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:
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
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
“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
“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
“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.
