Deductive logic

Chad Nilep

Nagoya University

Arguments

It is often said that academic writing is a kind of writing that makes an argument. According to writing instructor Lennie Irvin, "In college, everything's an argument" (2010, p. 9). What does it mean to say that academic writing makes an argument, or is an argument? What is an argument?

We use the word argument in at least two different ways. In one sense, an argument is what two people do when they disagree about something. If I tell my friend, "Go get me a drink," he might tell me, "No way. Get it yourself." This is a kind of disagreement, but it is not what we mean when we say that academic writing makes an argument.

An argument, in the sense we mean, is a set of facts that lead to a conclusion. Cognitive science professor Bram van Heuveln suggests this example of an easy to understand argument. If you want to order a pizza and need to decide what kind of toppings to order, you might think something like this:

"I should not get pepperoni on my pizza, because the last time I got pepperoni on my pizza I got very sick." (Van Heuveln 2011)

This is an argument. It offers a reason (because last time I got very sick), and that reason supports a conclusion (I should not get pepperoni on my pizza).

An argument is a set of one or more reasons – in logic we call them premises – that support one conclusion. In every good argument, the premises need to support the conclusion, and they need to be true. (We'll define these terms in Validity and truth, below.) These are necessary requirements for all good arguments: the reasons must be relevant to the conclusion, and to the best of our knowledge they must be true. There is also a third quality that many good arguments satisfy: they account for all of the information relevant to the conclusion. Van Heuveln offers this example of an argument that does not satisfy the third criteria:

Let us point out something about the pizza that the reader may have noted him or herself already: was it really because of the pepperoni that I became sick last time, or was that just a coincidence? Maybe I also had 6 glasses of coke, a dozen hot wings, and 2 pints of Ben and Jerry’s on that fateful day I ate the pepperoni pizza! Indeed, had we known that, then we may not have been as impressed with the original pizza argument. (Van Heuveln 2011)

Knowing all of the information relevant to the conclusion helps us judge whether the argument does a good job supporting the conclusion. Therefore, many good arguments include all of the relevant facts, and all good arguments need to include premises that are true and are relevant to the conclusion.

Deductive versus inductive logic

The system of reasoning with arguments, premises, and conclusions is called logic. There are different formal systems of logic – different ways of making an argument and different ways of representing these arguments in writing. Two major systems are called inductive logic and deductive logic.

Inductive logic (or inductive reasoning) constructs arguments that support a conclusion, but don't claim to show that the conclusion is necessarily true. A strong inductive argument gives us reason to think that its conclusion is most likely true. Some examples of inductive reasoning are generalizations, analogies, or statistical predictions. Many arguments in natural sciences, social science, the humanities, and other academic fields aim to show that a particular conclusion is likely to be true, and therefore they rely on inductive logic.

Deductive logic (also called deductive reasoning or deduction) is a precise and well-ordered system that aims to provide definite support for a conclusion. While inductive reasoning can show that a conclusion is probably true, deductive reasoning can show that a conclusion must be true. In other words, if we use deductive logic and if we have a valid argument with premises that are definitely true, then we can guarantee that our conclusion is true. Some examples of deductive reasoning are implication and syllogism.

Validity and truth

We said above that if a deductive argument is valid and its premises are true, then its conclusion must be true. What do we mean by valid, and what do we mean by true?

Validity – the quality of being valid – refers to how an argument is constructed, and the relationship among the premises and the conclusion. Remember that premises are the reasons that support the conclusion. An argument is valid if the premises "necessarily entail" the conclusion. In other words, based on the way the premises fit together, there is only one conclusion that can be made. Said another way, "In some sense 'the truth' of the conclusion is 'contained in' the truth of the premises" (Van Heuveln 2011). That may be difficult to understand, so let's look at an example. Consider this syllogism:

All licensed doctors in Japan passed the Licensing Board Exam.
My doctors are licensed doctors in Japan.
Therefore, my doctors passed the Licensing Board Exam.

We can represent this argument as a picture. In the diagram below, the blue circle represents my doctors. The yellow circle represents all licensed doctors in Japan. The green circle represents people who have passed the Licensing Board Exam. Since all of the people who are licensed to practice as doctors in Japan have passed the licensing board, and since all of the people who are my doctors are in that group, then it must be true that all of my doctors have passed the Licensing Board Exam. It is necessarily the case that the blue circle is inside the green circle, since all of the blue circle is inside the yellow circle, and all of the yellow circle is inside the green circle. That is what we mean by "necessarily entailed".

The group "my doctors" are inside the group "licensed doctors", which is inside the group "peole who passed the Board exam".

A valid argument is one whose conclusion is "contained" within the premises, or "necessarily entailed" by the argument structure. In other words, validity is a fact about the argument itself, and the relationship between the premises and the conclusion.

Truth, unlike validity, does not only refer to the argument itself. Premises are true if they accurately reflect the real world. In the example above the premise "My doctors are licensed doctors in Japan" is true if there are some people who are my doctors, and all of them are licensed doctors in Japan. If that is not the case – for example, if I have a doctor who is licensed in another country, or if one of my doctors is practicing without a license – then the premise is not true.

It is possible for an argument to be valid even if not all of its premises are true. If one of my doctors never passed the Licensing Board Exam and is practicing without a license, then the conclusion "All my doctors passed the Licensing Board Exam" is not true. The argument is still valid – the blue circle is still inside the green one – but the conclusion is only proven true if the argument is valid and the premises are true.

On the other hand, the truth of the conclusion doesn't guarantee that the argument is valid. Consider this syllogism:

All humans are animals.
Some animals live in Japan.
Therefore, some humans live in Japan.

Each of the premises is true: All humans are animals, in the sense that they are living things that eat and move around. It's also true that some animals live in Japan. It happens to be true, too, that some humans live in Japan. But this argument is not valid. To see this, lets try to represent it with circles like we did before.

If some, but not all animals are in Japan, we can't conclude where the humans are.

We know that all humans are animals, but not all animals are in Japan. The fact that some animals are in Japan does not necessarily mean that some humans are in Japan. In the real world, we know that some humans are in Japan, but this fact does not follow logically from our argument. The "conclusion" turns out to be true, but only accidentally; the argument is not valid.

Syllogism

One common type of deductive argument is the syllogism (sometimes called standard, categorical, Aristotelian, or deductive syllogism). A syllogism is an argument with two premises that support a conclusion. This form of syllogism was developed by the philosopher Aristotle more than two thousand years ago, and is very well developed and widely used. (For much, much more history and detailed description see e.g. Smith 2017, Lagerlund 2016, or Parsons 2012.) A syllogism of this type consists of a set of categorical propositions, which consist of terms. We will define categorical propositions and their terms in the next section.

Categorical propositions

The types of statements used in syllogisms like those described here are called "categorical propositions". Categorical propositions contain two "terms", a subject and a predicate. (The meanings of "subject" and "predicate" are slightly different in logic than in grammar, but we won't worry about the specific differences here.) In the following list, the subjects are underlined and the predicates are in italic.

As we said, all categorical propositions have two terms, the subject and the predicate. In addition, we need to pay attention to the quantity and the quality of a categorical proposition. Quantity can be either universal (true for all or nothing) or particular (true for some, which means at least one thing). Quality can be either positive (is) or negative (no, not). This leads to four types of categorical propositions. In the chart below, S means a subject and P means a predicate.

  Positive Negative
Universal All S is P
e.g. All dogs are mammals.
No S is P
e.g. No dogs are fish.
Particular Some S is P
e.g. Some mammals are carnivores.
Some S is not P
e.g. Some fish are not carnivores.

 

If we want to evaluate the logic of a piece of writing or speech, we can often think about what it means and try to represent this as a set of categorical propositions. The chart below shows several categorical statements that we can understand from the ordinary-language sentence about plankton.

Plankton consists of tiny, often single-celled plants such as diatoms or algae, as well as small or microscopic animals known as zooplankton.
  • Some plankton is plants.
  • Some plankton is diatoms.
  • Some plankton is algae
  • Some plankton is zooplankton.
  • All plankton is small.

 

We can use categorical statements to construct syllogisms. A syllogism is a type of argument in which a conclusion is inferred from two premises. The premises are categorical statements that share one term. Let's look again at the example syllogism mentioned above.

All licensed doctors in Japan passed the Licensing Board Exam.
My doctors are licensed doctors in Japan.
Therefore, my doctors passed the Licensing Board Exam.

Each line in the syllogism is a categorical statement, consisting of two terms. Notice that one of the terms, "licensed doctors in Japan", appears in both of the premises. Also, each of the terms in the conclusion (the subject "my doctors" and the predicate "passed the licensing board exam") appears in one of the premises. This is the form of a standard categorical syllogism. Traditionally the term that appears in both premises is called the "middle term", the subject of the conclusion is called the "minor term", and the predicate of the conclusion is the "major term". But you don't need to worry about memorizing those labels; the useful thing is to be able to recognize a valid argument.

Let's look at more of the description of plankton and see if we can find a logical argument.

Plankton consists of tiny, often single-celled plants such as diatoms or algae, as well as small or microscopic animals known as zooplankton. Since plants absorb carbon dioxide in the process of photosynthesis, plankton can reduce the level of CO2 in the atmosphere. Although most of the carbon eventually escapes back into the atmosphere, some sinks deep in the ocean as dead plants or zooplankton.
Some plankton is plants.
All plants absorb carbon dioxide.
Therefore, some plankton absorb carbon dioxide.

 

The word since in the ordinary-language sentences suggests that the text is making an argument. We can represent that argument as the syllogism in the lower box. The form of that syllogism is valid. If the argument is valid and the premises are true, we know that the conclusion must be true. Therefore, if it is true that some plankton is plants and all plants absorb carbon dioxide, it must be true that some plankton absorb carbon dioxide. Can you make other arguments of this type?

There are hundreds of possible argument forms that could be made with three categorical propositions of two terms each, but for most of those forms the conclusion does not necessarily follow. Only a few of the possible forms turn out to be valid arguments. The chart below shows several patterns for valid syllogisms. (If you are interested in how to prove that these are valid, and in other arguments that are valid in certain conditions, you can find more information in the references and further reading below.) Can you think of arguments using these forms?

All M is P
All S is M
All S is P
No P is M
All S is M
No S is P
Some M is P
All M is S
Some S is P
All P is M
No M is S
No S is P
No M is P
All S is M
No S is P
All P is M
No S is M
No S is P
All M is P
Some M is S
Some S is P
Some P is M
All M is S
Some S is P
All M is P
Some S is M
Some S is P
No P is M
Some S is M
Some S is not P
Some M is not P
All M is S
Some S is not P
No P is M
Some M is S
Some S is not P
No M is P
Some S is M
Some S is not P
All P is M
Some S is not M
Some S is not P
No M is P
Some M is S
Some S is not P
 

 

Evaluating a logical argument

We know that if a deductive argument is valid and its premises are true, then its conclusion must be true. Using that knowledge, we have a way to evaluate the truth of a conclusion. When someone makes an argument, you can check to see whether it is valid by examining its premises and their relation to the conclusion. If the argument is valid, you should then examine whether the premises are true. But if the argument is not valid, you can reject it without needing to examine the premises.

Let's look at an ordinary-language statement that draws a conclusion and try to represent it as a logical argument.

[American paper] money is green and so are trees, so money must grow on trees.
-a psychology patient described by Duffy and Campbell (1994)
Some money is green.
Trees are green.
Therefore, trees are money.
Some P is M
All S is M
All S is P

 

This conclusion is obviously not true. The premises seem to be true: some money is green. At least some trees are green, and maybe we could agree that all trees are green if we define "green" and "trees" in the right way. But the argument is not one of the valid forms. The conclusion does not necessarily follow from the argument, even if the premises are true. Showing that an argument form is not valid can be a useful way to reject an argument.

Let's look at one more example of ordinary language that makes an argument.

I do not believe that we can have any freedom in the philosophical sense, for we act not only under external compulsion but also by inner necessity.
-Albert Einstein
No one under compulsion and necessity is free.
All people are under compulsion and necessity.
Therefore, no people are free.
No M is P
All S is M
No S is P

 

Notice two things about what Albert Einstein said. First, the argument is not completely stated. The conclusion is stated ("we can [not] have any freedom" basically means "no people are free"), and one of the premises is stated ("we act not only under external compulsion but also by inner necessity"). But the other premise, written here as "No one under compulsion and necessity is free," is not directly stated. It is implied in what Einstein said. It is actually quite common for arguments in everyday speech to leave one of their premises implicit.

The other thing to notice about what Einstein said is that it follows one of the valid patterns we looked at earlier. That means that his argument is valid. So, is his conclusion true? The conclusion must be true if the argument is valid and the premises are true. Remember that "true" in an argument means "accurately reflecting the real world". Whether or not Einstein's premises are true in that sense may be a matter of belief.

Implication

Another type of deductive argument is an argument from implication, sometimes called by the Latin name modus ponens. Argument from implication starts with an if-then statement like these.

When the first part of the statement (the "if" part) is true, this implies that the second part (the "then" part) is also true. The argument continues by stating that the "if" part is true. This leads to the necessary conclusion that the "then" part must be true.

If a river is narrow, then it is easy to cross.
Yada River is narrow.
Therefore, Yada River is easy to cross.

The general structure of modus ponens can be written like this, where p and q represent any statement like the ones shown above:

If p, then q
p          
q

The reverse of modus ponens is called modus tollens in Latin, or in English "denying the consequent". Like a modus ponens implication, it starts with an if-then statement. The second premise states that the "then" part is not true. Since a true "if" implies a true "then", a false "then" implies a false "if".

If p, then q
not q       
not p

If anyone walks by, the dog barks.
The dog did not bark.
Therefore, no one walked by.

Be careful with the order of the terms in an argument from implication. A true "if" implies a true "then", but a true "then" does not necessarily imply a true "if". The statement, "If anyone walks by, the dog barks" is true in the real word, at least for my neighbor's dog. It is always barking at children, delivery people, neighbors, or anyone else who walks by. But if the dog barks, that doesn't necessarily mean that anyone walked by; sometimes that dog seems to bark for no reason.

NOT VALID
If p, then q
q          
p
If p, then q
not p       
not q
If anyone walks by, the dog barks.
The dog barked.
X Therefore, someone walked by.
If a river is narrow, it is easy to cross.
Sumida River is not narrow.
X Therefore, Sumida River is not easy to cross.

 

We will talk more about these kinds of invalid arguments later in the course when we discuss formal fallacies.

References, acknowledgements and further reading

Copi, Irving. 1962. Introduction to Logic. New York: Macmillan.

Duffy, J.D. and J.J. Campbell. 1994. "The regional and prefrontal syndromes: A theoretical and clinical overview." Journal of Neuropsychiatry and Clinical Neurosciences 6(4), 379-387.

Irvin, L. Lennie. 2010. "What is academic writing?" in C. Lowe and P. Zemliansky, Writing Spaces: Readings on Writing. Parlor Press.

Lagerland, Henrik. 2016. "Medieval theories of the syllogism." Stanford Encyclopedia of Philosophy.

Parsons, Terence. 2012. "The traditional square of opposition." Stanford Encyclopedia of Philosophy.

Smith, Robin. 2017. "Aristotle's logic." Stanford Encyclopedia of Philosophy.

Van Heuveln, Bram. 2011. "Critical Wisdom." http://www.cogsci.rpi.edu​/~heuveb​/teaching​/CriticalWisdom​/Introduction.htm

The explanations sketched here are influenced by several people's ideas about logical argumentation and deductive reasoning. Irving Copi's popular textbook, Introduction to Logic, was an important part of my education in logic. Bram van Heuveln's "Critical Wisdom" provides much more detail on critical thinking, argument, and deductive and inductive logic. The Stanford Encyclopedia of Philosophy covers not only topics in logic but also many other aspects of philosophy. My colleagues in the Department of Academic Writing at Nagoya University also provide courses in logical thinking, academic presentations, and academic writing that have inspired me. Some of their materials and information are also available from Nagoya University OpenCourseWare.