Are there cases or arguments where the Truth Table is not reliable?

Question

Below is a basic Truth Table where p can represent the first argument and q for the second argument.

+-----+-----+-------+-------+
|  p  |  q  | p ∧ q | p ∨ q |
+-----+-----+-------+-------+
|  T  |  T  |   T   |   T   |
|  T  |  F  |   F   |   T   |
|  F  |  T  |   F   |   T   |
|  F  |  F  |   F   |   F   |
+-----+-----+-------+-------+

Suppose that we use the operators AND ∧ and OR ∨ to verify arguments like so:

p - Samson is longhaired

q - Petrucci is longhaired

p∧q - Samson and Petrucci are longhaired

This works well for the above example.

However, are there cases where the truth table is not reliable as a basis for proving the validity of arguments?

Answer 1

Truth tables are used to define what validity is. For that reason, they cannot fail to be reliable indicators of validity.

There may be cases in which what seems intuitively invalid comes out as valid according to the truth table. In these cases, you will likely want to examine your intuitions or make some clarifications as to your language.

For example, this might happen if you treat a counterfactual conditional in the same way as a normal conditional. In this case, people considered the matter and decided that the intuitive problem was due to the counterfactual nature of the conditional, not to a problem with the truth table.

Philosophers are very reluctant to conclude that truth tables are not guides to validity because they are definitional of what validity is, so challenging them would require a wholly new conception of validity (not that there aren't such conceptions around, but I assume that's outside the scope of this question).

Answer 2

One may think that problems arise in a scheme similar to the following:

• p: Peter is a brother.
• q: Quentin is a brother.
• p ∧ q: Peter and Quentin are brothers.

Here, p and q may both be true, for example if Peter is Alice's brother and Quentin is Bob's brother. However, p ∧ q is not necessarily true, because Peter and Quentin may be from completely different families.

But this is of course playing with the translation in natural language. Strictly speaking, p ∧ q means "Peter is a brother and Quentin is a brother", or in your example "Samson is longhaired and Petrucci is longhaired".

The validity of an argument is usually defined as: an argument is valid if the conclusion necessarily follows from the premises; i.e., there is no situation that all the premises are true but the conclusion is not. This is a simple definition - nothing magical -, as are the definitions of ∧ and ∨. The reliability of a truth table is a mathematical fact.

Answer 3

Yes, there is propositions that can't use this truth table.

The logic you are using only works for things that are either true or false. If the proposition allows a "maybe", "depends on...", "is possible", "is likely", that logic does not apply.

This is very tricky because most human level statements are not binary. Then such binary truth tables does not apply.

When you have statements that allow more than true or false responses you have to reduce them using other kinds of logics to be able to use classical logic. For example, modal logics allow possible and necessary reasonings.