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Truth tables (part two)

A complex truth table

We can use the above truth tables to create a more complex truth table. For example, “Socrates is a human and he’s not a vampire.” We can translate this into logical form as “A ∧ ¬B.” (We use capital letters because they stand for something specific.) In this case “A” stands for “Socrates is a human” and “B” stands for “Socrates is not a vampire.”
The truth table for this is the following:
AB
¬B
A ∧ ¬B
TTFF
TFTT
FTFF
FFTF
We have “¬B” on the truth table because simpler statements must be resolved before we can find the truth values of more complicated statements. “¬B” is contained in “A ∧ ¬B.”
The truth value for “¬B” is the opposite of the truth vale for “B,” so we just write in the opposite values there.
The truth value for “A ∧ ¬B” will be true whenever both “A” and “¬B” are true. There’s only one place on the truth table where “A ∧ ¬B” is true.

Tautologies

A tautology is a statement that’s always true because of it’s logical form. For example, “there are life forms on other planets or there are no life forms on other planets.” That statement has the form “P ∨ ¬P.”
We can identify a tautology by looking at the truth table because all the possible truth values of a tautology are true. When looking at all the above truth tables, you will notice that none of the statements are tautologies because there’s always one possibility of each of the above statements to be false.
A truth table of a tautology:
P
¬P
P ∨ ¬P
TFT
FTT
“P ∨ ¬P” is true whenever at least on of those statements is true. That’s why it’s always true—Whenever P is false, ¬P is true and vise versa.

Self-contradictions

A self-contradiction is a statement that’s always false. For example, “there are life forms on other planets and there are no life forms on other planets.” That statement has the form “A ∧ ¬A.”
We can identify a self-contradiction on a truth table by seeing when a statement is always false. None of the above truth tables contain self-contradictions because none of those statements are always false.
A truth table of a self-contradiction:
P
¬P
P ∧ ¬P
TFF
FTF
“P ∧ ¬P” is false whenever either “P” or “¬P” is false. One of those simple statements is always false.

Consistent statements

Statements are logically consistent as long as they can all be true at the same time, and contradictory (or inconsistent) whenever they can’t be. “P” and “Q” are consistent because it’s possible they are both true, but “P” and “¬P” are inconsistent because it’s not possible that they’re both true.
Consider the following two statements:
¬P → Q
¬(P ∨ Q)
We can make a truth table for them:
PQ
¬P
¬P → Q
P ∨ Q
¬(P ∨ Q)
TTFTTF
TFFTTF
FTTTTF
FFTFFT
“¬P” must be given truth values before we can find the truth values of “¬P → Q” because it is part of that more complex statement. We need to find the truth values for “P ∨ Q” before “¬(P ∨ Q)” because it’s also part of that more complex statement.
“¬P → Q” will only be false when “¬P” is true and “Q” is false. That only happens on the bottom row.
“¬(P ∨ Q)” will only be true when “P ∨ Q” is false. That only happens on the bottom row as well.
The truth table above shows that “¬P → Q” and “¬(P ∨ Q)” are contradictory statements because they’re never both true at the same time.

Equivalent statements

Equivalent statements always have the same truth values. For example, “all humans are mammals and all humans are animals” is logically equivalent to “all humans are animals and all humans are mammals.” “P ∧ Q” is logically equivalent to “Q ∧ P.”
Consider the following two statements:
¬(P ∧ Q)
¬P ∨ ¬Q
We can make a truth table for them:
PQ
P ∧ Q
¬(P ∧ Q)
¬P
¬Q
¬P ∨ ¬Q
TTTFFFF
TFFTFTT
FTFTTFT
FFFTTTT
This truth table shows why they are equivalent—they always have the same truth values. Whenever “¬(P ∧ Q)” is false “¬P ∨ ¬Q” is also false, and they are both always true at the same time.

Valid arguments

An argument is logically valid whenever it’s impossible for the premises to be true and the conclusion false at the same time. They are logically invalid whenever it is possible for the premises to be true and the conclusion to be false at the same time.
Argument 1
Consider the following argument:
  1. Socrates is a human and Socrates isn’t a vampire.
  2. If Socrates is a human and Socrates isn’t a vampire, then Socrates is mortal.
  3. Therefore, Socrates is mortal.
We can translate this argument into the following logical statements:
A ∧ ¬B
(A ∧ ¬B) → C
C
Each letter stands for a specific statement:
A: Socrates is a human.
B: Socrates is a vampire.
C: Socrates is mortal.
The truth table for this argument is the following:
ABC
¬B
A ∧ ¬B
(A ∧ ¬B) → C
C
TTTFFTT
TTFFFTF
TFTTTTT
TFFTTFF
FTTFFTT
FTFFFTF
FFTTFTT
FFFTFTF
There’s only one spot on the table where both premises are true, and the conclusion is also true. Therefore, this argument is logically valid.
Argument 2
Consider the following argument:
  1. Socrates is an animal.
  2. If Socrates is a mammal, then Socrates is an animal.
  3. Therefore, Socrates is a mammal.
We can translate this argument into logical form:
A
B → A
B
The truth table for this argument is the following:
ABB → AB
TTTT
TFTF
FTFT
FFTF
“B → A” is only false when “B” is true and “A” is false. That’s only on the second row from the bottom.
This truth table proves that the argument is invalid because there’s a row of true premises and a false conclusion.

Now, what about our first argument?, is it valid or invalid:
"they will kill me if I work
and they will kill me if I don´t work
either if I work or I don´t work
They will kill me."

1- firstly we will formalize it:
p= I work
q= they will kill me.
p → q
¬p → q
(p ∨ ¬p) → q
2- then we built the truth table:
pq¬p
p → q
¬p → q
∨ ¬p
(p ∨ ¬p) → q
TT




TF




FT




FF





3- Finally I figure out whether there are any   row where the premises are true and the the conclusion is also true


Truth tables (part one)

Remember: our target is being  able to distinguish between a valid or an invalid argument, for instance: 
they will kill me if I work
and they will kill me if I don´t work
either if I work or I don´t work
They will kill me.

 In a quick overview this argument  seems to be valid, but how could we be sure?. Truth tables can help us. Let´s see!

A statement can be true o false: P = T v F, in other words, each single statement has two truth values.
The truth value In a complex statement  is the result of the  combination of truth values that single logical statements  which compose it, can have.


In this example, two statements have four possible combinations of truth values, but  If the truth values ​​of three propositions combine eight outcomes are obtained. In general the total number of possible combinations is 2ⁿ,  considering "n" the number of singles statements contained in a complex statement.

A complex statement is composed by many single statements linked with different connectives, so we have to take them in consideration, and figure up the way that each connective link affects the final complex statement´s truth value.

There is a column (vertical area) under each statement, which contains every possible truth value. The column under “p” has “T, T, F, F” (true, true, false, false). The column under “q” is “T, F, T, F” (true, false, true, false). The column under “p ∧ q” contains “T, F, F, F” (true, false, false, false).
Every row (horizontal area) beneath the statements contains every combination of truth values. The first row of truth values states that “p,” “q,” and “p ∧ q” are all true. The second row states that “p” is true, “q” is false, and “p ∧ q” is false. The third states that “p” is false, “q” is true, and “p ∧ q” is false. The fourth states that “p,” “q” and “p ∧ q” are all false.

Negation: 
p
¬p
TF
FT
Generally  speaking, “p” is any possible statement and “¬p” means “it’s not the case that p.”
Each box on the top row contains a logical statement. (In this case “p” or “¬p.”) Each box below a statement tells us the possible truth values of that statement. “p” can be true or false, and “¬p” can be false or true.
Each row of boxes below the logical statements contains the possible combinations of truth values of the statements above. The first row down says “p” is true and “¬p” is false. Whenever “p” is true, “¬p” will be false. For example, “p” can stand for “rocks exist.” In that case the statement is true, and “¬p” is false because it stands for “it’s not the case that rocks exist.”
The final row says “p” is false and “¬p” is true. Whenever “p” is false, “¬p” will be true. For example, “p” could stand for “1+1=3,” which is false. In that case “¬p” is true because it means “it’s not the case that 1+1=3.”
Truth tables provide every possible combination of truth values that logical statements can have. The only two truth values needed here are true and false, so there are only two rows beneath the logical statements.

Conjunction
pqp ∧ q
TTT
TFF
FTF
FFF
Lower case letters represent “predicate constants.” These lower case letters stand for any possible statement, such as “rocks exist” or “if rocks exist, then bananas are pink.”
The first row contains various statements (“p,” “q,” and “p ∧ q”). “p ∧ q” roughly translates to mean “both p and q.” For example, “p” can mean “rocks exist” and “q” can mean “bananas exist.” In that case “p ∧ q” means “rocks and bananas exist.”
You can notice that the result is "true" just  in the case that both single statements have the same  truth value. 

Disjunction
pqp ∨ q
TTT
TFT
FTT
FFF

“p ∨ q” generally  translates as “either p or q.” For example, “p” can be “dogs are mammals” and “q” can be “dogs are reptiles.” In that case “p or q” will be “dogs are mammals or dogs are reptiles.
The truth table indicates that every “p ∨ q” statement is true unless both “p” and “q” are false, which is shown on the final row down. For example “p” can be “dogs are reptiles” and “q” can be “dogs are lizards.” In that case “p ∨ q” stands for “either dogs are reptiles or they’re lizards.” That statement is false.

Conditional
pqp → q
TTT
TFF
FTT
FFT
“p → q” generally translates as “if p, then q.” For example, “if humans are mammals, then humans are animals.”
The truth table indicates that “p → q” is true unless “p” is true and “q” is false. “p” can be “the President of the USA is a human” and “p” can be “the President of the USA is a reptile.” In that case “p → q” will mean “if the President of the USA is a human, then the President of the USA is a reptile.” That statement is false.
We can also consider a true statement where “p” is false and “q” is false. For example, “p” can be “the President of the USA is a lizard” and “q” can be “the President of the USA is a reptile.” In that case “p → q” will be “if the President of the USA is a lizard, then the President of the USA is a reptile.” That statement is true.
Finally, let’s consider a conditional statement where “p” is false and “q” is true. “p” can stand for “the President of the USA is a lizard” and “q” can stand for “the President of the USA is an animal.” In that case the statement is “if the President of the USA is a lizard, then the President of the USA is an animal.” That statement is true.

Biconditional
pqp ↔ q
TTT
TFF
FTF
FFT
“p ↔ q” translates as “p if and only if q.” For example, “p” can stand for “1+2=3” and “q” can stand for “2+1=3.” In that case “p ↔ q” stands for “1+2=3 if and only if 2+1=3.”
The table above makes it clear that “p ↔ q” is only true when “p” and “q” have the same truth values. They must both be true or false. If not, the statement is false.
Consider when “p” stands for “dogs are animals” and “q” stands for “dogs are reptiles.” In that case “p ↔ q” stands for “dogs are animals if and only if dogs are reptiles.” That statement is false.


 
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