1. The two broad methods of reasoning.
2. It is the "top-down" approach of reasoning.
3. It is called the "bottom up" approach of reasoning.
4. The type of inductive reasoning which proceeds from a premise about a sample to a conclusion about the population.
5. Formal logic, as most people learn it, is deductive rather than inductive.(True or false)
6. The classic philosophical treatment of the problem of induction, meaning the search for a justification for inductive reasoning, was by the Scottish philosopher David Hume. .(True or false)
7. Inferences about the past from present evidence – for instance, as in archaeology, count as induction. (True or false)
8. A generalisation (more accurately, an inductive generalisation) proceeds from a premise about a sample to a conclusion about the population. (True or false)
9. An (inductive) analogy proceeds from known similarities between two things to a conclusion about an additional attribute common to both things(True or false)
10. A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect. (True or false)
11. Inductions are open; deductions are closed. (True or false)
12. The following is a weak induction: (True or false)
All observed crows are black.
Therefore:
All crows are black.
13. This is a strong induction: (True or false)
I always hang pictures on nails.
Therefore:
All pictures hang from nails.
14. Ultimately, inductive reasoning is reliable than deduction (True or false)
15. Induction is sometimes framed as reasoning about the past to the future. (True or false)
Wednesday, June 25, 2008
Tuesday, June 24, 2008
?
1. The two broad methods of reasoning.
2. It is the "top-down" approach of reasoning.
3. It is called the "bottom up" approach of reasoning.
4. The type of inductive reasoning which proceeds from a premise about a sample to a conclusion about the population.
2. It is the "top-down" approach of reasoning.
3. It is called the "bottom up" approach of reasoning.
4. The type of inductive reasoning which proceeds from a premise about a sample to a conclusion about the population.
Monday, June 23, 2008
quiz
1. is the branch of philosophy that studies knowledge.
2.sees knowledge as the product of sensory perception
3.sees it as the product of rational reflection.
4.According to this philosopher, knowledge results from the organization of perceptual data on the basis of inborn cognitive structures, which he calls "categories".
5. and 6. the nervous system of an organism cannot in any absolute way distinguish between these two.The only basic criterion is that different mental entities or processes within or between individuals should reach some kind of equilibrium.
7. . It notes that knowledge can be transmitted from one subject to another, and thereby loses its dependence on any single individual.
8. This Greek Philospher view knowledge is merely an awareness of absolute, universal Ideas or Forms, existing independent of any subject trying to apprehend to them.
9. Reasoning from General to specific.
10. Reasoning from specific to universal.
2.sees knowledge as the product of sensory perception
3.sees it as the product of rational reflection.
4.According to this philosopher, knowledge results from the organization of perceptual data on the basis of inborn cognitive structures, which he calls "categories".
5. and 6. the nervous system of an organism cannot in any absolute way distinguish between these two.The only basic criterion is that different mental entities or processes within or between individuals should reach some kind of equilibrium.
7. . It notes that knowledge can be transmitted from one subject to another, and thereby loses its dependence on any single individual.
8. This Greek Philospher view knowledge is merely an awareness of absolute, universal Ideas or Forms, existing independent of any subject trying to apprehend to them.
9. Reasoning from General to specific.
10. Reasoning from specific to universal.
Quiz
Philosophy 3 geniuses kindly read the article on this site as well as the photocopied notes given you for a quiz on monday...and a long quiz on Friday. Thanks
Wednesday, June 18, 2008
Monday, June 16, 2008
Notes no. 3: Validity, Truth and Knowledge
Validity, Truth, Knowledge, and the Good Life
Keith Burgess-Jackson
Validity is a property (feature, characteristic, attribute) of argument forms. Some argument forms have it; some do not. Valid argument forms are those that are truth-preserving. This means that if you put truths into them (as premises), you will get a truth out of them (as the conclusion). If we did not value truth, we would not value valid-ity. Like currency, validity has only extrinsic (instrumental) value. It is valued as a means to something else we value (namely, truth) but not as an end in itself. This is not to say that validity has no value; it is to specify the type of value it has. (Some things, such as friend-ship and knowledge, are valued both intrinsically and extrinsically.)
By definition, no valid argument has both true premises and a false conclusion. (If it did, it would not be truth-preserving.) Think of it this way. It is logically impossible (i.e., ruled out by definition) for an argument to have all three of the following properties:
------------------------
| is valid |
| has true premises |
| has a false conclusion |
------------------------
Suppose a given argument is known to have two of these properties. Then it can be inferred immediately that it lacks the third. There are three cases:
1. If a valid argument has true premises, then it does not have a false conclusion (i.e., it has a true conclusion).
2. If a valid argument has a false conclusion, then it does not have true premises (i.e., it has at least one false prem-ise).
3. If an argument has true premises and a false conclusion, then it is not valid (i.e., it is invalid).
1
We can go further. Suppose a given argument is known to have one of these properties. Then it can be inferred immediately that it lacks at least one of the others. There are three cases:
4. If an argument is valid, then either (a) it has a false premise or (b) it has a true conclusion (or both).
5. If an argument has true premises, then either (a) it is in-valid or (b) it has a true conclusion (or both).
6. If an argument has a false conclusion, then either (a) it is invalid or (b) it has a false premise (or both).
What this shows is that if you remember the definition of “validity,” you will be in a position to draw inferences about arguments from what you know about them—even if you know very little. Inference, whether inductive or deductive, mediate or immediate, is a means of extending knowledge. Because we value knowledge, we value truth (which is essential to it). Because we value truth, we value validity (which is a means to it). Validity turns out to be the key to knowl-edge! But wait; it gets better. If knowledge is essential to the good life, as the Greek philosopher Socrates implied when he said that the unexamined life is not worth living, then validity is the key to the good life. And you wondered why we were studying validity! 2
Keith Burgess-Jackson
Validity is a property (feature, characteristic, attribute) of argument forms. Some argument forms have it; some do not. Valid argument forms are those that are truth-preserving. This means that if you put truths into them (as premises), you will get a truth out of them (as the conclusion). If we did not value truth, we would not value valid-ity. Like currency, validity has only extrinsic (instrumental) value. It is valued as a means to something else we value (namely, truth) but not as an end in itself. This is not to say that validity has no value; it is to specify the type of value it has. (Some things, such as friend-ship and knowledge, are valued both intrinsically and extrinsically.)
By definition, no valid argument has both true premises and a false conclusion. (If it did, it would not be truth-preserving.) Think of it this way. It is logically impossible (i.e., ruled out by definition) for an argument to have all three of the following properties:
------------------------
| is valid |
| has true premises |
| has a false conclusion |
------------------------
Suppose a given argument is known to have two of these properties. Then it can be inferred immediately that it lacks the third. There are three cases:
1. If a valid argument has true premises, then it does not have a false conclusion (i.e., it has a true conclusion).
2. If a valid argument has a false conclusion, then it does not have true premises (i.e., it has at least one false prem-ise).
3. If an argument has true premises and a false conclusion, then it is not valid (i.e., it is invalid).
1
We can go further. Suppose a given argument is known to have one of these properties. Then it can be inferred immediately that it lacks at least one of the others. There are three cases:
4. If an argument is valid, then either (a) it has a false premise or (b) it has a true conclusion (or both).
5. If an argument has true premises, then either (a) it is in-valid or (b) it has a true conclusion (or both).
6. If an argument has a false conclusion, then either (a) it is invalid or (b) it has a false premise (or both).
What this shows is that if you remember the definition of “validity,” you will be in a position to draw inferences about arguments from what you know about them—even if you know very little. Inference, whether inductive or deductive, mediate or immediate, is a means of extending knowledge. Because we value knowledge, we value truth (which is essential to it). Because we value truth, we value validity (which is a means to it). Validity turns out to be the key to knowl-edge! But wait; it gets better. If knowledge is essential to the good life, as the Greek philosopher Socrates implied when he said that the unexamined life is not worth living, then validity is the key to the good life. And you wondered why we were studying validity! 2
Note no. 2: Deduction and Induction
Deduction & Induction
Deductive and Inductive Thinking
In logic, we often refer to the two broad methods of reasoning as the deductive and inductive approaches.
Deductive reasoning works from the more general to the more specific. Sometimes this is informally called a "top-down" approach. We might begin with thinking up a theory about our topic of interest. We then narrow that down into more specific hypotheses that we can test. We narrow down even further when we collect observations to address the hypotheses. This ultimately leads us to be able to test the hypotheses with specific data -- a confirmation (or not) of our original theories.
Inductive reasoning works the other way, moving from specific observations to broader generalizations and theories. Informally, we sometimes call this a "bottom up" approach (please note that it's "bottom up" and not "bottoms up" which is the kind of thing the bartender says to customers when he's trying to close for the night!). In inductive reasoning, we begin with specific observations and measures, begin to detect patterns and regularities, formulate some tentative hypotheses that we can explore, and finally end up developing some general conclusions or theories.
These two methods of reasoning have a very different "feel" to them when you're conducting research. Inductive reasoning, by its very nature, is more open-ended and exploratory, especially at the beginning. Deductive reasoning is more narrow in nature and is concerned with testing or confirming hypotheses. Even though a particular study may look like it's purely deductive (e.g., an experiment designed to test the hypothesized effects of some treatment on some outcome), most social research involves both inductive and deductive reasoning processes at some time in the project. In fact, it doesn't take a rocket scientist to see that we could assemble the two graphs above into a single circular one that continually cycles from theories down to observations and back up again to theories. Even in the most constrained experiment, the researchers may observe patterns in the data that lead them to develop new theories.
Deductive and Inductive Thinking
In logic, we often refer to the two broad methods of reasoning as the deductive and inductive approaches.
Deductive reasoning works from the more general to the more specific. Sometimes this is informally called a "top-down" approach. We might begin with thinking up a theory about our topic of interest. We then narrow that down into more specific hypotheses that we can test. We narrow down even further when we collect observations to address the hypotheses. This ultimately leads us to be able to test the hypotheses with specific data -- a confirmation (or not) of our original theories.
Inductive reasoning works the other way, moving from specific observations to broader generalizations and theories. Informally, we sometimes call this a "bottom up" approach (please note that it's "bottom up" and not "bottoms up" which is the kind of thing the bartender says to customers when he's trying to close for the night!). In inductive reasoning, we begin with specific observations and measures, begin to detect patterns and regularities, formulate some tentative hypotheses that we can explore, and finally end up developing some general conclusions or theories.
These two methods of reasoning have a very different "feel" to them when you're conducting research. Inductive reasoning, by its very nature, is more open-ended and exploratory, especially at the beginning. Deductive reasoning is more narrow in nature and is concerned with testing or confirming hypotheses. Even though a particular study may look like it's purely deductive (e.g., an experiment designed to test the hypothesized effects of some treatment on some outcome), most social research involves both inductive and deductive reasoning processes at some time in the project. In fact, it doesn't take a rocket scientist to see that we could assemble the two graphs above into a single circular one that continually cycles from theories down to observations and back up again to theories. Even in the most constrained experiment, the researchers may observe patterns in the data that lead them to develop new theories.
Notes No. 1: Introduction to Inductive Reasoning
nductive reasoning
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other article subjects named induction, see Induction.
Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument are believed to support the conclusion but do not entail it; i.e. they do not ensure its truth. Induction is a form of reasoning that makes generalizations based on individual instances.[1] It is used to ascribe properties or relations to types based on an observation instance (i.e., on a number of observations or experiences); or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is employed, for example, in using specific propositions such as:
This ice is cold. (or: All ice I have ever touched was cold.)
This billiard ball moves when struck with a cue. (or: 100/100 billiard balls struck with a cue moved.)
...to infer general propositions such as:
All ice is cold.
All billiard balls move when struck with a cue.
Inductive reasoning has been attacked several times. Historically, David Hume denied its logical admissibility. During the twentieth century, thinkers such as Karl Popper and David Miller have disputed the existence, necessity and validity of any inductive reasoning, including probabilistic (Bayesian) reasoning[citation needed].
Note that Mathematical induction is not a form of inductive reasoning. Mathematical induction is a form of deductive reasoning.
Contents
[hide]
* 1 Strong and weak induction
o 1.1 Strong induction
o 1.2 Weak induction
* 2 Validity
* 3 Types of inductive reasoning
o 3.1 Generalisation
o 3.2 Statistical syllogism
o 3.3 Simple induction
o 3.4 Argument from analogy
o 3.5 Causal inference
o 3.6 Prediction
* 4 Bayesian inference
* 5 Bibliography
* 6 Sources
* 7 References
* 8 External links
* 9 See also
[edit] Strong and weak induction
[edit] Strong induction
All observed crows are black.
Therefore:
All crows are black.
This exemplifies the nature of induction: inducing the universal from the particular. However, the conclusion is not certain. Unless we can systematically falsify the possibility of crows of another colour, the statement (conclusion) may actually be false.
For example, one could examine the bird's genome and learn whether it's capable of producing a differently coloured bird. In doing so, we could discover that albinism is possible, resulting in light-coloured crows. Even if you change the definition of "crow" to require blackness, the original question of the colour possibilities for a bird of that species would stand, only semantically hidden.
A strong induction is thus an argument in which the truth of the premises would make the truth of the conclusion probable, but not definite.
[edit] Weak induction
I always hang pictures on nails.
Therefore:
All pictures hang from nails.
Assuming the first statement to be true, this example is built on the certainty that "I always hang pictures on nails" leading to the generalisation that "All pictures hang from nails". However, the link between the premise and the inductive conclusion is weak. No reason exists to believe that just because one person hangs pictures on nails that there are no other ways for pictures to be hung, or that other people cannot do other things with pictures. Indeed, not all pictures are hung from nails; moreover, not all pictures are hung. The conclusion cannot be strongly inductively made from the premise. Using other knowledge we can easily see that this example of induction would lead us to a clearly false conclusion. Conclusions drawn in this manner are usually overgeneralisations.
Many speeding tickets are given to teenagers.
Therefore:
All teenagers drive fast.
In this example, the premise is built upon a certainty; however, it is not one that leads to the conclusion. Not every teenager observed has been given a speeding ticket. In other words, unlike "The sun rises every morning", there are already plenty of examples of teenagers not being given speeding tickets. Therefore the conclusion drawn can easily be true or false, and the inductive logic does not give us a strong conclusion. In both of these examples of weak induction, the logical means of connecting the premise and conclusion (with the word "therefore") are faulty, and do not give us a strong inductively reasoned statement.
[edit] Validity
Main article: Problem of induction
Formal logic, as most people learn it, is deductive rather than inductive. Some philosophers claim to have created systems of inductive logic, but it is controversial whether a logic of induction is even possible. In contrast to deductive reasoning, conclusions arrived at by inductive reasoning do not necessarily have the same degree of certainty as the initial premises. For example, a conclusion that all swans are white is false, but may have been thought true in Europe until the settlement of Australia, when Black Swans were discovered. Inductive arguments are never binding but they may be cogent. Inductive reasoning is deductively invalid. (An argument in formal logic is valid if and only if it is not possible for the premises of the argument to be true whilst the conclusion is false.) In induction there are always many conclusions that can reasonably be related to certain premises. Inductions are open; deductions are closed. It is however possible to derive a true statement using inductive reasoning if you know the conclusion. The only way to have an efficient argument by induction is for the known conclusion to be able to be true only if an unstated external conclusion is true, from which the initial conclusion was built and has certain criteria to be met in order to be true (separate from the stated conclusion). By substitution of one conclusion for the other, you can inductively find out what evidence you need in order for your induction to be true. For example, you have a window that opens only one way, but not the other. Assuming that you know that the only way for that to happen is that the hinges are faulty, inductively you can postulate that the only way for that window to be fixed would be to apply oil (whatever will fix the unstated conclusion). From there on you can successfully build your case. However, if your unstated conclusion is false, which can only be proven by deductive reasoning, then your whole argument by induction collapses. Thus ultimately, inductive reasoning is not reliable.
The classic philosophical treatment of the problem of induction, meaning the search for a justification for inductive reasoning, was by the Scottish philosopher David Hume. Hume highlighted the fact that our everyday reasoning depends on patterns of repeated experience rather than deductively valid arguments. For example, we believe that bread will nourish us because it has done so in the past, but this is not a guarantee that it will always do so. As Hume said, someone who insisted on sound deductive justifications for everything would starve to death.
Instead of approaching everything with unproductive skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.
Induction is sometimes framed as reasoning about the future from the past, but in its broadest sense it involves reaching conclusions about unobserved things on the basis of what has been observed. Inferences about the past from present evidence – for instance, as in archaeology, count as induction. Induction could also be across space rather than time, for instance as in physical cosmology where conclusions about the whole universe are drawn from the limited perspective we are able to observe (see cosmic variance); or in economics, where national economic policy is derived from local economic performance.
Twentieth-century philosophy has approached induction very differently. Rather than a choice about what predictions to make about the future, induction can be seen as a choice of what concepts to fit to observations or of how to graph or represent a set of observed data. Nelson Goodman posed a "new riddle of induction" by inventing the property "grue" to which induction does not apply.
[edit] Types of inductive reasoning
Sources for the examples that follow are: (1), (2), (3).
[edit] Generalisation
A generalisation (more accurately, an inductive generalisation) proceeds from a premise about a sample to a conclusion about the population.
The proportion Q of the sample has attribute A.
Therefore:
The proportion Q of the population has attribute A.
How great the support which the premises provide for the conclusion is dependent on (a) the number of individuals in the sample group compared to the number in the population; and (b) the randomness of the sample. The hasty generalisation and biased sample are fallacies related to generalisation.
[edit] Statistical syllogism
A statistical syllogism proceeds from a generalization to a conclusion about an individual.
A proportion Q of population P has attribute A.
An individual I is a member of P.
Therefore:
There is a probability which corresponds to Q that I has A.
The proportion in the first premise would be something like "3/5ths of", "all", "few", etc. Two dicto simpliciter fallacies can occur in statistical syllogisms: "accident" and "converse accident".
[edit] Simple induction
Simple induction proceeds from a premise about a sample group to a conclusion about another individual.
Proportion Q of the known instances of population P has attribute A.
Individual I is another member of P.
Therefore:
There is a probability corresponding to Q that I has A.
This is a combination of a generalization and a statistical syllogism, where the conclusion of the generalization is also the first premise of the statistical syllogism.
[edit] Argument from analogy
An (inductive) analogy proceeds from known similarities between two things to a conclusion about an additional attribute common to both things.
P is similar to Q.
P has attribute A.
Therefore:
Q has attribute A.
An analogy relies on the inference that the properties known to be shared (the similarities) imply that A is also a shared property. The support which the premises provide for the conclusion is dependent upon the relevance and number of the similarities between P and Q. The fallacy related to this process is false analogy.
[edit] Causal inference
A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect. Premises about the correlation of two things can indicate a causal relationship between them, but additional factors must be confirmed to establish the exact form of the causal relationship.
[edit] Prediction
A prediction draws a conclusion about a future individual from a past sample.
Proportion Q of observed members of group G have had attribute A.
Therefore:
There is a probability corresponding to Q that other members of group G will have attribute A when next observed.
[edit] Bayesian inference
Of the candidate systems for an inductive logic, the most influential is Bayesianism. This uses probability theory as the framework for induction. Given new evidence, Bayes' theorem is used to evaluate how much the strength of a belief in a hypothesis should change.
There is debate around what informs the original degree of belief. Objective Bayesians seek an objective value for the degree of probability of a hypothesis being correct and so do not avoid the philosophical criticisms of objectivism. Subjective Bayesians hold that prior probabilities represent subjective degrees of belief, but that the repeated application of Bayes' theorem leads to a high degree of agreement on the posterior probability. They therefore fail to provide an objective standard for choosing between conflicting hypotheses. The theorem can be used to produce a rational justification for a belief in some hypothesis, but at the expense of rejecting objectivism. Such a scheme cannot be used, for instance, to decide objectively between conflicting scientific paradigms.
Edwin Jaynes, an outspoken physicist and Bayesian, argued that "subjective" elements are present in all inference, for instance in choosing axioms for deductive inference; in choosing initial degrees of belief or prior probabilities; or in choosing likelihoods. He thus sought principles for assigning probabilities from qualitative knowledge. Maximum entropy – a generalization of the principle of indifference – and transformation groups are the two tools he produced. Both attempt to alleviate the subjectivity of probability assignment in specific situations by converting knowledge of features such as a situation's symmetry into unambiguous choices for probability distributions.
Cox's theorem, which derives probability from a set of logical constraints on a system of inductive reasoning, prompts Bayesians to call their system an inductive logic.
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other article subjects named induction, see Induction.
Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument are believed to support the conclusion but do not entail it; i.e. they do not ensure its truth. Induction is a form of reasoning that makes generalizations based on individual instances.[1] It is used to ascribe properties or relations to types based on an observation instance (i.e., on a number of observations or experiences); or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is employed, for example, in using specific propositions such as:
This ice is cold. (or: All ice I have ever touched was cold.)
This billiard ball moves when struck with a cue. (or: 100/100 billiard balls struck with a cue moved.)
...to infer general propositions such as:
All ice is cold.
All billiard balls move when struck with a cue.
Inductive reasoning has been attacked several times. Historically, David Hume denied its logical admissibility. During the twentieth century, thinkers such as Karl Popper and David Miller have disputed the existence, necessity and validity of any inductive reasoning, including probabilistic (Bayesian) reasoning[citation needed].
Note that Mathematical induction is not a form of inductive reasoning. Mathematical induction is a form of deductive reasoning.
Contents
[hide]
* 1 Strong and weak induction
o 1.1 Strong induction
o 1.2 Weak induction
* 2 Validity
* 3 Types of inductive reasoning
o 3.1 Generalisation
o 3.2 Statistical syllogism
o 3.3 Simple induction
o 3.4 Argument from analogy
o 3.5 Causal inference
o 3.6 Prediction
* 4 Bayesian inference
* 5 Bibliography
* 6 Sources
* 7 References
* 8 External links
* 9 See also
[edit] Strong and weak induction
[edit] Strong induction
All observed crows are black.
Therefore:
All crows are black.
This exemplifies the nature of induction: inducing the universal from the particular. However, the conclusion is not certain. Unless we can systematically falsify the possibility of crows of another colour, the statement (conclusion) may actually be false.
For example, one could examine the bird's genome and learn whether it's capable of producing a differently coloured bird. In doing so, we could discover that albinism is possible, resulting in light-coloured crows. Even if you change the definition of "crow" to require blackness, the original question of the colour possibilities for a bird of that species would stand, only semantically hidden.
A strong induction is thus an argument in which the truth of the premises would make the truth of the conclusion probable, but not definite.
[edit] Weak induction
I always hang pictures on nails.
Therefore:
All pictures hang from nails.
Assuming the first statement to be true, this example is built on the certainty that "I always hang pictures on nails" leading to the generalisation that "All pictures hang from nails". However, the link between the premise and the inductive conclusion is weak. No reason exists to believe that just because one person hangs pictures on nails that there are no other ways for pictures to be hung, or that other people cannot do other things with pictures. Indeed, not all pictures are hung from nails; moreover, not all pictures are hung. The conclusion cannot be strongly inductively made from the premise. Using other knowledge we can easily see that this example of induction would lead us to a clearly false conclusion. Conclusions drawn in this manner are usually overgeneralisations.
Many speeding tickets are given to teenagers.
Therefore:
All teenagers drive fast.
In this example, the premise is built upon a certainty; however, it is not one that leads to the conclusion. Not every teenager observed has been given a speeding ticket. In other words, unlike "The sun rises every morning", there are already plenty of examples of teenagers not being given speeding tickets. Therefore the conclusion drawn can easily be true or false, and the inductive logic does not give us a strong conclusion. In both of these examples of weak induction, the logical means of connecting the premise and conclusion (with the word "therefore") are faulty, and do not give us a strong inductively reasoned statement.
[edit] Validity
Main article: Problem of induction
Formal logic, as most people learn it, is deductive rather than inductive. Some philosophers claim to have created systems of inductive logic, but it is controversial whether a logic of induction is even possible. In contrast to deductive reasoning, conclusions arrived at by inductive reasoning do not necessarily have the same degree of certainty as the initial premises. For example, a conclusion that all swans are white is false, but may have been thought true in Europe until the settlement of Australia, when Black Swans were discovered. Inductive arguments are never binding but they may be cogent. Inductive reasoning is deductively invalid. (An argument in formal logic is valid if and only if it is not possible for the premises of the argument to be true whilst the conclusion is false.) In induction there are always many conclusions that can reasonably be related to certain premises. Inductions are open; deductions are closed. It is however possible to derive a true statement using inductive reasoning if you know the conclusion. The only way to have an efficient argument by induction is for the known conclusion to be able to be true only if an unstated external conclusion is true, from which the initial conclusion was built and has certain criteria to be met in order to be true (separate from the stated conclusion). By substitution of one conclusion for the other, you can inductively find out what evidence you need in order for your induction to be true. For example, you have a window that opens only one way, but not the other. Assuming that you know that the only way for that to happen is that the hinges are faulty, inductively you can postulate that the only way for that window to be fixed would be to apply oil (whatever will fix the unstated conclusion). From there on you can successfully build your case. However, if your unstated conclusion is false, which can only be proven by deductive reasoning, then your whole argument by induction collapses. Thus ultimately, inductive reasoning is not reliable.
The classic philosophical treatment of the problem of induction, meaning the search for a justification for inductive reasoning, was by the Scottish philosopher David Hume. Hume highlighted the fact that our everyday reasoning depends on patterns of repeated experience rather than deductively valid arguments. For example, we believe that bread will nourish us because it has done so in the past, but this is not a guarantee that it will always do so. As Hume said, someone who insisted on sound deductive justifications for everything would starve to death.
Instead of approaching everything with unproductive skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.
Induction is sometimes framed as reasoning about the future from the past, but in its broadest sense it involves reaching conclusions about unobserved things on the basis of what has been observed. Inferences about the past from present evidence – for instance, as in archaeology, count as induction. Induction could also be across space rather than time, for instance as in physical cosmology where conclusions about the whole universe are drawn from the limited perspective we are able to observe (see cosmic variance); or in economics, where national economic policy is derived from local economic performance.
Twentieth-century philosophy has approached induction very differently. Rather than a choice about what predictions to make about the future, induction can be seen as a choice of what concepts to fit to observations or of how to graph or represent a set of observed data. Nelson Goodman posed a "new riddle of induction" by inventing the property "grue" to which induction does not apply.
[edit] Types of inductive reasoning
Sources for the examples that follow are: (1), (2), (3).
[edit] Generalisation
A generalisation (more accurately, an inductive generalisation) proceeds from a premise about a sample to a conclusion about the population.
The proportion Q of the sample has attribute A.
Therefore:
The proportion Q of the population has attribute A.
How great the support which the premises provide for the conclusion is dependent on (a) the number of individuals in the sample group compared to the number in the population; and (b) the randomness of the sample. The hasty generalisation and biased sample are fallacies related to generalisation.
[edit] Statistical syllogism
A statistical syllogism proceeds from a generalization to a conclusion about an individual.
A proportion Q of population P has attribute A.
An individual I is a member of P.
Therefore:
There is a probability which corresponds to Q that I has A.
The proportion in the first premise would be something like "3/5ths of", "all", "few", etc. Two dicto simpliciter fallacies can occur in statistical syllogisms: "accident" and "converse accident".
[edit] Simple induction
Simple induction proceeds from a premise about a sample group to a conclusion about another individual.
Proportion Q of the known instances of population P has attribute A.
Individual I is another member of P.
Therefore:
There is a probability corresponding to Q that I has A.
This is a combination of a generalization and a statistical syllogism, where the conclusion of the generalization is also the first premise of the statistical syllogism.
[edit] Argument from analogy
An (inductive) analogy proceeds from known similarities between two things to a conclusion about an additional attribute common to both things.
P is similar to Q.
P has attribute A.
Therefore:
Q has attribute A.
An analogy relies on the inference that the properties known to be shared (the similarities) imply that A is also a shared property. The support which the premises provide for the conclusion is dependent upon the relevance and number of the similarities between P and Q. The fallacy related to this process is false analogy.
[edit] Causal inference
A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect. Premises about the correlation of two things can indicate a causal relationship between them, but additional factors must be confirmed to establish the exact form of the causal relationship.
[edit] Prediction
A prediction draws a conclusion about a future individual from a past sample.
Proportion Q of observed members of group G have had attribute A.
Therefore:
There is a probability corresponding to Q that other members of group G will have attribute A when next observed.
[edit] Bayesian inference
Of the candidate systems for an inductive logic, the most influential is Bayesianism. This uses probability theory as the framework for induction. Given new evidence, Bayes' theorem is used to evaluate how much the strength of a belief in a hypothesis should change.
There is debate around what informs the original degree of belief. Objective Bayesians seek an objective value for the degree of probability of a hypothesis being correct and so do not avoid the philosophical criticisms of objectivism. Subjective Bayesians hold that prior probabilities represent subjective degrees of belief, but that the repeated application of Bayes' theorem leads to a high degree of agreement on the posterior probability. They therefore fail to provide an objective standard for choosing between conflicting hypotheses. The theorem can be used to produce a rational justification for a belief in some hypothesis, but at the expense of rejecting objectivism. Such a scheme cannot be used, for instance, to decide objectively between conflicting scientific paradigms.
Edwin Jaynes, an outspoken physicist and Bayesian, argued that "subjective" elements are present in all inference, for instance in choosing axioms for deductive inference; in choosing initial degrees of belief or prior probabilities; or in choosing likelihoods. He thus sought principles for assigning probabilities from qualitative knowledge. Maximum entropy – a generalization of the principle of indifference – and transformation groups are the two tools he produced. Both attempt to alleviate the subjectivity of probability assignment in specific situations by converting knowledge of features such as a situation's symmetry into unambiguous choices for probability distributions.
Cox's theorem, which derives probability from a set of logical constraints on a system of inductive reasoning, prompts Bayesians to call their system an inductive logic.
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