Common Mistakes and Correct Solutions for
1. Misunderstanding the concept of truth functional indeterminacy.
To say that a sentence is t-f indeterminate is not to say that its truth value is unknown.
The truth value of every sentence of SL is always exactly one or the other of T or F, and
which it is, is always known. You see this on the truth table. There are no blank spaces
on the truth table. On every line of every truth table, every sentence gets either a T or
an F assigned to its main connective. The value is always known. What makes a
sentence t-f indeterminate is not that it has an unknown truth value, but that it has
different truth values on different truth value assignments. On the truth table, the
column of truth values under its main connective contains at least one T and at least
one F. This is what differentiates it from t-f true sentences (which are true on every tva
and on every line of the truth table) and from t-f false sentences (which are false on
every tva and on every line of the truth table).
2. Confusing truth on a truth value assignment with truth-functional truth.
To say that a sentence is truth functionally true is to say that it is true on every truth
value assignment, not just that it is true on some truth value assignment (or just “true”
as some people ambiguously put it). On the truth table, there will be a column of only
T’s under its main connective. From the fact that there is one truth value assignment
that assigns a T to the sentence, you cannot infer that the sentence is truth functionally
true, but just that it is true on that one truth value assignment. For this reason, you
should never say that a sentence is just true or just false. No sentence of SL is ever
just true or just false. They are either true on some truth value assignment or false
on that truth value assignment, or true on all truth value assignments or false on
all truth value assignments.
3. Giving arguments where examples are called for and examples where
arguments are called for.
Whenever you are trying to show that something does not have to be the case, an
example is called for. When you are trying to show that something does have to be the
case, an argument appealing to the semantic rules and definitions is called for. Since
4f, 4j, and 5b are all false, examples are called for. The other questions all call for
4. Only doing half the job.
Question 4h does not just ask you to prove that if a sentence is t-f indeterminate then its
negation is t-f indeterminate, but the converse as well, that if a negation is t-f
indeterminate then its immediate component is t-f indeterminate as well. 5. Trying to use truth tables to answer questions about metavariables.
The metavariables, P and Q, stand for arbitrarily complex sentences of SL.
Consequently there is no way of telling how many lines there are on their truth tables. If
the sentences are t-f indeterminate, there is also no way of telling how the T’s and F’s
are distributed under their main connectives.