Inductive Logic

Definition: An argument is a list of statements, one of which is the conclusion and the rest of which are the premises.

The conclusion states the point being argued for and the premises state the reasons being advanced in support the conclusion. They may not be good reasons. There are good and bad arguments.

Tip: You may often recognize arguments in English by the appearance of conclusion indicators like "therefore", "consequently", "it follows that", and so on. In each case, the sentence that immediately follows the conclusion indicator is the conclusion of an argument. There are also premise indicators like "because", "since", and so on.

Q 1: Is "for that reason" a conclusion indicator or a premise indicator?

Q 2: If you see a sentence like "Therefore the butler is the murderer because his fingerprints were on the knife", what is the conclusion?

A look ahead: Most examples of reasoning, or inference, may be represented as arguments. However, perceptual judgement, such as "I see a blue square", or the conclusions of scientific experts reading in X-rays, or looking through a microscope, involve the support of tacit knowledge, which is not easily articulated as a set of statements. Nevertheless, they appear to be examples of reasoning. If so, then not all reasoning is easily represented as arguments. (Thomas Kuhn [60] suggests that such reasoning should be analyzed in terms of neural networks [42]).

Argument evaluation: In logic, we assume that any reasoning is represented as an argument, and the evaluation of an argument involves two questions:

    1. Are the premises true?
    2. Supposing that the premises are true, what sort of support do they give the conclusion?

Acceptable answers to question 2: Consider the difference between the following arguments.

  1. All planets move on ellipses. Pluto is a planet. Therefore, Pluto moves on an ellipse.
  2. Mercury moves on an ellipse. Venus moves on an ellipse. Earth moves on an ellipse. Mars moves on an ellipse. Jupiter moves on an ellipse. Saturn moves on an ellipse. Uranus moves on an ellipse. Neptune moves on an ellipse. Therefore, Pluto moves on an ellipse.

If you suppose that the two premises in argument 1 are true, then the conclusion must be true. So, the premises give the strongest possible support for the conclusion. However, if we suppose that the eight premises of argument 2 are true, the conclusion may still be false. The premises provide some reason for believing the conclusion, but they do not guarantee the truth of the conclusion.

Definition: An argument is deductively valid if and only if it is impossible that its conclusion is false while its premises are true.

Argument 1 is deductively valid, while argument 2 is not.

What’s possible? The sense of "impossible" needs clarification. Consider the example:

  1. George is a human being. George is 100 years old. George has arthritis. Therefore, George will not run a four-minute mile tomorrow.

Suppose that the premises are true. In logic, it is possible that George will run a four-minute mile tomorrow. It is not physically possible. But logicians have a far more liberal sense of what is "possible" in mind in their definition of deductive validity. Argument 3 is not deductively valid on their definition.

Key idea: In any deductively valid argument, there is a sense in which the conclusion is contained in premises. Deductive reasoning serves the purpose of extracting information from the premises. In a non-deductive argument, the conclusion ‘goes beyond’ the premises. Inferences in which the conclusion amplifies the premises is sometimes called ampliative inference.

Note: Whether an argument is deductively valid or not, depends on what the premises are.

‘Missing’ premises?: We can always add a premise to turn an invalid argument into a valid argument. For example, if we add the premise "No 100-year-old human being with arthritis will run a four-minute mile tomorrow" to argument 3, then the new argument is deductively valid. (The original argument, of course, is still invalid).

Definition: An argument is inductively strong if and only if it is improbable that its conclusion is false while its premises are true.

Remember: This definition is the same as the definition of "deductively valid" except that "impossible" is replaced by "improbable."

The degree of strength of an inductive argument may be measured by the probability of that the conclusion is true given that all the premises are true. This is known as a conditional probability because the probability of the conclusion is conditional on the truth of the conclusion. This does not mean that the premises must be true, or probably true, for the conditional probability to be high. For example, the probability that there are single-celled organisms living on Mars given that there are multi-celled organisms living on Mars is high even though it is improbable that there are multi-celled organisms living on Mars.

The probability of the conclusion of a deductively valid argument given the premises is one, so deductively valid arguments may be thought of as having the strongest possible degree of inductive strength. However, they are not usually referred to as inductively strong because we have a separate, more informative, term to describe them.

Inductive logicians have made a distinction between inductive, or logical, probability and epistemic probability. Inductive probability was intended to be an objective probability of the conclusion of an inductive argument given its premises that depends solely on the strength of the evidence that the premises furnish for the conclusion of the argument.

The epistemic probability of the conclusion given the premises measures the degree of conviction that a person would have for the conclusion were they to learn that the premises were true. It does change from person to person, and it depends on the stock of relevant knowledge possessed by a given person at a given time.

It is now widely agreed that there is no such thing as inductive probability. But if the definition of inductive strength is understood in terms of epistemic probability, then the evaluation of inductive arguments in inductive logic is a subjective affair. In fact, it is no longer clear how it is to be understood as an evaluation at all.

Summary: In response to question 2, we may give answers like "the argument is valid", "the arguments is inductively strong" or "the argument is inductively weak."

Remark: No answer to question 1 implies an answer to question 1, and no answer to question 2 implies an answer to question 1. For example, in arguments 1 and 2, the fact that planets do not move (exactly) in ellipses does not change our answers to question 2. For question 2 requires us to suppose (hypothetically) that the premises are true.

Exercise: In light of this remark, discuss the following examples (all statements are understood to refer to the year 1998 AD):

  1. There are multi-celled organisms living on Mars. Therefore, there is intelligent life on Mars.
  2. There are multi-celled organisms living on Mars. Therefore, there are single-celled organisms living on Mars.
  3. There are multi-celled organisms living in Lake Mendota. Therefore, there is intelligent life living in Lake Mendota.
  4. There are multi-celled organisms living in Lake Mendota. Therefore, there are single-celled organisms living in Lake Mendota.
  5. Nevertheless, in logic, it is assumed that the answer to question 1 is relevant to the evaluation of an argument. But it is a question that needs to be asked in addition to question 2. So, if the premises of an inductively strong argument are false, then logicians are forced to say that the argument is not a good one. It is confusing to say that an inductively strong argument is weak, but this is how the terms are defined.

Tip: Defined terms must be used as defined. You can’t use the term differently just because you don’t like the definition.

A look ahead: In the evaluation of scientific arguments, it is not clear that question 1 is exactly the right question to ask. Are we to say that arguments 1 and 2 are bad arguments because their premises are false. It seems that the premises are at least approximately true, in which case, the argument appears to provide strong evidence that the conclusion is approximately true (though see reading [66] for a word of caution about this). Any talk of approximate truth goes beyond logic.

Another look ahead: It is also not clear that the answers that logicians give to question 2 are the right answers. In inference to the best explanation [13], or in curve-fitting [35, 36, 37], scientists make no attempt to evaluate the probability to the conclusion of the inference. Moreover, if they are only interested in the approximate truth, or closeness to the truth, of the inductive conclusion, then the probability of truth is not the appropriate measure. Popper [68] argues for this.

The General and the Specific

Mill (1874, p.208) says that induction may be defined as the operation of discovering and proving general propositions. The term ‘general’ contrasted with ‘specific’ but it is not obvious what this means.

Definition: Roughly, a specific statement is one that refers only to a list of named particulars, where a particular is a thing that is located in a bounded region of space and time.

For example: "The dime in my pocket is silver" refers to the named particulars ‘the dime in my pocket’.

Note: ‘The dime’ and ‘my pocket’ are not names but descriptions, but this is sufficient for them to count as named particulars.

"Mars is in opposition to the Sun," refers only to the particulars ‘Mars’ and ‘the Sun’. Yet not all the vocabulary used refers to particulars. The term ‘silver’ refers to a property possessed by many things not in my pocket, just as the relationship ‘in opposition to’ applies to many pairs of objects besides Mars and the Sun. But the generality of concepts used does not make these statements general in the sense understood by logicians because none of the other particulars to which the concepts apply are named in the statement.

Another issue is whether the property ‘silver’ or the relationship ‘in opposition to’ are themselves particulars. This is a matter of debate in philosophy today. But either way, a statement like "The volt meter reads 110 volts" from being a specific statement in the sense intended in inductive logic.

The contrast is between specific statements and general statements. A general statement in inductive logic has the form "All A’s are B’s." For example: "All the dimes in my pocket are silver," "All planets move on ellipses," or "All bodies attract one another by a force proportional to the product of their masses and inversely to the square of the distance between them." One might think that these should be specific statements as well, since they apply to a specific list of particulars: the dimes in my pocket, the planets in the universe, and all material objects in the universe, respectively. But the point is that this list of particulars is not a named list.

Question: Is the statement "A sodium salt burns yellow" specific or general? (This is a general statement because it can be rephrased as "All sodium salts burn yellow.") Is the statement "All emeralds previously found have been green " specific or general? (This is also general.)

Simple enumerative induction goes from a list of observations of the form "this A is a B" to the conclusion "All A’s are B’s". The example Hume made famous is like this:

  1. Billiard ball 1 moves when struck. Billiard ball 2 moves when struck. Billiard ball 3 moves when struck… Billiard ball 100 moves when struck. Therefore, all billiard balls move when struck.
  2. Having made this distinction, Skyrms (1966, section 1.5) goes on to point out that not all inductive arguments go from the specific to the general. Analogical arguments go from specific to specific (e.g., the grapes I just sampled from this bunch taste sweet, therefore the next grape I sample from the same bunch will taste sweet). And many inductive arguments in science go from the general to the general, like:

  3. All bodies freely falling near the surface of the Earth obey Galileo’s law. All planets obey Kepler’s laws. Therefore, all material objects obey Newton’s laws.

And others go from the general to the specific. For example:

  1. All emeralds previously found have been green. Therefore, the next emerald to be found will be green.

So, is there any point in mentioning this distinction at all? Mill thought so, while other inductive logicians do not (e.g., Skyrms (1966)). Skyrms’ argument is that there are exceptions to the rule that all inductive arguments go from the specific to the general. However, there seems to be some connection with our earlier characterization of inductive arguments as being ampliative. Arguments from specific statements to general statements are ampliative. Moreover, analogical arguments from specifics to specifics seem to proceed via an inference to a general proposition when spelt out in full detail. For example, in inferring that the next grape sampled will taste sweet, I have implicitly concluded that all grapes in the bunch are sweet because there have no information about the next grape apart from its being from that bunch. From that general conclusion, the specific conclusion that the next grape will taste sweet is a deductive inference, and is therefore unproblematic. Example 10 is similar.

Example 9 is a case in which a more general proposition is inferred from less general propositions. But presumably those less general propositions are inferred from other premises, and we expect such premises will contain specific statements if traced back far enough.

So, all inductive logicians all agree to the following statement:

If we understand how to infer general propositions (in the sense defined) from specific premises, then we will understand all forms of inductive inference.

Note: Mill does not insist that induction always starts from specific statements alone. So, he has no hesitation in saying that Example 10 is an example of induction.

A look ahead: William Whewell agrees with Mill’s statement that induction may be defined as the operation of discovering and proving general propositions, but he appears to have a different idea about what counts as ‘general’. He thinks that an inductive conclusion should introduce a more general concept not already contained in the premises, such as in Example 10, where Newton’s law introduces the concept of ‘gravity’, which does not appear in Galileo’s or Kepler’s laws. Whewell does not think that Hume’s billiard ball example is a genuine induction because the concepts of ‘billiard ball’, ‘move’ and ‘struck’ already appear in the premises. This disagreement is at the heart of the Mill-Whewell debate.

Summary: Inductive logic makes the following assumptions about scientific inference:

      1. Whether the premises are true.
      2. The probability that the conclusion is true given that the premises are all true.

I believe that all of these commitments should be abandoned.