1.3.1 Deductive Reasoning
Deductive reasoning is characterized by the certainty that can be guaranteed by the conclusion. A few common argument forms typically associated with deductive reasoning are described here.
Syllogisms make claims about groups of things, or categories. They use statements that refer to the quantity of members of a category (all, some, or none]) and denote membership or lack thereof of members of one category in another category. These are examples of categorical statements:
- No vegetarians are pork-chop lovers.
- Some meat eaters are not pork-chop lovers.
- Some mosquitoes are disease carriers.
- All mice are rodents.
Syllogisms are broadly characterized as arguments with two premises supporting the conclusion. Each premise shares a common term with the conclusion, and the premises share a common term (the middle term) with each other.
Some examples provided for valid deductive arguments in section 1.2.2 Attributes of Deductive Arguments are categorical syllogisms. Recall this one:
Premise 1: All cats are mammals.
Premise 2: All mammals are animals.
Conclusion: Therefore, all cats are animals.
This well-known categorical syllogism refers to a specific member of the class of “men”:
Premise 1: All men are mortal.
Premise 2: Socrates is a man.
Conclusion: Therefore, Socrates is mortal.
This type of syllogism has a “disjunction” as a premise, that is, an “either-or” statement. Here’s an example:
Premise 1: Either my pet is a dog, or my pet is a cat.
Premise 2: My pet is not a cat.
Conclusion: Therefore, my pet is a dog.
A hypothetical statement is an “if-then” statement. Hypothetical statements have two components:
- The “if” portion is referred to as the antecedent. It is the precipitating factor.
- The “then” portion is called the consequent. It is the resulting condition.
A pure hypothetical syllogism has two hypothetical premises. Here’s an example:
Premise 1: If it rains on Sunday, then the concert will be canceled.
Premise 2: If the concert is canceled, then the band will go to the movies.
Conclusion: Thus, If it rains on Sunday, the band will go to the movies.
The next two common argument forms use a hypothetical statement as one of the premises.
This argument form has one premise that is a hypothetical (if-then) statement, and another premise that affirms the antecedent of the hypothetical premise. The conclusion then claims the truth of the consequent. In symbolic form, modus ponens looks like this:
if A then C
Here’s an example:
Premise 1: If we get up before sunrise, then we have time for a run.
Premise 2: We get up before sunrise.
Conclusion: So, we have time for a run.
This argument form also has one premise that is a hypothetical (if-then) statement, and the other premise denies (indicates untruth of) the consequent of the hypothetical premise. The conclusion then claims that the antecedent is not the case (that is, denies it.) In symbolic form, modus tollens looks like this:
if A then C
therefore not A
Here’s an example:
Premise 1: If we win today’s game, then we qualify for the final match.
Premise 2: We did not qualify for the final match.
Conclusion: We did not win today’s game.
Arguments based on Mathematics
Arguments supported by arithmetic or geometry lead to necessary conclusions and thus are deductive: Here’s an example:
Premise 1: Twenty-five eggs were left by the Easter bunny in the front yard.
Premise 2: Twenty eggs have been found in the front yard. so far.
Conclusion: Therefore, five eggs remain to be found.
It is important to keep in mind that math-based arguments do not include statistical arguments, because statistics usually suggest probable, not certain, conclusions.
1.3.2 Inductive Reasoning
Inductive reasoning is characterized by the lack of absolute certainty that can be guaranteed by the conclusion. Several of the types of inductive reasoning are described here.
As we have just pointed out, statistical reasoning, though based on numbers like mathematical reasoning, is not deductive because it can offer only probability. Statistics suggest likely outcomes or conclusions but cannot guarantee certainty. The larger process of statistical reasoning often includes complex analysis of properties and populations; in the end, the conclusions can be derived with probability but not certainty. Here’s an example; it did not involve complex analysis:
Premise 1: Of the Easter eggs hidden in the front yard 95% are chocolate.
Premise 2: This egg was found in the front yard.
Conclusion: So, this egg probably is chocolate.
There is no certainty that the egg just found is a chocolate one; but it is highly likely.
An analogy involves highlighting perceived similarities between two things as grounds for transferring further attributes or meanings from one (the source analog) to the other (the target analog.) Here’s an example:
Premise 1: Bandicoots and opossums are marsupials with extra upper teeth.
Premise 2: Opossums are omnivorous and eat small animals and plant matter.
Conclusion: Therefore, bandicoots probably eat small animals and plant matter.
Analogical reasoning is used extensively in making arguments in philosophy; we will see such arguments in later units involving other branches of philosophy. Analogical reasoning is also the core practice in making legal decisions; cases that have already been decided become precedents (source analogs) for deciding subsequent similar cases (target analogs.)
Arguments that advance from knowledge about a subset of members of a particular group of things to conclusions about all such things make generalizations. Such conclusions use inductive reasoning. They may have high probability of being true, but good generalizations are made cautiously. For example, it may be a well reasoned generalization to infer that because rabbits you have seen have whiskers, that all rabbits whiskers. On the other hand, it may be risky to conclude that every Democrat favors gun control, because the democrats you know do so.
Sometimes patterns of inductive reasoning overlap. If 100% of the 85 jelly beans removed so far from the 100-count box have been licorice, one might infer that all jelly beans in that box are licorice. This argument might be characterized as a statistical claim or a generalization.
A causal argument supports a conclusion about a cause-and-effect relationship. Essentially, it asserts a connection between two events. In a particular argument, either the cause or the effect may be known, and the one that is not known is claimed to be the case.
- If I go to the refrigerator and find that my leftover pizza is gone, I conclude that my roommate ate it. This is a move from knowing the effect to inferring the cause.
- If wash my car in the afternoon, and that evening I’m aware of a rain shower, I conclude that my car will be speckled with dirt spots when I go out in the morning. This is a move from knowing the cause to inferring its effect.
Arguments that are predictions make claims about the future. No matter how certain a claim about the future seems – that the sun will rise tomorrow – it is still inductive reasoning. If I have seen giraffes at the zoo each time I was there in the past, I might reasonably conclude that I will see giraffes when I go there tomorrow. But it is not a certainty.
Some Comments on Inductive Reasoning
Despite the lack of total certainty that inductive arguments may offer, inductive reasoning is in no way less valuable or useful than deductive logic. Reasoning we in do philosophy involves making arguments that while plausible, do no lead to absolute certainty. The process of science is based on inductive reasoning; it involves formulating hypotheses that infer connections, not yet proven, between events. When we make moral judgment about a particular actions, our conclusions may be based on our regard for comparable (analogical) actions. Weather forecasts, political polls, and legal investigations are further examples of how inductive reasoning abounds in our world.