Reasoning

We have seen so far that, as rational beings, we must support the claims we think are true with evidence. We have also seen that we need to, at times, convince or persuade others of the truth of a claim through an argument whereby we try to show that the claim we think is true - the conclusion - follows from or can be inferred from another claim or claims we take to be true - the premise(s).

It is the logical move from premise(s) to conclusion - inference - that pri­marily concerns us here in this book.

In fact, the very essence of reasoning is concerned with this logical move. There are plenty of times when we reason correctly. You may come to the immediate conclusion that “I have to take Main St. to work today” from the claims, “There are only two ways for me to get to work, either Highway 1 or Main St.” and “Highway 1 is closed due to construction.” In argument form, this piece of reasoning looks like the following:

(1) There are only two ways for me to get to work - either Highway 1 or Main St.

(2) Highway 1 is closed due to construction.

(3) I have to take Main St. to get to work today.

And if it really is the case there are only two ways for you to get to work - either Highway 1 or Main St. - then we can see that it follows that, or we can infer, deduce, or conclude that, you have to take Main St. to work today. That move, the one from premise (or premises) to a conclusion, is a straightforward example of reasoning - and, in this case, reasoning cor­rectly. Here are a couple more examples of reasoning correctly.

(1) Politicians in the Crack Down Party are all conservative about spending.

(2) Congresswoman Smith is in the Crack Down Party.

(3) Congresswoman Smith is conservative about spending.

If you found out that it is true that Congresswoman Smith is in the Crack Down Party and you know that it is true politicians in the Crack Down Party are conservative about spending, then you would reason (infer, deduce, conclude) correctly that Congresswoman Smith is conservative about spending.

Now, let’s say you’re watching your favorite courtroom drama on TV and the attorney for the defendant puts forward this argument:

(1) If my client committed the crime, then he would have been in Chicago.

(2) But he was not in Chicago; check the gas receipts, GPS, and witness testimony.

(3) Therefore, my client did not commit the crime.

Again, if the two premises are absolutely, positively true, and there is noth­ing else to consider in the case (in other words, this was the so-called linchpin), then the defense attorney, jury, judge, and any of us familiar with the case would be reasoning correctly that the defendant did not commit the crime.

Sometimes, we are confronted with claims that require us to reason through their connections slowly, in a more conscious fashion. Consider a certain apartment lease agreement. Let’s say that you got a new job in Denver, Colorado, six months into a 12-month lease agreement you signed with an apartment in San Francisco, California, so you need to move from your apartment and would like to break your lease. The contract states:

The leasee may be let out of the 12-month lease agreement if and only if:

(1) He can find another leasee to take his place for the remainder of the agreement;

(2) or He is moving out of the state for work purposes only.

However, if (1), then the new leasee (the one replacing the leasee who broke his lease agreement) must sign an additional 12-month lease agree­ment. If (2), then the leasee must provide a notarized copy of the offer of employment. If neither of these conditions can be met, then the leasee will not be let out of the 12-month lease agreement.

Since you have a notarized copy of your offer of employment for the job in Denver, Colorado, you reason correctly that you, the leasee, will be let out of your 12-month lease agreement.

Deductive Reasoning and Arguments

Broadly speaking, there are two types of reasoning: deductive reasoning, which utilizes deductive arguments, and inductive reasoning, which utilizes inductive arguments.

The conclusion is supposed to follow necessarily (absolutely, apodictically) from the premise(s);

If the conclusion does, in fact, follow from the premise(s) and if, in fact, all of the premises are true, then it is impossible for the conclusion to be false - the conclusion must/has to be true;

The premise(s) is/are supposed to entail the conclusion - in other words, the conclusion can already be found in the premise(s).

These arguments are all deductive arguments utilizing deductive reasoning:

(1) There are only two ways for me to get to work - either Highway 1 or Main St.

(2) Highway 1 is closed due to construction.

(3) So, I have to take Main St. to work today.

(1) Politicians in the Crack Down Party are all conservative about spending.

(2) Congresswoman Smith is in the Crack Down Party.

(3) This shows us that Congresswoman Smith is conservative about spending.

(1) If my client committed the crime, then he would have been in Chicago.

(2) But he wasn’t in Chicago (check the gas receipts, GPS, and witness testimony).

(3) Therefore, my client did not commit the crime.

(1) If it’s raining we’ll stay inside, and if it snowing we’ll go outside to play·

(2) It’s either raining or snowing·

(3) Hence, we’ll either stay inside or go outside to play.

We can see that the conclusions drawn in each argument are the only ones that could possibly follow, so the conclusions follow necessarily (absolutely, apodictically) from the premises. Also, given the truth of the premises, it has to be the case that the conclusions are true as well. Finally, if the conclusions were not explicitly stated, they still are there implicitly or implied by the premises - we can logically “see” (so to speak) that the conclusions are already present in the premises or, understood differently, we can “read” the conclusion from the premises.

In this book, we will be concerned with two standard forms of deduc­tive reasoning: the first considers reasoning with categories of things that dates back to Aristotle, called categorical logic, traditional logic, syllogis­tic logic, or simply Aristotelian logic; the second considers the reasoning associated with claims, which are statements, propositions, or declarative sentences (or parts of declarative sentences), aptly called propositional logic or sentential logic.

Categorical Logic

In his Prior Analytics, Aristotle (384-322 âñå) laid out a system of logic that would dominate Western and much of Middle Eastern thought for some 2,000 years. Aristotle’s logic concerns categories of things (kind of like sets, groups, or types/kinds), the characteristics that the members (individuals, instances) of those categories possess or do not possess, and what appropri­ate inferences can be made from our knowledge of the categories and the characteristics of their members. Aristotle was able to capture these rela­tionships in four categorical claims, which have come to be known as:

A Claim: All A are B.

E Claim: No A are B.

O Claim: Some A are not B

The fundamental instrument of reasoning in Aristotle’s system of logic is the syllogism, which he defines in the Prior Analytics as “discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so.” In its simplest form, a syllogism is an argu­ment composed of exactly two premises and one conclusion, the claims of which are A, E, I, and/or O claims. The following are examples of properly formed syllogisms that should be intuitively obvious:

(1) All cats are mammals.

(2) All mammals are warm-blooded animals.

(3) All cats are warm-blooded animals.

(1) No Christians are atheists.

(2) All Catholics are Christians.

(3) No atheists are Catholics.

(1) Some cats are black.

(2) All black things are harder to see at night.

(3) Some cats are harder to see at night.

(1) No Republicans are Democrats.

(2) Some Republicans are socially liberal people.

(3) Some socially liberal people are not Democrats.

(1) All men are mortal.

(2) Socrates is a man.

(3) Socrates is mortal.

Propositional Logic

There is a tradition of reasoning with propositions or sentences in the West that goes back to Aristotle as well but received its first serious treatment with the Stoic philosopher Chrysippus (ca. 280-205 âñå). Whereas categori­cal logic deals with the attributes of categories of things and what can be inferred from the relationships between and among these things, proposi­tional logic concerns the attributes of claims (propositions, sentences) and the reasoning possible when claims are related to one another. Among oth­ers, Chrysippus noted these pieces of correct reasoning:

(A) If the 1st, then the 2nd. But, the 1st. Therefore, the 2nd.

(B) If the 1st, then the 2nd. But, not the 2nd. Therefore, not the 1st.

(C) Either the 1st or the 2nd. But not the 1st. Therefore, the 2nd.

Examples of these arguments, which we saw above already, are all in the realm of propositional logic:

(1) There are only two ways for me to get to work - either Highway 1 or Main St.

(2) Highway 1 is closed due to construction.

(3) So, I have to take Main St. to work today.

Notice that this argument is an example of the form of reasoning utilized by Chrysippus in (C).

(1) If my client committed the crime, then he would have been in Chicago.

(2) But he wasn’t in Chicago (check the gas receipts, GPS, and witness testimony).

(3) Therefore, my client did not commit the crime.

Notice that this argument is an example of the form of reasoning utilized by Chrysippus in (B).

(1) If it’s raining we’ll stay inside, and if it snowing we’ll go outside to play.

(2) It’s either raining or snowing.

(3) Hence, we’ll either stay inside or go outside to play.

You can see how the conclusions of these arguments can be deduced (inferred) from the premises.

Inductive Reasoning and Arguments

In an inductive argument, on the other hand:

The conclusion is supposed to follow likely or probably from the premise(s), not necessarily - in other words, the inference to the conclusion is one of degree and on a percentage scale from 0.000...1, which is the least likely or probable case where the conclusion may be drawn from the premise(s), through 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 to 0.9999., which is the most likely or probable case where the conclusion may be drawn from the premise(s).

If all of the premises are true, then it is still possible for the conclusion to be false.

The premise(s) does/do not entail the conclusion - the conclusion cannot already be found in the premise(s), at least not in the same way the con­clusion is entailed by the premise(s) of a deductive argument.

These arguments are all inductive arguments utilizing inductive reasoning:

(1) My old running shoes were Brand X, genuine leather, and lasted me a year.

(2) These new shoes I’m thinking of buying are Brand X and genuine leather.

(3) So, these new shoes will last me a year, too.

Note that it’s still possible that the new shoes won’t last you a year. Also, it’s more appropriate to conclude, “So, it’s likely that or probably the case that these new shoes will last me a year, too.” Finally, you’d feel more confident

about the conclusion if you had more cases/examples of other running shoes lasting a year - one case offers some support for the conclusion, while 100 similar cases would offer more support for the conclusion, and 1,000 simi­lar cases would offer even more support for the conclusion, etc.

(5) (1) Team X won the championship game the last two years.

(2) Team X has had the best statistics of all of the teams this season.

(3) Thus, Team X will win the championship game against Team Y today.

Again, it’s still possible that Team X loses the game for any number of rea­sons. And again, it’s better to state, “Thus, Team X probably or likely will win the championship game against Team Y today.” Finally, you’d feel more confident about the conclusion if we knew this information, too: “Team Y’s star player has an injury and won’t be playing in the game” and “Team Y is playing in Team X’s arena.”

(6) (1) I know there are 10 beans total in this bag, but I don’t know

their color.

(2) The last nine beans I pulled from this bag have been red.

(3) The next bean I pull out will be red, too.

Could the last bean you pull out be a different color? Of course. So, you might want to conclude that the last one has a chance of being red and not that it will absolutely, positively, definitely, without a shadow of a doubt, be red. Notice that you’d feel less confident about the conclusion following from the premises if the second premise was “The last five beans I pulled out of this bag have been red” and even less confident if that premise was “The last two beans I pulled out of this bag have been red.”