Tautologies, Contradictions and Contingencies

To determine if an argument is valid or invalid, we’ve used truth tables and the truth assignment test. The same methods can be used to investigate the logical qualities of individual propositions as well as the logical relationships between them. Propositional qualities are discussed here, and several significant logical relations are introduced on the following page. There are three types of propositions:

1. Tautologies

2. Contradictions

3. Contingencies. 

The logical structure of a proposition determines whether it is a tautology, contradiction, or contingency.

A tautology

It is also known as a tautologous proposition, which is a logical form that cannot be proven wrong (no matter what truth values are assigned to the sentence letters).

The following statements are all tautologies:

(A ⊃ A)

(A ∨ ~A)

~ (A • ~A)

((A • B) ⊃ (A ∨ B))

((A ∨ B) ≡ (B ∨ A))

A contradiction 

It is also known as a self-contradictory proposition, and has a logical form that can’t be true (no matter what truth values are assigned to the sentence letters).

The following statements are incompatible:

(A • ~A)

~ (A ∨ ~A)

~ (A ⊃ A)

((A ∨ ~A) ⊃ (B • ~B))

~ ((A ∨ B) ≡ (B ∨ A))

 A contingent proposition 

Contingency has a logical form that can be true or untrue (depending on what truth values are assigned to the sentence letters).

Contingencies include the following propositions:

A

~A

(A ∨ B)

~ (A • B)

(A ⊃ B)

((A ∨ B) ⊃ (C • D))

((A • B) ≡ (C ∨ D))

When described in propositional logic, some tautologies and contradictions appear as contingencies because their logical structure cannot be represented using truth-functional connectives. Other types of logical structure will be discussed in later chapters.

A truth table can be used to determine whether a proposition is a tautology, contradiction, or contingency. A tautology is a statement that is true in every row of the table. It’s a contradiction if it’s false in every row. The statement is a contingency if it is neither a tautology nor a contradiction—that is, if there is at least one row where it is true and at least one row where it is untrue.

Consider the following statement:

If violets are blue and roses are red, roses aren’t red.

This may appear to be a contradiction—a statement that cannot possibly be true. Our intuitions concerning logical characteristics, on the other hand, are frequently incorrect. Let’s symbolise the statement and build a truth table for it to see which form of the proposition it is:

R           B       ((R • B) ⊃ ~R)

0            0            1

0            1             1

1             0            1

1             1            0

The proposition is not a contradiction, as we can see from the truth table above. In fact, there are more possibilities for it to be true than false: every row except the final is true. It is a contingency since it is true in at least one row and untrue in at least one row.

Consider a few more examples. The truth values for three compound propositions are shown in the table below. One statement is a tautology, another is a contradiction, and the third is a condition. Can you figure out which one is which?

Because it is untrue in every row, the proposition ((~A ꓦ B) • B) is a contradiction. (There are no people in its table column.) 

Because it is true in some rows and untrue in others, the statement ((A • B) ꓦ C) is a contingency.

 Because it is true in every row, the proposition (B ᴝ (B ꓦ C)) is a tautology. (Its column has no zeros.)  

To assess whether a claim is a tautology, contradiction, or contingency, we can utilise the truth assignment approach. Rather than constructing the complete truth table, we can simply check whether the proposition is possible to be false, and then whether the proposition is possible to be true. A more detailed description of the technique follows:

To evaluate whether a proposition is a tautology, we must first determine whether the proposition can be proven wrong. (Keep in mind that a tautology has a form that can’t possibly be incorrect.) As a result, we start by assigning “0” to the primary connective, then calculate the truth values of any other connectives and sentence letters that can be identified using that assumption. If some letters are impossible to calculate, try all feasible combinations of values for those letters to see if the statement can be proven incorrect. It’s a tautology if there’s no method to make the proposition wrong. However, if you succeed in making the proposition incorrect, it isn’t a tautology, and you should move on to step 2: 

To see if a statement is a contradiction, assign “1” to its main connective and then calculate the truth values of any additional connectives and sentence letters that can be determined using that assumption. If some letters are impossible to calculate, try all potential values for those letters. It’s a contradiction if there’s no method to make the proposition true. It is not a contradiction, though, if you can find a means to make the proposition true. 

A statement is a contingency if it is neither a tautology nor a contradiction, as defined by stages 1 and 2.

Is the sentence below a tautology?

Roses are either red and violets are blue, or roses are red and violets aren’t.

Remember that a tautology has a logical form that can’t be wrong. To see if this statement is a tautology, we need to see if there is any method to make it false. Let’s try to figure it out by symbolising it and assigning “0” to its major connective:

((R • B) ∨ (R ⊃ ~ B))

             0

Only one way for the ” ” to be false is for both disjuncts to be false:

(( R • B ) ∨ ( R ⊃ ~ B ))

0              0           0

Furthermore, only one way for the “” to be false is if R is true and B is false (and thus B is true). This means that if the proposition is untrue as a whole (as we anticipated), R and B must both be true. However, if R and B are both true, then (R • B) is no longer false, and so the “” is no longer false:

((R • B) ∨ (R ⊃ ~ B))

1   0 1    0   1 0 0 1

As a result, there is no way to refute this assertion. It’s a contradiction in terms.

Conclusion:

Remember that propositional logic allows us to determine if an argument is valid or invalid if the form (logical structure) of the argument can be stated using truth-functional connectives. However, because some arguments rely on non-truth-functional logical structure, their form cannot be properly analysed using simply propositional logic tools. Similarly, propositional logic allows us to investigate logical properties of individual propositions (such as whether a proposition is a tautology, contradiction, or contingency) as well as logical relations between propositions (such as entailment, equivalence, and consistency), but only when those properties and relations are based on truth-functional connectives.