Recall that an argument is valid when there is no truth value assignment that makes true and false. There are different ways to express that an argument is valid. The following three statements are equivalent (meaning that 1 is true if, and only if, 2 is true; 2 is true if and only if 3 is true; and 3 is true if, and only if, 1 is true):
- The argument with premise and conclusion is valid (denoted )
- is a tautology.
- is a contradiction.
In this section, we are interested in arguments that are inductively strong. This is weaker than being valid. For instance, the following argument is not valid:
Ann brought her laptop to the first 10 lectures. Therefore, Ann will bring her laptop to the next lecture.
However, the above argument is a strong argument.
The first thing to note is that whether an argument is inductively strong will depend on the stochastic truth table. This is very different than when evaluating whether an argument is valid. To determine validity of an argument , you need to construct a truth table with columns for and . This truth table is completely determined by the formulas and . Evaluating whether is inductively strong requires more information than the truth table with columns for and . An argument may be inductively strong according to one way of assigning probabilities to the rows of its truth table, but not inductively strong according to a different way of assigning probabilities to the rows of its truth table.
The first idea that one might have to define inductively strong arguments is: is inductively strong when is probable. To say that a formula is "probable" means that the probability of the formula is above some threshold. For instance, the probability of the formula is greater than . However, this definition does not capture the notion of inductive strength that we are after. The following well-known examples from Skyrms, Choice and Chance, Chapter II illustrate the problem with classifying as inductively strong when the is probable. First of all, recall that is tautologically equivalent to . So, is probable if, and only if, is probable.
Let be "there is a man in Cleveland that is 2,000 years old", and be "there is a man in Cleveland with three heads that is 2,000 years old". Intuitively, the argument is not inductively strong. However, the disjunction is probable. Since is very unlikely likely to be true, is very likely to be true, so is very probable. So, the above definition would incorrectly classify this as an inductively strong argument.
Let be "there is a man in Cleveland that is 1,999 years and 11 months old and in good health", and be "there will never be a man in Cleveland that is 2,000 years old". Intuitively, the argument is not inductively strong. However, the conjunction is probable. Since is very likely to be true, is also very likely to be true. So, the above definition would incorrectly classify this as an inductively strong argument.
The crucial problem in both examples that may be probable (and so is probable) even though there is no evidential relationship between and . This suggests the following definition of inductive strength: is probable given that is true (or under the supposition that is true). That is, the argument with premise and conclusion is inductively strong when is probable. This definition matches the intuitive judgements in the above two examples: In both of the above examples, is low (e.g., the probability that there is 2,000 year old man with three heads is low under the supposition that there is a 2,000 year old man in Cleveland). This suggests the first component of an inductively strong argument:
Suppose that and are formulas and is a number between and . We say that evidentially supports to degree in a stochastic truth table if . When , we say that is probable and that evidentially supports .
Evidential support is not the only property that inductively strong arguments satisfy. To illustrate, consider the following argument:
- Let the premise be "Bob (who is male) is taking a new birth control pill."
- Let the conclusion be "Bob (who is male) does not get pregnant."
Then, , so according to the above definition evidentially supports . However, is not an inductively strong argument. As in the above two examples, there is no evidential relationship between and . In this case, the problem is that . That is, learning that is true does not change the probability that is true. In such a case we say that and are independent.
Given a stochastic truth table, we say that and are independent when .
There are alternative characterizations of independence:
Given a stochastic truth table, for all formulas and , the following are equivalent:
- and are independent ().
The second aspect of an inductively strong argument is that the premises are relevant to the conclusion:
Given a stochastic truth table, we say that:
- is positively relevant to when ; and
- is negatively relevant to when .
Putting everything together, the argument is inductively strong when:
- evidentially supports (i.e., ).
- is positively relevant to (i.e., ).
- is not valid.
The reason that we do not define valid arguments as inductively strong is because an argument can be valid because its premises are contradictory. For instance, is valid. However, since , is undefined, so does not evidentially support and is not positively relevant to .
In the stochastic truth table below, you can change the probabilities assigned to each row. Recall that the sum of the numbers assigned to each row must be 1 and each number must be greater than or equal to 0. The probabilities of the (conditional) formulas are updated when the stochastic truth table changes. When you hover over a formula with your mouse, the rows of the truth table where that formula is true is highlighted. Use this interactive tool to solve the following problems:
- Find a stochastic truth table such that and are independent.
- Find a stochastic truth table such that is positively relevant to .
- Find a stochastic truth table such that is negatively relevant to .
- Find a stochastic truth table such that evidentially supports , but does not evidentially supports .