Relationships in Truth Statements
Read these sections to learn more about relationships among truth statements and using and constructing logical proofs.
These sections review materially equivalent propositions and three other relationships among statements: tautological, contradictory, and contingent relationships. They also review the eight valid forms of inference: modus ponens, modus tollens, hypothetical syllogism, simplification, conjunction, disjunctive syllogism, addition, and constructive dilemma. They show how to construct proofs, including strategies for working forward or backward, depending on which is easier according to your premises. Finally, they summarize everything you have learned about sentential and propositional logic.
Complete the exercises as you study, then check your answers against the key.
Material equivalence
As we saw in the last section, two different symbolic sentences can translate the
same English sentence. In the last section I claimed that "~S ⊃ R" and "S v R"
are equivalent. More precisely, they are equivalent ways of capturing the truth-
functional relationship between propositions. Two propositions are materially
equivalent if and only if they have the same truth value for every assignment of
truth values to the atomic propositions. That is, they have the same truth values
on every row of a truth table. The truth table below demonstrates that "~S ⊃ R"
and "S v R" are materially equivalent.
| R | S | ~S ⊃ R | S v R |
|---|---|---|---|
| T | T | F T |
T |
| T | F | T T |
T |
| F | T | F T |
T |
| F | F | T F |
F |
If you look at the truth values under the main operators of each sentence, you can see that their truth values are identical on every row. That means the two statements are materially equivalent and can be used interchangeably, as far as propositional logic goes.
Let’s demonstrate material equivalence with another example. We have seen
that we can translate "neither nor" statements as a conjunction of two
negations. So, a statement of the form, "neither p nor q" can be translated:
~p ⋅ ~q
But another way of translating statements of this form is as a negation of a
disjunction, like this:
~(p v q)
We can prove these two statements are materially equivalent with a truth table (below).
| p | q | ~p ⋅ ~q | ~(p v q) |
|---|---|---|---|
| T | T | F F F |
F T |
| T | F | F F T |
F T |
| F | T | T F F |
F T |
| F | F | T T T |
T F |
Again, as you can see from the truth table, the truth values under the main
operators of each sentence are identical on every row (i.e., for every assignment
of truth values to the atomic propositions).
In fact, there is a fifth truth functional connective called "material equivalence" or the "biconditional" that is defined as true when the atomic propositions share the same truth value, and false when the truth values different. Although we will not be relying on the biconditional, I provide the truth table for it below. The biconditional is represented using the symbol "≡" which is called a "tribar".
| p | q | p ≡ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Some common ways of expressing the biconditional in English are with the
phrases "if and only if" and "just in case". If you have been paying close
attention (or do from now on out) you will see me use the phrase "if and only if"
often. It is most commonly used when one is giving a definition, such as the
definition of validity and also in defining the "material equivalence" in this very
section. It makes sense that the biconditional would be used in this way since
when we define something we are laying down an equivalent way of saying it.
Source: Matthew J. Van Cleave
This work is licensed under a Creative Commons Attribution 4.0 License.