Truth Tables - artificial intelligence:
In propositional logic, where we are limited to expressing sentences where propositions are true or false - we can check whether a specific statement is true or false by working out the truth of ever larger sub statements using the truth of the propositions themselves. To tests the truth of sub statements, in the presence of connectives, we have to know how to deal with truth assignments. For instance, if we know that isis_male(barack_obama) and is president(barack_obama) are true, then we know that the sentence:
is_president(barack_obama)∧ is_male(barack_obama)
is also true, because we know that a sentence of the form P∧Q is true when P and Q both are true.
This table allows us to read the truth of the connectives in the following manner. Imagine we are looking at row 3. This says that, if Q is true and P is false, then
1. ¬P is true
2. P∧Q is true
3. P ∨Q is false
4. P -> Q is true
5. P <-> Q is false
Note down that, if P is false, then regardless of whether Q is true or false, the statement P->Q is true. This takes a little getting used to, but may be a very valuable tool in theorem proving: if we know that something is false, it may imply anything we want it to. So, the following sentence is true: "Barack Obama is female" implies that "Barack Obama is an alien", because thesentence that Barack Obama is female was false, so the result that Barack Obama is an alien may be deduced in a sound way.
Each row of a truth table describes the connectives for a specific assignment of true and false to the individual propositions in a sentence. We say each assignment a model: it represents a specific possible state of the world. For 2 propositions P and Q there are 4 models.
In general, for propositional sentences, a model is also only a specific assignment of truth values to its distinct propositions. A sentence which contains propositions will have 2n possible models and so 2n rows in its truth table. A sentence S will be false or true for a given model M - when S is true we say 'M is a model of S'.
Sentences which are always true, regardless of the truth of the distinct propositions, are known as tautologies (or valid sentences). For all models,Tautologies are true. For illustration, if I said that "Tony Blair is prime minister or Tony Blair is not prime minister", this is basically a content-free sentence, because we could have replaced the predicate of being Tony Blair with any predicate and the sentence would still have been correct.
Tautologies are not always as simple to notice as the one above, and we may use truth tables to be sure that a statement we have written is true, regardless of the truth of the distinct propositions it contains. For doing this, the columns of our truth table will be headed with ever larger sections of the sentence, till the final column contains the complete sentence. As before, the rows of the truth table will represent all the possible models for the sentence, for example each possible assignment of truth values to the individual propositions in the sentence.In the truth table,we will use these initial truth values to assign truth values to the subsentencesand then use these new truth values to assign truth values to bigger subsentences and so on. If the last column (the entire sentence) is always assigned true, then its means, whatever the truth values of the propositions being discussed, the complete sentence will turn out to be true.