Inductive Reasoning Explained
What is Inductive Reasoning
We explain inductive reasoning, a bottom-up reasoning method that reasons by consistency, comparing particulars and probabilities to find likely truths.^{[1]}^{[2]}
- Inductive reasoning is a type of logical reasoning that deduces likely truths by comparing likely truths and/or facts about specific things.
- Inductive reasoning is a reasoning type that reasons by consistency.
- Inductive reasoning starts from the bottom-up, comparing facts/observations/measures about specific things (or likely truths about a class of things gleaned from comparing facts about specific things) and reasoning toward a generalized and likely conclusion about a class of things (or in some cases, a likelihood about a specific thing).
- Inductive reasoning is reasoning is based on recognizing patterns in data and drawing likely conclusions based on those patterns.
- Inductive reasoning is a reasoning method that speaks to comparing empirical evidence (that which is observed and measured in the real world) and rationalized ideas to find likely rational and empirical truths.
- Inductive reasoning is the process of deducing likelihoods, as opposed the process of deducing necessarily certain logical truths from facts about classes of things compared to specific observations (like the other main reasoning type deduction does.)
- Inductive reasoning synthesizes data, looking for relations to draw likely conclusions, it doesn’t just analyze necessarily truths about systems like deductive reasoning does.
- Inductive reasoning, given that it deals with likelihoods, can produce logically strong and cogent arguments with false conclusions (and can also produce weak arguments with accidentally true conclusions as well).
- Inductive reasoning produces multi-value truth-values, and since it expresses likelihood, it is best to state findings along side qualifiers like confidence and likelihood (see image below for a visual).
All the points above speak to what inductive reasoning is, but these points do not alone explain every aspect of induction. Below we explain a number of different ways to understand inductive reasoning and offer a number of examples.
TIP: The image below is an example of how we can state confidence and likelihood for inductive inferences (conclusions to arguments made using inductive evidence). The idea is to use “multi-value” truth-values to communicate to a reader how likely a truth is and how confident the author is of the findings. If confidence and likelihood are stated, then a statement which contains probable truth can itself be considered true.
An Introduction to Inductive Reasoning
We can use inductive reasoning in math to find patterns, or in empirical science to draw inferences from observations, or in philosophy to deduce likely truths from other likely truths.
No matter how we use inductive reasoning, it is the same basic idea.
Inductive reasoning is determining what is possible given the data we have (which by extension can help us deduce what isn’t possible using deduction). The more useful data we have, the more certain we can be about our conclusion.
If we say:
- Most Greeks had beards.
- Plato was a Greek.
- Therefore, Plato likely had a beard.
That would be inductive reasoning. The conclusion didn’t necessarily follow from the premises, it was just likely.
No matter what example we use, the basics are the same, inductive reasoning involves the inferring of B from A where B does not necessarily follow from A.
In other words, induction involves inferring an inference (conclusion) from a premise or set of premises (statement or set of statements) in cases where the inference doesn’t necessary follow from the premise(s).
Since it doesn’t necessarily follow, we therefore have to state qualifiers like confidence and/or likelihood to make our conclusions themselves are factual (f we say “therefore, Plato had a bread – period”… we are essentially lying, even though our data suggested it).
Meanwhile, to help illustrate induction and deduction, one should note that when the inference does follow from the premise, it is deduction.
If we say:
- Exactly 90% of Greeks had beards.
- Plato was a Greek.
- Therefore there is a 90% chance Plato had a beard.
That is deductive reasoning (even though we used a probability, the conclusion was logically certain, so it is deductive).
In other words, dealing with probability doesn’t itself mean we are doing induction. The above argument is deductive and not inductive because all we are doing is stating a logical certainty about a probability (we aren’t stating a probable conclusion, so it isn’t induction).
90% of Greeks had beards, and Plato was a Greek, therefore there is exactly a 90% chance he had a beard (that is a logically certain conclusion given the premises).
The takeaway is:
- Even if we are dealing with a probability, if the conclusion we are making is logically certain, it is deduction.
- Meanwhile, even if we aren’t dealing with probability, if say we are only dealing with two specific observations, if the inference doesn’t necessarily follow from the premise, it is induction.
For another example:
If I say 1, 2, …, then ask what the next number in the sequence is. You’ll use inductive reasoning to conclude “3” based on the pattern. However, you won’t necessarily be right. The answer is “probably 3,” but it isn’t “certainly 3.” Instead it could be literally any number. Maybe it is “2” again, or maybe it is “1,” we don’t know the method behind the sequence for sure, so we don’t know the number for sure. That is induction in logic, and it forms the basis of inductive arguments used in propositional logic.
If I say 1+1=X, then ask what X is. You’ll say “2,” and in this case, you’ll necessarily be correct as the answer is logically certain given the statement. That is deduction in logic, and it forms the basis of deductive arguments used in propositional logic.
Due to the inherent uncertainty of induction, we can state simply that inductive reasoning is any reasoning that deals with probability rather than certainty in its inference (regardless of the argument form used or the structure of the argument).
Inductive reasoning is, speaking metaphorically, the considering of possible worlds given the facts at hand to reasonably predict what might be the case. Then, from there we can use deduction to exclude impossible worlds.
For example, if we know X + Y = 0, we can be sure that both numbers are either zero, or that one positive and the other negative (using deduction we know this is necessarily the case). However, beyond that we don’t know what the values of X and Y are. Now, if we get more data, and learn that X is a negative number between -5 and -1, we can narrow down our guesses (and necessarily exclude other possibilities). Now, if we run a series of tests that can somehow give us a value for X, and 90% of the time the result shows us X is -3, we can reasonably state that we are 90% positive that X=-3 (using inductive reasoning). Then we can from there assume Y is 3 (and we can run tests based on that hypothesis). We can also deduce that X and Y are not zero using deduction, thus our inductive method will have provided us grounds to employ deductive reasoning and narrow down our search. This helps us understand how induction works and how it pairs with deduction.
The gist of induction is the simple part, it is dealing with inferences that don’t necessarily follow from the premise (and thus dealing with probabilities, especially in the inference), but the many different ways inductive arguments can work in different fields add complexity… which is probably why this page is long-ish.
Inductive Reasoning.TIP: Most of the reasoning we do as humans is informal inductive reasoning. We learn a new fact, then we compare it to many other facts and generalizations in our heads in an instant, drawing conclusions that aren’t certain, and forming analogies between the new facts and past experiences. When we run tests in a lab, to test a hypothesis, it is simply a formal version of this inductive method. We can get very useful information using induction, but we can never achieve total certainty. That is the probable nature of induction.
Understanding Inductive Reasoning Via Examples
One way to wrap your head around inductive reasoning, especially in terms of semantic arguments, is through some examples. So let’s look at some examples:
An example of an inductive argument that compares a likelihood about a class of things to an observation about a specific thing might look like this:
- 1. Since most A are B, and 2. since this C is A, 3. therefore this C is likely also B.
- Or this, 1. Since most Greeks have beards, and 2. since Socrates is a Greek, therefore 3. it is likely Socrates also has a beard.
For another example, consider this inductive argument which compares particular facts to infer a generalization about a class of things (which may or may not be true):
- 1. Since this A is B, and 2. since this C is A, 3. perhaps all C are also B as a general rule.
- And this, 1. Since Socrates is a man, and 2. since Socrates is a Greek, 3. perhaps all Greeks are men as a general rule (that isn’t true of course, but that makes sense, as the argument was “weak;” more on that below).
For another example, consider an inductive argument that compares a general rule with a specific observation, phrased as an if…then statement:
- 1. If it’s A then it’s probably B, it’s A, so it’s probably B.
- Or this, If it’s raining then it’s probably cloudy (general rule-of-thumb, a likelihood about a class of things), it’s raining (the case), therefore it’s probably cloudy (a probable fact, a generalization about a specific thing; its only correct if it is the case that it is cloudy; it is conditional).
All of these examples, which compare facts about specific things and/or general rule-of-thumbs to draw out a likely conclusions, illustrate the probable nature of induction and some of the different ways inductive arguments can work.
TIP: See more Examples of Inductive Reasoning.
Semantics: Inductive reasoning/logic/argument/method are all synonyms for the inductive method of reasoning (the subject of this page). Whether we are applying it to formal logic, informal arguments, or statistical equations, it is always the same underlying concept (comparing particulars and likelihoods to infer likelihoods about other specific facts and classes of things).
More on Inductive Reasoning in General
With the above in mind, Inductive Reasoning (AKA Reasoning By Consistency or Bottom-up Reasoning) is first and foremost a reasoning method that deals with probability (as opposed to certainty).
Generally, induction starts from the “bottom up,” beginning with specific facts or observations/measurements about specific things, and then uses those likelihoods to predict a likely truth about a class of things.
In other words, induction generally starts with a set of likelihoods and particulars and then looks for probable conclusions and general rules. This is true if we are working with a full argument with many propositions (claims and it is true if we working with a single proposition.
In more complete terms, induction generally starts with facts that contain uncertainty or facts about specific things (for example: statistics, general rules-of-thumb, or observations/measurements about a specific thing as opposed to a necessary truth about a class of things), and compares them to other likelihoods and particulars, and then finds what is likely true (often about a class of things) based on comparing that “inductive evidence.”
This can be done formally, with formal logic using the rules of logic, reason, and inference, or it can be done informally (much of human reason is informal induction, when we learn a new fact, we compare it to all the facts sitting in our head to make sense out of it, and the end result is often a generalized rule).
When all the evidence is compared, especially in formal terms, if the inductive evidence is “strong” enough, the argument is considered cogent and certainty can be stated as a multi-value truth value (for example: very likely false, likely false, likely true, very likely true) or a simple 2-value truth value (for example: probably true and probably false).
With that said, there are two general ways to go about induction.
- We can do bottom-up induction (induction in its classical form): Starting with facts that contain uncertainty (like statistics) -> detecting patterns -> and then making a conclusion of likelihood (which produces a hypothesis/theory),
- or we can do inverse induction: Starting with a hypothesis -> then comparing observations about specific things and/or probable facts -> then comparing the hypothesis and observations -> then drawing out certain conclusions (potentially resulting in a new hypothesis/theory).
Given the different ways induction can work, and given the ways in which it can essentially mirror top-down deductive reasoning in its formatting, it helps to understand Induction simply as: Inductive reasoning describes any reasoning method that results in a probable conclusion rather than a certain truth-value.
TIP: In the next section we’ll discuss the other major reasoning method, deduction, and we’ll compare deduction and induction. With this in mind, almost all reasoning methods are based on empirical data and ideas are at their core, and are therefore somewhat inductive (at least in the metaphysical sense; as they are all essentially dealing with likelihoods based on the observation and consideration of particulars at their core; consider, to know “all men or mortal” one must have first observed many particular men being mortal and confirmed a general rule-of-thumb, one can’t confirm each instance). This is to say, deductive reasoning only deals with necessary logical truths (generally necessary logical truths about language), while inductive reasoning deals with probability (and the empirical and rational world is wrought with probabilities… after-all our senses may be tricking us, and each of us only truly know what we can empirically sense). Luckily, by pairing deduction and induction, and by following the logical, mathematic, and scientific methods, we can get pretty darn close to knowing (we can be pretty darn “positive”).
Understanding Induction By Understanding Inductive vs. Deduction
With the above in mind, there are only really two main reasoning methods (so if you understand them, then you’ll understand induction and the foundation of all other reasoning methods).
Deduction is the one that deals with necessarily certain facts and rules only (and produces necessary 2-value truth-values, True and False specifically), and induction is the one that deals with likelihoods (and produces probable multi-value truth-values, which can be transposed to the simple 2-value probably True and probably False).
NOTE: There is a third truth value to consider, “unknown.” We don’t need to deal with that here, but it is essentially the main factor that creates multi-value truth values in induction.
With that above covered, let’s compare deduction and induction in more detail:
- Deductive reasoning deduces certain logical truths from other certain truths to produce certain truth-values, generally proceeding from general premisses to a specific conclusion (top-down), based on logical rationalism.
- Inductive reasoning deduces the likelihood of truth by comparing probable truths to other probable truths to produce a probable truth-value, generally proceeding from specific premisses to a general conclusion (bottom-up), based on observation, speculation, and empiricism.
Now consider:
- Where deductive reasoning might look like this: As a rule all men are mortal (a certain rule), it is a fact that Socrates is a man (certainly the case), therefore it is a fact that Socrates is necessarily mortal (a certain truth; a logically necessary fact).
- Inductive reasoning looks like this: As a rule-of-thumb most Greeks had beards in Socrates’ time (a general rule-of-thumb; a probable fact about a class of things), and since it is the fact that Socrates was a Greek (a specific fact), therefore it is likely the case that Socrates had a beard (a likely truth).
The Semantics of Rule, Case, and Fact: Above we used the terms rule (something that is always true), case (something that is true or is suspected to be true “in this case”), fact (something that is observed to be true or was deduced as a logical truth).
NOTE: Below is one of many ways to illustrate the difference between deduction and induction.
Deductive | Inductive | |
Major Premise | A certain fact about a class of things (a generalization or rule).
Ex. All Humans are Mortal. |
A certain or probable fact about a specific thing or a probable fact about a class of things.
Ex. Socrates is a Greek. |
Minor Premise | A certain fact about a specific thing or class of things (a fact or rule).
Ex. Socrates is a Human, or All Greeks are Human. |
A certain or probable fact about a specific thing or a probable fact about a class of things.
Ex. Socrates is a Man, or Most Greeks have Beards. |
Conclusion (Inference) | Deduce a certain fact about a specific thing or class of things; produces a certainty. Ex. Socrates is a Mortal, All Greeks are Mortal (this is necessarily true supposing the premisses are true; tautology). | Infer a likelihood about a specific thing or class of things; produces a likelihood or generalization. Ex. Since Socrates is Greek and he is a man, therefore all Greeks are likely men (a generalization about a class of things; and a false one); or Since Socrates is a Greek he also likely has a beard (a likelihood about a specific thing). |
TIP: In general, the order of the major and minor premisses doesn’t matter (although those terms have conations). The only time that could change is in a complex equation where Order of Operations said otherwise.
Deductive and Inductive Reasoning (Bacon vs Aristotle – Scientific Revolution).Bottomline on the above: Deduction and induction don’t produce compelling arguments on their own. Deduction produces tautological (redundant) facts about ideas. Meanwhile, induction, while based on observation, data, and experiment, produces only probabilities. For reasons like this, all good arguments contain a mix of reasoning types and seek enough data to make “strong” (likely) arguments and testable theories and hypotheses.
Understanding the Rules of Induction and Probable Truth Values
To reiterate, the most important fact to know about induction is that it deals with probability and not certainty, generally because it deals with likelihoods and particulars in an attempt to infer a likelihood about a class of things, and this nature of probability will color everything induction touches.
Consider, for deductive arguments, if the premises are true then the inference is always true (and if even one premise is false, the argument is logically unsound and invalid… even if the inference is true).
Meanwhile inductive arguments are more complex, as the premises can be true and the conclusion can still be false (if say the data isn’t “strong” enough).
The beard argument above works well enough, Socrates probably did have beard.
However, the other argument about all Greeks being men doesn’t work. Socrates is a Greek, and Socrates is a man, but inferring that all Greeks are men from this is obviously not right. So what is up?
The answer, as we’ll see below, is that this argument is “weak” (and therefore not cogent AKA uncogent), as the conclusion lacks significant supporting evidence.
Only two data points were considered, and so we unsurprisingly drew a demonstrably false conclusion about the Greeks using our inductive method!
If we had also considered Athena, we would have seen that all those specific facts together pointed to a the a general truth, that is: Since Athena is a female, and Socrates is a male, and since both are Greek, all Greeks are either male and female.
The moral here, we should remain skeptical when dealing with induction and constantly seek the best data.
The goal of science would be to find find a Greek who was neither male or female (to falsify our hypothesis that all Greeks are either male or female), and to produce a better theory, not simply to find more evidence to support the conclusion. After-all an inability to find the Greek who was neither male or female would itself be a type of evidence of absence, and would make for a “strong” inductive argument. Meanwhile, our ability to find a Greek neither male nor female would help to create a better theory… either way, it is a win for logic and science.
Deductive Reasoning Vs. Inductive Reasoning and the Syllogistic Structure
Putting aside other argument forms (which we will discuss below), we can use the same argument structure for essentially any argument, inductive or deductive. This structure is called a syllogism.
The basic syllogistic form, of which all the fundamental laws of logic apply in a simple way, is called a “categorical syllogism.”
A categorical syllogism is an argument consisting of exactly three categorical propositions (two premisses and a conclusion) in which there appear a total of exactly three categorical terms, each of which is used exactly twice.
The syllogism below is an example of a “categorical” syllogism. Here categorical means the term (subject term or predicate) represents a category things, not a specific thing.^{[3]}
- Premise 1: All humans (subject) are (logical connective) mortal (predicate); or, A = B.
- Premise 2: All Greeks are a human; or, C = A.
- Conclusion: All Greeks are mortal; or, Therefore, B = C.
That syllogism above is deductive, because it uses necessary truths to deduce other necessary truths. But we are working with induction, so let’s look at that same format, but this time use an inductive argument.
An inductive syllogism (a non-deductive or statistical syllogism) might look like this:^{[4]}
- Almost all Adult Humans are taller than 25 inches; or, Almost all A are B; or, A probably equals B.
- Socrates is an Adult Human; or, This specific C is A; or, C = A.
- Therefore, it is “highly likely” Socrates is taller than 25 inches; or, Therefore this C is likely B; or, B probably equals C.
With deductive reasoning we can know whether an argument is true or not based on its “figure” and “mood” (as long as we confirm our logic is sound), which means we can create a logic rule-set that always works.
With the above said, things doesn’t work exactly the same way with inductive reasoning (as we aren’t just working with necessarily certain truths).
With inductive reasoning we can know a lot depending on the “figure” and “mood” of a proposition, but there is only so much we can know for sure (see the truth-table below).
In other words, there are different metrics that apply to deductive and inductive reasoning respectively. So let’s cover those now to further illustrate the difference between these two main logic types.
Deductive Reasoning and Validity and Soundness Vs. Inductive Reasoning and Cogency and Strength
Below we will explain the general rule-set behind inductive and deductive arguments for determining truth values (knowing deductive is vital for understanding induction, so we’ll continue to cover both on this page).
- Deductive arguments are either sound or unsound and either valid and invalid.
- Inductive arguments are either cogent or uncogent and either strong or weak.
All of those terms speak to whether or not the parts (subject, premisses, predicates, propositions, etc) of the argument make sense together (that they connect logically).
TIP: For more reading see: Deduction and Induction from Patrick J. Hurley, A Concise Introduction to Logic, 10th ed.
The following is true for deductive arguments only:^{[5]}
- A valid deductive argument is an argument in which it is impossible for the conclusion to be false given that the premises are true.
- An invalid deductive argument is a deductive argument in which it is possible for the conclusion to be false given that the premises are true.
- A sound argument is a deductive argument that is valid and has all true premises (if it isn’t true for all premisses, it is “unsound”).
- An unsound argument is a deductive argument that is invalid, has one or more false premises, or both.
The relationship between the validity of a deductive argument and the truth or falsity of its premises and conclusion can be illustrated by the following “deductive reasoning truth table”:
TIP: Below the table is saying “if the premises are true and if the conclusion is true… then the statement is or isn’t valid.” It is, in other words, a type of if…then statement. It is also a hard a fast rule that applies to any deductive argument.
Premises | Conclusion | Validity |
T | T | ? |
T | F | Invalid |
F | T | ? |
F | F | ? |
Meanwhile, the following is true for inductive arguments only:
Unlike the validity and invalidity of deductive arguments, the strength and weakness of inductive arguments is expressed in degrees of probability.
- To be considered “strong,” an inductive argument must have a conclusion that is more probable than improbable (there must have a likelihood of greater than 50% that the conclusion is true).
- The inverse is also true (i.e. argument is therefore “weak” if it has less than 50% probability).
- Thus, an uncogent argument is an inductive argument that is weak, has one or more false premises, or both.
- Meanwhile, A cogent argument is an inductive argument that is strong and has all true premises; if either condition is missing, the argument is uncogent.
The relationship between the strength of an inductive argument and the truth or falsity of its premises and conclusion can be illustrated by the following “inductive reasoning truth table”:
TIP: As you can see below, with induction we never know if the conclusion is true for sure. It is always probably true or probably false. Otherwise, induction uses truth tables just like deduction to much the same effect.
Premises | Conclusion | Strength |
T | prob.T | ? |
T | prob. F | Weak |
F | prob.T | ? |
F | prob. F | ? |
TIP: As you can see inductive reasoning follows rule-sets like deduction does, but it doesn’t produce certainty like the sound and valid “moods of the deductive syllogisms” do. Instead, our rulesets of induction only offer insight. This is due to the probable nature of induction. Still, it really does help to know things like “if the premise is true, and the conclusion is probably false, then the strength of the argument is necessarily weak.” Induction is hard, so we need all the help we can get. With that in mind, we have a few more tricks up our sleeve for drawing more certainty out of induction (explained below).
TIP: With both deductive and inductive logic we should consider how the terms of propositions relate to each other, do they follow necessarily? Are they tautological (do we need to say All Greeks are mortal, isn’t mortality a property of the categorical class “All Greeks” in the first place)? You can learn more about that on our page on Hume’s Fork.
Exploring the World Through Induction: Using Conditional Reasoning, Bayesian Theorems, and the Scientific Method
As it stands now, we have a framework to put inductive arguments in (the syllogism), and if we explore the rules of logic and reason a bit more we can learn more about how to fact-check propositions, how if…then statements work, and dive into other aspects of induction.
For now, let’s assume all our propositions are logically sound and true and talk about some other methods we can apply induction to in order to learn about the world.
Good methods for using induction to learn about the world are the scientific method, if..then statements, equations, and Bayesian theorems.
If…then Statements and the General Structure of an Argument: General, Conditional, and Syllogistic
If…then statements are an alternative format to the syllogism we used above that we can use to state both inductive and deductive arguments.
The truth tables above featured if…then statements that treated all the premises and conclusions as “ifs” and then related it back to validity and soundness, but we can transpose single premises onto if…then statements as well.
With if…then statements, instead of saying A=B, we can say ” if A “→” B (where “→” means “then” or “implies”) or, more appropriately, we can say P → Q (where P stands for the subject term and P for the predicate term in a proposition whether it is a premise or inference).
Now the only note is, that since we are doing induction, we want to think of it as “P → prob. Q” (if P… then probably Q, where P stands for the argument or subject term and Q the inference or predicate for example).
TIP: In logic P, Q, and R are generally used in place of A, B, and C (especially when an equation needs to use all those symbols like inductive Bayesian equations do). They are all just placeholders like X, Y, and Z in algebra.
Typically an argument has a basic structure such as:
- a set of assumptions or premises
- a method of reasoning or deduction and
- a conclusion or point.
If the assumptions and premises are only about specific things or general rules of thumb, or if they state uncertainty, then the argument is inductive (regardless of what format or other sub-type of reasoning we use).
For inductive logic, the following formats will all work.
We can follow the law of detachment (a law behind if… then… conditional reasoning that uses a hypothesis):
- P → Q (a conditional statement that says “if P then Q”). So with induction, if P then probably Q.
- P (hypothesis stated; assigns a value to P). With induction our P value will be a specific fact, a probability, or a rule-of-thumb.
- Q (conclusion deduced; therefore Q). With induction it will be “probably Q.”
Or, in plain English:
- Premise 1: If it’s raining then it’s probably cloudy.
- Premise 2: It’s raining.
- Conclusion: It’s probably cloudy.
Or, we can follow the law of contrapositive (a law behind if… then… conditional reasoning that uses a variable):
- P → Q (conditional, if P then Q). So with induction, if P then probably Q.
- ~Q (~Q means if it is Q in this case; it is a type of variable). If it is Q in this case.
- Therefore, we can conclude ~P (we can conclude it will be P in this case). Then it is probably P in this case.
Or, in plain English:
- Premise 1: If it’s raining then it’s probably cloudy.
- Premise 2: If it is cloudy in this case.
- Conclusion: It is probably raining.
Or, we can follow the law of the syllogism which can be stated in a conditional form (a law behind if… then… that works with two certain statements):
- P → Q (if P then Q). So with induction, if P then probably Q.
- Q → R (if Q then R). And, if Q then probably R
- Therefore, P → R (if P then R). Again, like we covered above, it would be: Therefore, if P then probably R.
Or:
- Premise 1: If it’s raining then it’s probably cloudy.
- Premise 2: If it is cloudy then it’s probably humid.
- Conclusion: It is raining so it implies it is probably humid.
Or, a statement can be transposed to the classical “syllogistic” form in its inductive form which shows equivalence:
- A = prob. B
- B = prob. C
- Therefore, A = prob. C
… and that syllogism above can, like the if…then statements, be transposed to an inductive form, where, for example, A = B is either a specific fact, or is stated as a probability such as A likely equals B, or A often equals B.. (like we showed above).
This argument can be phrased exactly like the “P → prob. Q, Q → prob. R, Therefore P → prob. R” statement above!
- Premise 1: If it’s raining then it’s probably cloudy.
- Premise 2: If it is cloudy then it’s probably humid.
- Conclusion: It is raining so it implies it is probably humid.
In other words, the above if…then statements are all somewhat deductive in nature (like the classical syllogism), but they also work as a format in which to place inductive arguments.
This is sort of like it is with math where 1+2=3 is deductive in general, but a statement like X+2=Z can see X and Z change or be uncertain if the value of X changes or is uncertain.
Any statement where the conclusion necessarily follows from one or more premises is deductive, and any statement in which the conclusion likely follows the premise is inductive.
As you can see above, both types can be stated in the same format, but they follow slightly different rules (in one we draw probable conclusions, in the other we draw certain ones).
Using Bayesian Theory to Calculate Probable Truth Values
Bayes’ Theorem shows that we can compare conditional probabilities to find the “likelihood” something is true or false (even when we can’t know it for sure).
When we use bayesian theory, we use inductive logic to find likelihood.
Bayesian theorem looks like this:
Where:
- Where A and B are events and P means probability.
- and are the probabilities of observing the events and without regard to each other.
- , a conditional probability, is the probability of observing event given that event is true.
- is the probability of observing event given that is true.
That looks a bit intimidating on its own, and when multiple values are used, it can get a little more intimidating, but logically you should understand the gist of what is happening above.
All we have done is taken a syllogism and transposed it onto a mathematic equation that makes an inductive argument. We could just as easily write something like:
- Since the result of observing and without regard to each other is X.
- And since the probability of observing event given that is true is Y.
- Therefore, the probability of observing event given that is true is Z.
We could also generally transpose this to any of the forms above… and, we can transpose it to essentially any other argument form that uses any logical connector and rule of inference in propositional calculus (formal propositional logic).
Learn more about What Bayesian Probability theory Can Be Used For here.
The point here isn’t to master logic-as-mathematics, the point is to show that no matter how we deal with probability, we are always practicing induction, and that induction can be used to help bring us toward knowing (even though it can never tell us anything with 100% certainty).
Using Induction to “Do Science”
The Scientific Method shows that we can create a rock solid and usable theory based on a collection of related facts, testing, and a strong hypothesis by pairing deduction and induction (and the not yet noted abduction which speculates a hypothesis).
We don’t need to know F=ma directly, and we don’t need to know it with certainty, we just need to know it works every time when subjected to rigorous test (we can glean it is correct from repeated testing, in the spirit of Bayesian logic).
We can also use hypothesis and theory to great effect. In other words, we can use induction to great effect despite its probable nature.
A hypothesis, a theory, an observation in a test, a statistic… its all essentially inductive.
Conclusion
We may not even know “I think therefore I am” for 100% certain (as even that could be a trick), but it doesn’t mean we can’t come up with great theories that haven’t failed us yet.
Sure, deduction can tell us that all red balls are red, and that is useful when considering the redness of a ball, but induction lets us put that red ball to the test, measuring it in labs across the country, comparing data, and determining certainty about how red balls in general will likely behave in terms of physics when interacted with, based on careful testing and analysis of past results.
With that all covered, you should check out our full section on logic and reason to explore our pages on different reasoning methods and ontological and epistemological theories, as those pages all tie back into each other to help explain the importance of the inductive reasoning method and its strengths and limitations (especially when paired with the other reasoning methods and the general rules of logic and reason).
So bottom line, in a word, induction compares facts about likelihood and particulars to determine likelihoods about particulars or classes of things… but there are many different formal and informal ways this can be done, and many different ways to express this using formal or informal language.