David Hume presented philosophers with a problem. His claim was that there is no justification in the assertion that, “instances of which we have had no experience resemble those of which we have had experience”. In the following paper, I will show that observations of instances can justify assertions of instances that have not been experienced.I will begin, in part A, by defining what it means to be an observation of an instance. Next, in part B, I will define what it means to be an assertion of an instance that has not been experienced. Then, in part C, I will define what it means for an assertion to be justified. In part D, I will show how experiences (as defined in part A) meet the criteria for justification (as defined in part C) for non-experienced instances (as defined in part B). Finally, in corollary part E, I will demonstrate how generalized statements about future events can be justified using the methodology from part D and the conceptualized framework of the (epsilon, delta) proofs that are employed in calculus.
Part A
An observation of instance, is a statement that can be made about the Universe. For example, “there is at least one swan that is white. (this sort of assertion can be made while looking at a white swan that is found in nature.)”
Part B
An instance that has not been experienced, is an observation that is to be made in the future. For example, “the next swan that we see will be white. (with regard to the next swan that will be observed in nature.)” An assertion of such instance, is a statement that is made at a time (t1) that predicates an observation that occurs at a time (t2), such that, the time (t1) is before the time (t2).
Part C
An assertion (p) is justified if the negation of (p) implies a contradiction. An assertion is also justified if it is directly observed. For example, if a white swan is observed, we are justified in making the assertion, “a white swan has been observed.”
Part D
Consider a case where an assertion (p1) is made, (the next swan that we see will be white.), at a given time (t1). According to the criteria in part C, an observation at time (t2) of an object that holds both properties (the next swan seen) and (white) justifies the assertion (p1). It is important to understand that at time (t1) the assertion (p1) was not justified, but at time (t2) the assertion (p1) was justified.
There is a substantial statement that is presented in Hume’s work; primarily that, “Assertions that are made about future instances, cannot be justified at the time they are made.”, however, this substantial statement does not entail that; “Assertions that are made about future instances can never be justified.” which subsequently entails the more general statement that; “Assertions that are made about the future cannot be justified.” On the contrary, it is my intention to illustrate that assertions of instances that have not been experienced (with respect to their assertion at t1) can be justified in the future in which they are observed (with respect to their observation at t2).
Part E
A generalized statement is an assertion that predicates properties to an unlimited number of observational cases. For example, “all objects that will be observed to have the property (swan) will also have the property (white).” It is the justification of this kind of generalized statement that I intend to establish a basis for approximation. In other words, there are observations which can be made that imply a partial justification of a generalized statement.
A formalized notion of partial justification is critical to the arguments presented here and thus, must be rigorously defined. A partial justification for the assertion of a given statement (q) is the affirmation of a premise (p) such that; the premise (p) is a member of the set of premises necessary to entail (q). This implies that partial justification for a statement (q), in the form of a true statement (p1), reduces the set of undetermined premises (p2, p3, p4, …) necessary to entail (q). In other words, if the set of premises necessary to entail (q), after the observation of (p1), is a proper subset of the set of premises necessary to entail (q), before the observation of (p1), then it follows that (q) is partially justified by (p1).
Consider the non-generalized statement, “the next n objects that are observed to have the property (swan) will also have the property (white).”
In the case; n = 2, we have;
(q1) The next two objects that are observed to have the property (swan) will also have the property (white). (stated at t1).
The statement (q1) is justified by the truth of both statements;
(p1) The first object to have the property (swan) also has the property (white). (observed at t2)
and
(p2) The second object to have the property (swan) also has the property (white). (observed at t3).
It is important to note that the justification of (q1) at t3 relies equally on both observations (p1 and p2). In other words, if either of the observations were to be false, q1 would necessarily be false. At t2, when the observation (p1) occurs, the set of unobserved premises necessary to entail (q1) transforms from the set A = ((p1) AND (p2)) to the set B = (p2). Since B is a subset of A, it follows that; (q1) is partially justified by (p1).
At this point, I will introduce a method for representing the degree of partial justification. To determine the degree of partial justification that a given set of observations (p1, p2, p3, …) makes on a statement (q), the cardinality of a set B = (the set of unobserved premises that are necessary to entail (q)) is subtracted from the cardinality of a set A = (the set of premises that were necessary to entail (q) at the time (q) was asserted) and then divided by the cardinality of A to produce a value between 0 and 1.
For the case of (q1) above (at time t2) we have;
In the cases; n = 3, n = 4 and n = 5 we have;
(q2) The next three objects that are observed to have the property (swan) will also have the property (white). (stated at t1)
(q3) The next four objects that are observed to have the property (swan) will also have the property (white). (stated at t1)
(q4) The next five objects that are observed to have the property (swan) will also have the property (white). (stated at t1)
Given the following observations;
The first object observed to have the property (swan) also has the property (white). (observed at t2)
The second object observed to have the property (swan) also has the property (white). (observed at t3)
The third object observed to have the property (swan) also has the property (white). (observed at t4)
The fourth object observed to have the property (swan) also has the property (white). (observed at t5)
The fifth object observed to have the property (swan) also has the property (white). (observed at t6)
It follows;
At t1, (q2) (q3) (q4) have a partial justification = 0.
It can now be shown, that for any value n in the statement (q1) “the next n objects that are observed to have the property (swan) will also have the property (white).”, there is a number of observations m that justify (q1).
Consider the case; statement (q1) is made at (t1) and (pm) “The (m)th object observed to have the property (swan) also has the property (white).” is made at (t(m+1)).
let m = the number of observations;
Since;
= partial justification
and
we have;
= the partial justification of each observation (pm) for (q1).
we can now show;
Therefore, for all (q1), as n approaches infinity, it is the case that; as m approaches n, m/n approaches one (total justification). It is this notion that allow us to approximate the justification of generalized statements.
Justification for Induction
David Hume presented philosophers with a problem. His claim was that there is no justification in the assertion that, “instances of which we have had no experience resemble those of which we have had experience”. In the following paper, I will show that observations of instances can justify assertions of instances that have not been experienced. I will begin, in part A, by defining what it means to be an observation of an instance. Next, in part B, I will define what it means to be an assertion of an instance that has not been experienced. Then, in part C, I will define what it means for an assertion to be justified. In part D, I will show how experiences (as defined in part A) meet the criteria for justification (as defined in part C) for non-experienced instances (as defined in part B). Finally, in corollary part E, I will demonstrate how generalized statements about future events can be justified using the methodology from part D and the conceptualized framework of the (epsilon, delta) proofs that are employed in calculus.
Part A
An observation of instance, is a statement that can be made about the Universe. For example, “there is at least one swan that is white. (this sort of assertion can be made while looking at a white swan that is found in nature.)”
Part B
An instance that has not been experienced, is an observation that is to be made in the future. For example, “the next swan that we see will be white. (with regard to the next swan that will be observed in nature.)” An assertion of such instance, is a statement that is made at a time (t1) that predicates an observation that occurs at a time (t2), such that, the time (t1) is before the time (t2).
Part C
An assertion (p) is justified if the negation of (p) implies a contradiction. An assertion is also justified if it is directly observed. For example, if a white swan is observed, we are justified in making the assertion, “a white swan has been observed.”
Part D
Consider a case where an assertion (p1) is made, (the next swan that we see will be white.), at a given time (t1). According to the criteria in part C, an observation at time (t2) of an object that holds both properties (the next swan seen) and (white) justifies the assertion (p1). It is important to understand that at time (t1) the assertion (p1) was not justified, but at time (t2) the assertion (p1) was justified.
There is a substantial statement that is presented in Hume’s work; primarily that, “Assertions that are made about future instances, cannot be justified at the time they are made.”, however, this substantial statement does not entail that; “Assertions that are made about future instances can never be justified.” which subsequently entails the more general statement that; “Assertions that are made about the future cannot be justified.” On the contrary, it is my intention to illustrate that assertions of instances that have not been experienced (with respect to their assertion at t1) can be justified in the future in which they are observed (with respect to their observation at t2).
Part E
A generalized statement is an assertion that predicates properties to an unlimited number of observational cases. For example, “all objects that will be observed to have the property (swan) will also have the property (white).” It is the justification of this kind of generalized statement that I intend to establish a basis for approximation. In other words, there are observations which can be made that imply a partial justification of a generalized statement.
A formalized notion of partial justification is critical to the arguments presented here and thus, must be rigorously defined. A partial justification for the assertion of a given statement (q) is the affirmation of a premise (p) such that; the premise (p) is a member of the set of premises necessary to entail (q). This implies that partial justification for a statement (q), in the form of a true statement (p1), reduces the set of undetermined premises (p2, p3, p4, …) necessary to entail (q). In other words, if the set of premises necessary to entail (q), after the observation of (p1), is a proper subset of the set of premises necessary to entail (q), before the observation of (p1), then it follows that (q) is partially justified by (p1).
Consider the non-generalized statement, “the next n objects that are observed to have the property (swan) will also have the property (white).”
In the case; n = 2, we have;
(q1) The next two objects that are observed to have the property (swan) will also have the property (white). (stated at t1).
The statement (q1) is justified by the truth of both statements;
(p1) The first object to have the property (swan) also has the property (white). (observed at t2)
and
(p2) The second object to have the property (swan) also has the property (white). (observed at t3).
It is important to note that the justification of (q1) at t3 relies equally on both observations (p1 and p2). In other words, if either of the observations were to be false, q1 would necessarily be false. At t2, when the observation (p1) occurs, the set of unobserved premises necessary to entail (q1) transforms from the set A = ((p1) AND (p2)) to the set B = (p2). Since B is a subset of A, it follows that; (q1) is partially justified by (p1).
At this point, I will introduce a method for representing the degree of partial justification. To determine the degree of partial justification that a given set of observations (p1, p2, p3, …) makes on a statement (q), the cardinality of a set B = (the set of unobserved premises that are necessary to entail (q)) is subtracted from the cardinality of a set A = (the set of premises that were necessary to entail (q) at the time (q) was asserted) and then divided by the cardinality of A to produce a value between 0 and 1.
For the case of (q1) above (at time t2) we have;
In the cases; n = 3, n = 4 and n = 5 we have;
(q2) The next three objects that are observed to have the property (swan) will also have the property (white). (stated at t1)
(q3) The next four objects that are observed to have the property (swan) will also have the property (white). (stated at t1)
(q4) The next five objects that are observed to have the property (swan) will also have the property (white). (stated at t1)
Given the following observations;
The first object observed to have the property (swan) also has the property (white). (observed at t2)
The second object observed to have the property (swan) also has the property (white). (observed at t3)
The third object observed to have the property (swan) also has the property (white). (observed at t4)
The fourth object observed to have the property (swan) also has the property (white). (observed at t5)
The fifth object observed to have the property (swan) also has the property (white). (observed at t6)
It follows;
At t1, (q2) (q3) (q4) have a partial justification = 0.
At t2, (q2) = ((3-2)/3) = 1⁄3, (q3) = ((4-3)/4)= 1⁄4, (q4) = ((5-4)/5)= 1⁄5.
At t3, (q2) = ((3-1)/3) = 2⁄3, (q3) = ((4-2)/4) = 1⁄2, (q4) = ((5-3)/5) = 2⁄5.
At t4, (q2) = ((3-0)/3) = 1, (q3) = ((4-1)/4) = 3⁄4, (q4) = ((5-2)/5) = 3⁄5.
At t5, (q3) = ((4-0)/4) = 1, (q4) = ((5-1)/5) = 4⁄5.
At t6, (q4) = ((5-0)/5) = 1.
It can now be shown, that for any value n in the statement (q1) “the next n objects that are observed to have the property (swan) will also have the property (white).”, there is a number of observations m that justify (q1).
Consider the case; statement (q1) is made at (t1) and (pm) “The (m)th object observed to have the property (swan) also has the property (white).” is made at (t(m+1)).
let m = the number of observations;
Since;
and
we have;
we can now show;
Therefore, for all (q1), as n approaches infinity, it is the case that; as m approaches n, m/n approaches one (total justification). It is this notion that allow us to approximate the justification of generalized statements.