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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

pq - 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?

  • Welcome to Philosophy.SE. I have attempted to answer your question, but it is unclear to me why you would think a truth table could be unreliable. If my answer is not what you are looking for, could you edit your question to clarify in what cases a truth table may be unreliable, preferably with an example? Thanks! – Keelan Sep 5 '16 at 14:00
  • Propositional logic is decidable : "the truth-table method can be used to determine whether an arbitrary propositional formula is logically valid" and the validity of a propositional argument can be reduced to the validity of a single formula. – Mauro ALLEGRANZA Sep 5 '16 at 14:18
  • I used to remember my former professor who said: - p: God is so powerful - assuming that is true - q: God can create a rock so heavy he can't carry - assuming that is true - p^q: ~ does this mean this is true? – Abel Melquiades Callejo Sep 5 '16 at 14:18
  • @AbelMelquiadesCallejo's comment: it is correct that that argument leads a contradiction. Some have used this to show that God is 'beyond' logic in some sense. They do not, however, typically conclude that truth tables are not good guides to validity (in fact they assume that truth tables are good guides, or else there would be no contradiction). Sorry if this is not the place to reply like this but I just thought I should make sure so OP is clear on this. – Kourosh Alizadeh Sep 5 '16 at 17:07
  • In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. tandfonline.com/doi/abs/10.1080/… – user6917 Sep 6 '16 at 13:34
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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).

  • What exactly do you mean with "truth tables are used to define what validity is"? In my answer I give the usual definition of validity, without using a truth table. – Keelan Sep 5 '16 at 17:15
  • 1
    @Keelan Your definition doesn't make explicit mention of a truth table, but it could be reformulated to do so. For example: to be logically valid is for it to be the case that on a truth table of the argument, whenever all the premises are true, the conclusion is also true (this is how Wittgenstein defines it in the Tractatus, I believe). I feel like these are largely interchangeable definitions of validity. Yours is admittedly more intuitive, but the truth table definition doesn't require us to talk about necessity (which is a kind of controversial concept). – Kourosh Alizadeh Sep 5 '16 at 17:27
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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.

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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.

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