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The Amateur Logician Tutorial:
Symbols & Translations | Basic Truth Tables | Complicated Translations | Introducing Rules of Inference | Equivalences (Replacement Rules) | Conditional & Indirect Proofs
Basic Truth Tables
– Introduction
– Common Sentential Connectives:
Negation, Conjunction, Disjunction
Conditional, Biconditional
– Less Common Sentential Connectives:
XOR, NAND, NOR
– Computing Truth Values in General
– Classifying Propositions via Truth Tables:
Contingency, Tautology, Self-Contradictory
– Comparing Propositions via Truth Tables:
Equivalent, Contradictory, Contrary, Consistent
Introduction
Propositional logic upholds the Principle of Bivalence.
That is, a proposition has the truth value of either being true or false (but not both).
Constructing a Truth Table is a method by which we can evaluate the truth or falsity of a molecular proposition as a whole given the possible truth values of the atomic proposition(s) that compose it. These tables allow us to see all possible truth values a given proposition can have.
We can, additionally, construct truth tables to determine if a given argument, in propositional logic, is valid or invalid. (These uses of the truth table will be covered elsewhere!)
Recall that any molecular proposition (a.k.a. compound proposition) is formed with at least one sentential connective. Sentential connectives include the “not,” “and,” “or,” “if, then” and “if and only if.” All sentential connectives are said to be “Truth Functional” because a molecular proposition formed by them will always have a truth value determined by the atomic proposition(s) that compose it.
In propositional logic, we can take a semantic or a syntactical point of view.
Semantic POV – examines the meaning behind propositions.
Syntactical POV – examines propositions without regard to their meaning.
It’s only from the semantic point of view that we can declare that “squares have four sides” is true and that “squares have five sides” is false (because it’s by understanding the meaning of those propositions that we know the former example is true and the latter example is false).
Negation
The “not” sentential connective corresponds to our “natural” language understanding of “not.” We use the word “not” to contradict, to deny. Now, any given proposition will be true or false. So, the truth or falsity of the molecular proposition ~Φ will depend upon the truth or falsity of the atomic proposition Φ that composes it.
A truth table is pedagogically useful in showing this.
Underneath the first header row, there will be a total of two rows.
In any truth table, T represents true and F represents false.
(Boolean algebra would use a 1 or a 0 instead, respectively.)
The first column underneath Φ states that Φ can be true or false.
These are the two possible input values.
The second column underneath ~Φ evaluates the truth of ~Φ based on the given input values. Here we get the output values.
So, if Φ is true, then ~Φ is false; and if Φ is false, then ~Φ is true.
Conjunction
The “and” sentential connective also corresponds to our “natural” language understanding of “and.” We use the word “and” to combine, to connect together.
Φ & ψ is only true when Φ is true and ψ is true.
Since Φ can be true or false and ψ can be true or false, there are two times two possible output values of Φ & ψ. As a result, the truth table will have a total of four rows underneath the first header row.
Disjunction
The “or” sentential connective doesn’t always correspond to our “natural” language understanding of “or.” That’s because we often use the “or” in natural language in an exclusive sense, not an inclusive sense.
The “or“ sentential connective takes the “or” in an inclusive sense. Φ v ψ is only false when Φ is false and ψ is false. Notice that, unlike an exclusive “or,” this implies that Φ v ψ is true when Φ is true and ψ is true.
Like the conjunction, we have two atomic propositions composing molecular proposition Φ v ψ. This means there must be a total of four rows underneath the first header row.
Conditional
The “if, then” sentential connective doesn’t correspond to our “natural” language understanding of “if, then.” We can look at propositional logic as an artificial language. Sometimes it models our natural language perfectly, other times it doesn’t.
Φ → ψ is defined to be false only when Φ is true and ψ is false.
Φ → ψ is otherwise true, and there are no other requirements.
Remember, Φ is called the antecedent and ψ is called the consequent.
Biconditional
The “if and only if” sentential connective doesn’t always correspond to our “natural” language understanding of “if and only if” (iff). To be sure, when people use the iff expression, they often do use it in the technical mathematical sense.
Φ ↔ ψ is true when Φ and ψ have the same truth values.
Φ ↔ ψ is false when Φ and ψ have different truth values.
Less Common Sentential Connectives
Φ ⊕ ψ uses the XOR sentential connective, which is the exclusive “or.”
Φ | ψ uses the NAND sentential connective.
Φ↓ψ uses the NOR sentential connective.
Computing Truth Values in General
We can use truth tables for longer propositions. A molecular proposition can have any arbitrary number of atomic propositions and various sentential connectives within it, as long as it is a wff (i.e., a well-formed formula). While there are ways to take “short cuts” to make truth tables less tedious, what follows is easiest to explain and implement.
Let’s generalize our understanding of the row size of a truth table.
There will be a column for each atomic proposition.
Each atomic proposition has two possibilities: true or false.
Applying the principle of multiplication, we get. . .
– One atomic proposition will bring in two rows.
– Two atomic propositions will bring in four rows.
– Three atomic propositions will bring in eight rows.
There will be 2^n rows, where n is the number of atomic propositions.
(This is not including the header row.)
Example 1.
“Tiffany will either have eggs and bacon, or chocolate for breakfast.”
We’ll stipulate that the “or” is inclusive. Tiffany can have all three for breakfast.
First, let’s translate it into the symbolic language of propositional logic. The use of the word “either” and the placement of the comma suggests that it is a disjunction overall.
We can thereby get the following: (E&B)vC
E stands for eggs, B stands for bacon, and C stands for chocolate.
Because this is a disjunction overall, the disjuctive connective is outside the parentheses.
Second, let’s work on its truth table.
By the way, how many rows will there be? E can be true or false, B can be true or false, and C can be true or false. Hence, there are 2*2*2 = 2^3 = 8. Although the molecular proposition as a whole will be true or false, it depends upon the inputs. Notice, there are eight different possibilities.
The input values have been filled. All eight possibilities are there.
Never miss a possibility. Everything must be there.
Let’s work our way to the main sentential connective!
E&B is a sub-molecular proposition that composes the overall molecular proposition.
We have to do that first before we can address (E&B)vC.
In general, each sub-molecular proposition will get its own column.
And the final column will be display the output values of or proposition as a whole.
To fill in that fourth column, all we had to do is to look at the input values of E and B to then compute the value of E&B row-by-row.
To fill in the final fifth column, all we had to do is look at the values of E&B to compare to C in order to get all outputs for the disjunction (E&B)vC.
Now that the truth table is complete, we have exhaustively shown what the truth value of the proposition will be given any possible input values.
Pop Quiz!
Assuming the truth values for having eggs, bacon, and chocolate are equally likely, what’s the probability that it is false that Tiffany will either have eggs and bacon, or chocolate for breakfast?
(The probability would be equal to the number of times the proposition can be false divided by the total possible truth values. Hence, the probability is 3/8 = 0.375 = 37.5%.)
Example 2.
~(PvQ)→~Q&~P
To construct this truth table, first notice that there are two atomic propositions involved; ergo, there will be 2^2 = 4 rows. Next, notice that there’s a lot of sub-molecular propositions within the overall molecular proposition. We’ll have to break this up part-by-part, which will give us multiple columns. Overall, we have a conditional.
The first two columns are for P and Q. Then, we take on the molecular proposition part-by-part. It’s a conditional overall. This is why the last column deals with that conditional sentential connective.
As we work on this column-by-column, we should get the following result. . .
We just have to remember our basic definitions to do this proficiently. As we do the third column, for example, we’re just applying what we know about disjunctions. The value that’s placed in, e.g., the first row and third column must have a value of T because the inputs are both T in that first row. Et cetera.
Example 2, by the way, got us only true values. This is called a “Tautology.”
Classifying Propositions via Truth Tables
A proposition will be (1) contingent, (2) a tautology, or (3) self-contradictory.
It’s contingent if its truth value varies based on the values of its components.
Example 1 is contingent, since the output values vary. The truth value of (E&B)vC is “contingent” upon the input values.
A tautological proposition has it that its truth value is always true, no matter the values of its components.
Example 2 is a tautology, since the output value is always true with ~(PvQ)→~Q&~P.
Self-contradictory propositions always have the truth value of being false, no matter what the values of its components.
Example 3 is self-contradictory. The output value is always false. P&~P is a contradiction; it cannot possibly be true.
What if we negated a tautology or negated a contradiction?
Well, the negation of a tautology must be a contradiction. Turning all the true values into false values will do that. And the negation of a contradiction must be a tautology. Turning all the false values into true values will do that.
Note that other jargon terms used in propositional logic include “satisfiable” and “unsatisfiable.” A proposition is “satisfiable” when the proposition is either a contingency or a tautology. It can be true under some conditions. A proposition is “unsatisfiable” if it’s self-contradictory. It will never be true under any condition.
Comparing Propositions via Truth Tables
Another thing we can do with a truth table is to compare or relate propositions. Two propositions may be (1) equivalent, (2) contradictory, (3) contrary, or (4) consistent.
Two propositions are equivalent when they have the same exact truth outputs.
This is highly useful to know! This implies that if one of those propositions is true, it then immediately follows that the other proposition must be true. Conversely, if one of those proposition is false, it then immediately follows that the other proposition must be false.
Proofs in propositional logic often require making use of this fact. We can “replace” one proposition with an equivalent proposition without making any fallacious move.
Example 1 has two equivalent propositions. “I will not study both physics and biology,” which translates as ~(P&B), is equivalent to the proposition “I will not study physics or I will not study biology,” which translates as ~Pv~B.
Two propositions are contradictory when they have the exact opposite truth outputs.
This is, likewise, highly useful. Think about it! Imagine two contradictory propositions. This means both cannot be true. When one is true, the other must be false (or vice versa).
Example 2 has two contradictory propositions. “I will not study both physics and biology,” which translates as ~(P&B), is contradictory to “I will study both physics and biology,” which translates as P&B.
While this example may be pretty obvious, the truth table verifies it. Propositional logic can apply to real life situations. It’s a tool that can help us think more clearly.
Two propositions are contrary when they can never be both true but can both be false.
Example 3 has two contrary propositions. “I will study physics and biology,” which translates as P&B, is contrary to “I will study physics and not biology,” which translates as P&~B.
Notice that neither can simultaneously be true, though both simultaneously can be false. In the truth table, each time one is true, the other is false. But there are occasions when both are false at the same time. That’s why they are contrary, not contradictory.
Warning! People often confuse contradictory with contrary opposition. Sometimes, in daily conversation, people unknowingly and fallaciously treat contrary propositions as if they were contradictory.
[This issue, in a broader context outside of propositional logic, is raised in the “Presumption Fallacies” page.]
Two propositions are consistent when both can be true simultaneously: there’s at least one instance where they can have the same truth output. All equivalent propositions are consistent propositions, therefore, and no contradictory propositions are consistent.
Example 4 has two consistent propositions. “I will study physics if and only if I study calculus,” which translates as P↔C, is consistent with “I will study physics and calculus,” which translates as P&C. Though they are consistent, they are not equivalent.
Being “consistent” in formal logic, if we think about it, is very minimal.
All sorts of unrelated, odd, and bizarre propositions will be “consistent” in this sense. Indeed, take the trivial atomic propositions P and Q. Well, P can be true or false and Q can be true or false. So, they are “consistent.” Finding out what’s true and false among them is a task that takes us outside of the confines of formal logic. Logic is necessary for good thinking, but it’s not sufficient.