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3.3: Logical Truths and Contradictions

Let us look at another interesting example:

A ~A Av~A
t f t
f t f

'Av~A' is true no matter what. Such a sentence is called a Logical Truth.

A sentence of sentence logic is a Logical Truth just in case it is true in all possible cases, that is, just in case it is true for all assignments of truth values to sentence letters.

Many authors use the word Tautology for a logical truth of sentence logic. I prefer to use the same expression, 'logical truth', for this idea as it applies in sentence logic and as it applies to predicate logic, which we will study in volume II.

   Clearly there will also be sentences which are false no matter what, such as

A ~A A&~A
t f f
f t f

Such a sentence is called a Contradiction.

A sentence of sentence logic is a Contradiction just in case it is false in all possible cases, that is, just in case it is false for all assignments of truth values to sentence letters.

   Later on in the course, logical truths and contradictions will concern us quite a bit. They are interesting here because they provide several further laws of logical equivalence:

The Law of Logically True Conjunct (LTC): If X is any sentence and Y is any logical truth, then X&Y is logically equivalent to X

The Law of Contradictory Disjunct (CD): If X is any sentence and Y is any contradiction, then XvY is logically equivalent to X.

You should be able to show that these laws are true. Furthermore, you should satisfy yourself that a conjunction is always a contradiction if one of its conjuncts is a contradiction and that a disjunction is always a logical truth if one of its disjuncts is a logical truth.

Exercise \(\PageIndex{1}\)

3-4. Explain why a disjunction is always a logical truth if one of its disjuncts is a logical truth. Explain why a conjunction is always a contradiction if one of its conjuncts is a contradiction.

3-5. Further simplify the sentence 'A&(Bv~B)', which was the last line of the first example in section 3-2.

3-6. Prove the following logical equivalences, following the same instructions as in exercise 3-2:

a) 'A&(~AvB)' is logically equivalent to 'A&B'.b) 'AvB' is logically equivalent to 'Av(~A&B)'.

c) 'A' is logically equivalent to '(A&B)v(A&~B)'. (This equivalence is called the Law of Expansion. You may find it useful in some of the &her problems.)

d) 'A' is logically equivalent to '(AvB)&(Av~B)'.

e) 'A&[Bv(~A&C)I' is logically equivalent to 'A&B'.

f) 'CvB' is logically equivalent to '(C&A)v(B&A)v(C&~A)v(B&~A)'.

g) 'C&B' is logically equivalent to '(CvA)&(Bv~D)&(~AvC)&(DvB)'.

h) '(A&B)v(-A&-B)' is logically equivalent to '(~AvB)&(~BvA)'.

i) '~Av~BvC' is logically equivalent to '~(~AvB)v~AvC'.

3-7. For each of the following sentences, determine whether it is a logical truth, a contradiction, or neither. (Logicians say that a sentence which is neither a logical truth nor a contradiction is Contingent, that is, a sentence which is true in some cases and false in others.) Simplify the sentence you are examining, using the laws of logical equivalence, to show that the sentence is logically equivalent to a sentence you already know to be a logical truth, a contradiction, or neither.

a) (B&A)v(B&~A)

b) B&[)~AvA)&~B)]

c) (A&B)v[(A&~B)&(~A&B)]

d) (AvB)v~(A&C)

e) (AvB)&(Av~B)&(~AVB)&(~Av~B)

f) ~A&~B&(AvB)

g) (AvC)&C&(AvB)

h) (A&B)v~Bv~A

i) (~B&A)v(~A&B)v(~B&~A)

j) (Av~AvB)&(Av~Av~C)

k) (A&C)v~(~BvC)v~(AvB)vC