and Q are both true). Scenario 2: Peggy goes to the party, but Quinn does NOT go. (P is true, Q is false). Scenario 3: Peggy does NOT go to the party, but Quinn does go. (P is false, Q is true). Scenario 4: Peggy does NOT go, and neither does Quinn. (P and Q are both false).
If I say that “Peggy will go if and ONLY if Quinn goes”, I have told a lie in scenarios 2 and 3, but I have NOT lied in scenarios 1 and 4.
In short, for “P Q” to be true, “P” and “Q” must either BOTH be false, or else they must BOTH be true. They must stand or fall together. We can write the truth table for “” as follows:
Bi-Conditional P Q P Q T T T T F F F T F F F T
Remember that “P Q” is just shorthand for “(P Q) (Q P)”. This is a conjunction. (This is because, as we will soon learn the “main operator” in this formula is the “”.) But, conjunctions such as “P Q” only comes out true when “P” and “Q” are BOTH true. So, likewise, “(P Q) (Q P)” only comes out true when BOTH conjuncts “(P Q)” AND “(Q P)” are true.
We now have a complete set of definitions for our 5 logical operators!
5 Compound Statements and Well-Formed Formulae
So far, we have (mostly) only looked at statements where there is ONE operator (for instance, “¬A”, “A B”, “A B”, “A B”, and “A B”). Each of these has only ONE operator. But, often, statements can be much more complicated than this, and require TWO or more operators. For instance, we already saw one such statement:
(F S) ¬(F S)
Recall that this meant, “He is either five or six, and he is not both five and six.” Notice the words in bold (“or”, “and”, “not”, and “and”). Each of these four words are operators, and so the symbolic translation requires FOUR operator symbols.
But, we can’t write the statement above just ANY old way. The statement above is a “well-formed formula”. A well-formed formula is a formula which does not violate any of the rules for symbolic formulas. Here are several rules of thumb for formulas:
1) Two statement letters can never appear side by side; they must always be separated by an operator (and they cannot be separated ONLY by a “¬”).
The following are NOT well-formed formulas:
AB ¬AB A BC NO! A¬B
To see that these sentences are counter-intuitive, consider how poorly they would look if we translated them into English:
Albert likes cheddar Brenda likes swiss. It is not the case that Albert likes cheddar Brenda likes swiss. Either Albert likes cheddar or Brenda likes swiss Charlie likes mozzarella. Albert likes cheddar not Brenda likes swiss.
2) All operators except “¬” must have something on either side of them, and the “¬” must have something on its right side.
The following are NOT well-formed formulas:
A B ( “” needs something on both sides of it) A¬ ( “¬” must go on the left) NO! A ( “” needs something on both sides of it) 6
3) Two operators can never appear side by side unless the second is a “¬”.
The following are NOT well-formed formulas:
A B A B NO! ¬A ¬A (note: this one violates rule 2: “” needs something on both sides of it)
Meanwhile, the following IS a well-formed formula: A ¬B
4) Statements with three or more letters must have parentheses or brackets.
Brackets are sort of like commas in English sentences. For instance, the following sentence is ambiguous:
“Buddy likes cheese and Peggy likes pepperoni or Sue likes Hawaiian.”
We might be tempted to symbolize this statement as the following:
B P S
But, which of the following is the speaker saying?
“Either Buddy likes cheese and Peggy likes pepperoni … OR Sue likes Hawaiian.” OR “Buddy likes cheese … AND Either Peggy likes pepperoni or Sue likes Hawaiian.”
It’s ambiguous, and there is an important difference. Commas would help to clarify things. If we add commas to these two interpretations, we get the following:
“Buddy likes cheese and Peggy likes pepperoni, or Sue likes Hawaiian.”
“Buddy likes cheese, and Peggy likes pepperoni or Sue likes Hawaiian.”
The commas represent separators, just like parentheses do. Here are the symbolizations:
(B P) S
B (P S)
7 In the first sentence, Buddy and Peggy occur together, while Sue is separated by a comma. This indicates that the claim about Buddy and Peggy come together as a single unit. In logic, we indicate this by putting parentheses around them. Meanwhile, in the second sentence, Buddy occurs alone, and Peggy and Sue occur together after the comma. So, we put parentheses around Peggy and Sue in the second formula.
But, there won’t always be a comma. The placement of the word “either” can be helpful in these cases. For instance, notice the difference between the following two sentences:
“Either Harry orders juice and Mark orders beer or John orders soda.”
“Harry orders juice and either Mark orders beer or John orders soda.”
These get symbolized as the following:
(H M) J
H (M J)
Here are a few more sentences and their translations:
“Harry likes juice or both Mark and John like soda.” H (M J)
“Harry likes juice and Mark or John like soda.” H (M J)
“If Harry orders juice, then, if Mark orders beer, then John will order soda.” H (M J)
“If Harry will order juice provided that Mark orders beer, then John will order soda.” (M H) J
“If both Harry and Mark order juice or John orders beer, then Larry will order water.” [(H M) J] L
8 Identifying the Main Operator
Before moving on, we must learn how to identify the “main operator” within a formula. The main operator determines which KIND of statement a statement is (e.g., negation, conjunction, disjunction, etc.). Here are two rules for finding the “main operator”:
1) If there ARE NO parentheses, then the main operator will be the ONLY operator—unless there is more than one operator, in which case the main operator is the operator that is the one that is not a “¬”.
2) If there ARE parentheses or brackets, then the main operator will be the ONLY operator which is outside of all of the parenthesis or brackets—unless there is more than one operator outside of the parenthesis/brackets, in which case the main operator is the one that is not a “¬”.
Here are some examples.
Negation: In the following formulas, the main operator is the “¬”. So the following compound statements are all negations:
¬A ¬(A B) ¬[(A B) (C D)]
Conjunction: In the following formulas, the main operator is the “”. So the following compound statements are all conjunctions:
A B ¬A ¬B (A B) (C D) ¬A [(B C) (D E)]
Disjunction: In the following formulas, the main operator is the “”. So the following compound statements are all disjunctions:
A B ¬A ¬B (A B) (C D) ¬A [(B C) (D E)]
9 Conditional: In the following formulas, the main operator is the “”. So the following compound statements are all conditionals:
A B ¬A B (A B) ¬(C D) [(A B) (C D)] E
Bi-Conditional: In the following formulas, the main operator is the “”. So the following compound statements are all bi-conditionals:
A B A ¬B ¬(A B) (B C) [A (B C)] [D (E F)]
Once we have found the main operator, we know what KIND of statement something is. This becomes very clear if you note that, we can represent all of the complicated symbolizations above as very simple ones with a single operator. If Δ and □ are wff’s:
ALL of the negations above have the form, “¬□” ALL of the conjunctions above have the form, “Δ □” ALL of the disjunctions above have the form, “Δ □” ALL of the conditionals above have the form, “Δ □” ALL of the bi-conditionals above have the form, “Δ □”
For instance, take the very last equation under the bi-conditionals above. Now:
Let ‘Δ’ = [A (B C)] and Let ‘□’ = [D (E F)]
In that case…
[A (B C)] [D (E F)]
…is really just an instance of a well-formed formula of the form:
Δ □ i.e., this is really just a bi-conditional with a wff on either side of the double-arrow.
10 Determining the Truth Values of Compound Statements
If I give you an easy statement, such as “A B” and tell you that “A” is true and “B” is false, then you would easily be able to tell me that “A B” is false. (Have a look at the truth table for “” and see that “P Q” ONLY comes out true if BOTH conjuncts are true.
Example #1: But, let’s try a harder one:
“If either Trump or Clinton wins the election, then the world will end in Jan., 2017.”
In symbolic form, this becomes:
(T C) W But we know that: T=True, C=False, and W=False (Trump DID win, Clinton DIDN’T, and the world DID NOT end)
Using “T” for “True” and “F” for “False”, we can re-write this as follows:
(T F) F
The main operator in the formula above is the “”. We get rid of one operator at a time, and save the main operator for last. So, first let’s focus on the “T F” portion of the formula. What is the truth value of “P Q” when “P” is true and “Q” is false? It’s true! So, we know that “T F” is true. We can replace the entire disjunction with a “T”, like this:
T F
Now, what is the truth value of “P Q” when “P” is true” and “Q” is false? It’s false! So, the WHOLE sentence that we originally started with is false; i.e., “If either Trump or Clinton wins the election, then the world will end in Jan., 2017” is FALSE.
Example #2: Here is another example:
¬[(A B) (C D)] A=true, B=true, C=true, D=false
Is this sentence true or false? Referring to our truth tables, we get the following steps:
¬[(T T) (T F)] (we replace the letters with “T” or “F”) ¬( T F ) (the first conjunction is true; the second one is false) ¬ T (the disjunction is true) F (the negation of a truth is a falsehood)
Answer: So “¬[(A B) (C D)]” is false when A=true, B=true, C=true, and D=false.
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Example #3: Let’s try one more example:
[A (B C)] [D (E F)] A=false, B=true, C=false, D=false, E=true, F=false
Is this sentence true or false? Using our truth tables above, we get the following steps:
[F (T F)] [F (T F)] (we replace the letters with “T” or “F”) (F F ) (F F ) (both of the conditionals in the parentheses are false) F F (both of the disjunctions are false) T (a bi-conditional is TRUE when both sides are false)
Answer: True
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