2.4. Deductive Arguments
By Stephanie Gibbons and Justine Kingsbury, adapted by Marc Chao and Muhamad Alif Bin Ibrahim
A deductive argument is one that is intended to guarantee the truth of its conclusion. The terms we use in evaluating deductive arguments are validity/invalidity and soundness/unsoundness.
First, let us discuss validity. A valid argument is one in which, if all its premises were true, the conclusion would also have to be true. For validity, it does not matter whether the premises are actually true. What matters is that there is a connection between the premises and the conclusion such that if the premises were true, the conclusion would necessarily follow. A valid argument is one where it is impossible for the premises to be true and the conclusion false.
The validity of an argument is independent of the actual truth of the premises. Therefore, you do not need to know anything about the subject matter of the argument to judge its validity. To say that an argument is valid is to comment on its structure, not its content. When we talk about validity, we are addressing the first of the two argument evaluation tasks: the connection between the premises and the conclusion.
A valid deductive argument is one in which, if all the premises were true, the conclusion would also have to be true.
For example:
| Premise 1: | No men are mothers. |
| Premise 2: | Some students are men. |
| Conclusion: | Some students are not mothers. |
and
| Premise 1: | All cats can fly. |
| Premise 2: | Whiskers is a cat. |
| Conclusion: | Whiskers can fly. |
Remember that when considering an argument’s validity, it does not matter whether the premises are actually true. So, it does not matter, for the moment, whether it is true that no men are mothers or that all cats can fly. What matters is the connection between the premises and the conclusion. A valid argument has the strongest possible connection between premises and conclusion, so strong that if the premises were all true, the truth of the conclusion would be guaranteed.
In the first example, to see why the argument is valid, think: suppose it is true that no men are mothers and that some students are men. Then, must it also be true (on that supposition) that some students are not mothers?
The answer is that, supposing those premises to be true, it must also be true that some students are not mothers. So, the argument is valid.
Applying the same reasoning to the Whiskers example: Suppose it were true that all cats can fly, and that Whiskers is a cat. Then the conclusion would also have to be true: Whiskers can fly. The argument is valid.
You may be thinking, “But that’s absurd! We all know that it is not true that all cats can fly! So how can the argument be valid?”
Bear in mind that validity is not the only consideration in determining whether an argument is good. It also matters whether the premises are true. The Whiskers argument is valid, but it is still not a good argument. We will address this issue shortly.
In everyday language, the word ‘valid’ is often used to mean ‘true’ or ‘reasonable’. In philosophy and in this book, ‘valid’ has a technical meaning. A valid argument is one where it is impossible for the premises to all be true and the conclusion false.
Here are several ways to define a valid argument. They all convey the same concept, so you can use whichever helps you best understand validity.
A valid argument is one in which:
- It is impossible for all the premises to be true and the conclusion to be false at the same time.
- The conclusion logically follows from the premises.
- If all the premises were true, the truth of the conclusion would be guaranteed.
- If all the premises were true, the conclusion would also have to be true.
When assessing whether an argument is valid, you are evaluating its structure, not its content. Consider the following argument:
| Premise 1: | All adlers are bobkins. |
| Premise 2: | All bobkins are crockers. |
| Conclusion: | All adlers are crockers. |
You can determine that this argument is valid even if you do not know what adlers, bobkins, or crockers are. The validity of the argument is based on its structure: if the premises were true, the conclusion would necessarily follow.
Typically, you cannot determine an argument’s validity or invalidity solely from the truth or falsity of its premises and conclusion. A valid argument can have false premises and a false conclusion, false premises and a true conclusion, or true premises and a true conclusion. An invalid argument can also have these combinations. The only scenario where you can determine an argument’s validity from the truth or falsity of its premises and conclusion is when an argument has true premises and a false conclusion. A deductive argument with true premises and a false conclusion must be invalid, as a valid argument can never have true premises and a false conclusion.
Therefore, when deciding whether an argument is valid, do not focus on the actual truth of the premises and conclusion. Instead, imagine or suppose that the premises are true, and then consider whether this would mean that the conclusion also has to be true.
You might be wondering why we should care about validity. Since validity does not concern the truth of the premises, what is its significance? We should not accept the conclusion of an argument if its premises are obviously false, as in the Whiskers the cat argument or the “sisters and brothers” argument mentioned earlier. So why point out that the argument is valid if it is clearly flawed?
In such cases, you might not need to in real-life contexts. A deductive argument must satisfy two conditions to be considered sound: it must be valid, and it must have true premises. Once we identify that an argument has false premises, we can already determine it is unsound, regardless of its validity.
However, it is crucial to assess the validity of an argument in other situations. One important scenario is when a deductive argument has all true premises. This alone does not make it a good argument; you must also check its validity. For example, consider the following argument:
| Premise 1: | February is the next month after January. |
| Premise 2: | Grass is green. |
| Conclusion: | Snow is white. |
Although the premises are true, they do not logically connect to the conclusion. Therefore, they provide no reason to believe the conclusion. This demonstrates that having true premises is not sufficient to make an argument sound.
Another scenario where assessing validity is important is when others disagree with you about the truth of the premises. In some contexts, it is useful to point out that even if you think the premises are false, the argument would still be invalid if the premises were true.
Common Argument Patterns
Some argument types are so common that they have their own names. Learning to recognise these patterns will help you identify valid arguments.
This section introduces four common argument patterns and some simple variations on them. All of the argument patterns in this section are valid:
- Modus ponens (affirming the antecedent)
- Modus tollens (denying the consequent)
- Disjunctive syllogism
- Hypothetical syllogism
- Some notes on conditionals and generalisations
Modus Ponens (Affirming the Antecedent)
Consider the following argument:
| Premise 1: | If Rover is a dog, then Rover is a mammal. |
| Premise 2: | Rover is a dog. |
| Conclusion: | Rover is a mammal. |
This argument follows the pattern:
| Premise 1: | If p then q |
| Premise 2: | p |
| Conclusion: | Therefore q |
The letters ‘p’, ‘q’, ‘r’, etc., are traditionally used to represent statements. Any statement can be substituted for ‘p’ and ‘q’, and the resulting argument will be valid. Recognising common argument forms makes it easier to identify valid (and invalid) arguments.
Here is another argument with the same pattern:
| Premise 1: | If Winston Peters is a Cabinet Minister, then New Zealand First must have entered into a coalition agreement with either the National Party or the Labour Party. |
| Premise 2: | Winston Peters is a Cabinet Minister. |
| Conclusion: | New Zealand First must have entered into a coalition agreement with either the National Party or the Labour Party. |
Although this argument has longer statements, the basic pattern remains the same. This pattern is known as Modus Ponens, and it is valid.
Modus Tollens (Denying the Consequent)
Modus Tollens is another common valid argument form. It follows this pattern:
| Premise 1: | If p then q |
| Premise 2: | Not q |
| Conclusion: | Therefore not p |
The first premise states that if p occurs, then q must also occur. The second premise points out that q has not occurred. Therefore, it must follow that p has not occurred either. If p had occurred, then q would also have occurred, but we know q has not occurred.
For example:
| Premise 1: | If Kamala Harris had won the last election, she would be president. |
| Premise 2: | Kamala Harris is not president. |
| Conclusion: | Kamala Harris did not win the last election. |
It is clear that this conclusion cannot be false while the premises are true. This is a valid argument.
Try this one. Remember, you are looking to see whether the pattern of Modus Tollens applies.
Try another one:
Disjunctive Syllogism
A ‘disjunction’ is a complex statement where two statements are joined with ‘or’ (or another word serving the same role). A disjunctive syllogism is a valid argument with the following form:
| Premise 1: | p or q |
| Premise 2: | Not p |
| Conclusion: | Therefore q |
This argument form is valid because the initial premise dictates that one of the two options must hold, and the second premise asserts that one does not hold. It follows that the other must hold.
For example:
| Premise 1: | Chess is the most challenging board game or Monopoly is the most challenging board game. |
| Premise 2: | Chess is not the most challenging board game. |
| Conclusion: | Monopoly is the most challenging board game. |
It does not matter whether it is what is before the ‘or’ or what is after the ‘or’ that is denied. But it must be denied.
You can practice applying the pattern in the following questions.
A disjunctive syllogism is often expressed using an ‘either… or…’ construction. For instance:
| Premise 1: | Either there will be a recession, or house prices will continue to rise. |
| Premise 2: | House prices will not continue to rise. |
| Conclusion: | There will be a recession. |
Sometimes a disjunctive syllogism uses ‘either’ along with ‘or’, and sometimes it does not. This does not change the force of ‘or’ either way. ‘Either’ is generally used rhetorically to emphasise the contrast between the two options.
Hypothetical Syllogism
A hypothetical syllogism creates a ‘chain’ of conditional claims. As long as the links of the chain occur in the right way, where each leads to the next, the intermediate links can be omitted. For example:
| Premise 1: | If housing prices continue to rise, then rents will continue to rise. |
| Premise 2: | If rents continue to rise, then rental accommodation will become unaffordable for the working poor. |
| Conclusion: | If housing prices continue to rise, then rental accommodation will become unaffordable for the working poor. |
This follows the general form:
| Premise 1: | If p then q |
| Premise 2: | If q then r |
| Conclusion: | Therefore, if p then r |
When checking an argument form, the order of the premises is irrelevant. Validity treats the premises as a collection of claims. You can change the order of the premises if it makes it easier for you.
Conditionals and Generalisations
Conditionals
Several of the basic argument patterns above use conditional claims. A conditional is an ‘if… then…’ statement. The ‘if…’ part of the statement is called the ‘antecedent’, and the ‘then…’ part is called the ‘consequent’.
Conditionals are sometimes expressed in a different order. The ‘antecedent’ is the ‘if…’ clause no matter what order the parts are presented in. For example, “If Borka is a goose, then Borka is a bird” means the same as “Borka is a bird if she’s a goose”.
One version of a conditional that people find especially tricky is ‘only if’. For example, a sign in a university car park that says, “Staff permit holders only” means “You can only park here if you are a staff permit holder”. This does not mean that if you are a staff permit holder you must park there. It means that if you are not a staff permit holder, you must not park there. This is equivalent to “If you park here, then you are (must be) a staff permit holder”.
The order of antecedent and consequent in a conditional statement is very important and cannot simply be reversed. For example, “If Borka is a goose, then Borka is a bird” is true, but “If Borka is a bird, then she is a goose” is false because some birds are not geese. The ‘only if’ claim equivalent to “If Borka is a goose, then she is a bird” is “Borka is a goose only if she is a bird”.
Conditional statements can also be expressed using ‘unless’. “If Borka is a goose, then she is a bird” is equivalent to “Borka is not a goose unless she is a bird”. We often use ‘unless’ in contrast to a ‘not’ claim. For example, “I won’t babysit for you unless you pay me” means the same as “If I babysit for you, then you are (must be) paying me”.
Conditionals and Generalisations
There is also an important relationship between conditionals and generalisations. The reason why “If Borka is a goose, then Borka is a bird” is true is because the generalisation “All geese are birds” is true.
Any hard generalisation can be expressed as a conditional. “All geese are birds” can be expressed as “If something is a goose, then it is a bird”. This means that the basic argument patterns that use conditionals all have forms that use generalisations instead.
| Premise 1: | All As are Bs |
| Premise 2: | x is an A |
| Conclusion: | Therefore, x is a B |
This is a variation on Modus Ponens, using a generalisation in place of a conditional.
Here is an example of a hypothetical syllogism using generalisations instead of conditionals:
| Premise 1: | All squares are quadrilaterals. |
| Premise 2: | All quadrilaterals are polygons. |
| Conclusion: | All squares are polygons. |
Any argument with this form will be valid.
Here are some for you to try:
Basic Structural Fallacies (Formal Fallacies)
A fallacy represents a flawed argument. These fallacies often resemble valid argument patterns and are frequently mistaken for them. However, these forms are invalid. It is essential to practice recognising the differences, paying close attention to the patterns.
This section examines three common structural fallacies:
- the fallacy of affirming the consequent
- the fallacy of denying the antecedent
- disjunctive fallacies.
The Fallacy of Affirming the Consequent
The fallacy of affirming the consequent is an invalid argument often confused with Modus Ponens or Modus Tollens. It follows this structure:
| Premise 1: | If p then q |
| Premise 2: | q |
| Conclusion: | Therefore p |
In this fallacy, the consequent of the first premise is affirmed in the second premise. Such an argument is invalid. The first premise asserts that if p occurs, q must also occur. However, it does not claim that the occurrence of q guarantees the occurrence of p.
Consider the following examples:
Example 1:
| Premise 1: | If Bernie Sanders is the president of the US, then Kamala Harris is not the president of the US. |
| Premise 2: | Kamala Harris is not the president of the US. |
| Conclusion: | Bernie Sanders is the president of the US. |
While the first premise is true (only one person can be president at a time), and the second premise is also true (Kamala Harris is not the president), the conclusion does not logically follow. The absence of Kamala Harris as president does not necessarily mean Bernie Sanders is the president.
Example 2:
| Premise 1: | If this shape is a square, then its sides are equal in length. |
| Premise 2: | This shape’s sides are equal in length. |
| Conclusion: | This shape is a square. |
Although any square will have sides of equal length, it is possible for a shape with equal sides to not be a square (e.g., an equilateral triangle). Thus, the conclusion does not follow from the premises.
Recognising the problem in these examples is relatively straightforward. However, some instances of affirming the consequent can be more challenging to identify. It is helpful to carefully analyse the argument’s form, often using letters instead of statements to avoid being distracted by known or believed truths.
Can you correctly identify the argument form?
The Fallacy of Denying the Antecedent
The fallacy of denying the antecedent is another invalid argument often mistaken for Modus Ponens or Modus Tollens. It follows this structure:
| Premise 1: | If p then q |
| Premise 2: | Not p |
| Conclusion: | Therefore not q |
Consider the following example:
| Premise 1: | If it is wrong to eat meat, then it is wrong to eat human beings. |
| Premise 2: | It is not wrong to eat meat. |
| Conclusion: | It is not wrong to eat human beings. |
While the first premise may be true, it does not follow that if eating meat is permissible, eating human beings is also permissible. There can be compelling reasons to avoid eating humans, even if eating other types of meat is acceptable.
Another example:
| Premise 1: | If Kamala Harris is president, then the president is a woman. |
| Premise 2: | Kamala Harris is not president. |
| Conclusion: | The president is not a woman. |
The conclusion does not follow from the premises. It is possible to imagine a scenario where Kamala Harris is not president, but another woman holds the office. Thus, the argument is invalid.
Disjunctive Fallacies
A disjunctive syllogism follows this valid form:
| Premise 1: | p or q |
| Premise 2: | Not p |
| Conclusion: | Therefore q |
| Premise 1: | p or q |
| Premise 2: | p |
| Conclusion: | Therefore not q |
There are situations where arguments of the second form might be accepted. Consider the following example:
| Premise 1: | Donald Trump or Kamala Harris will win the next presidential election. |
| Premise 2: | Kamala Harris will win the next presidential election. |
| Conclusion: | Donald Trump will not win the next presidential election. |
At first glance, this argument may appear valid: if both premises are true, the conclusion must also be true. This is because there can only be one winner of the election; it is impossible for both Trump and Harris to win. Thus, if Harris wins, Trump cannot. Here, the true meaning of Premise 1 is “Donald Trump or Kamala Harris will win the next election, but not both”.
In English, the word “or” can mean “or, and both are possible” (inclusive or) or “or, but not both” (exclusive or). The valid form of a disjunctive syllogism applies to either interpretation of “or”. However, the form:
| Premise 1: | p or q |
| Premise 2: | p |
| Conclusion: | Therefore not q |
is only valid if the “or” is exclusive, meaning it is not possible for both p and q to occur. This determination depends on the meaning of the claim, not merely the argument’s form.
When assessing such arguments, ask yourself, “Is it possible for both p and q to occur?” If it is possible, the argument is invalid. If it is impossible, the argument is valid. Adding “but not both” to the disjunctive premise can clarify this.
Sometimes, people assume that an “either… or…” construction indicates an exclusive “or”. While this might be useful, it is not always the case in English. For example, if someone says, “Bring either beer or wine to the party; I don’t mind”, and you bring both, it would be unreasonable for them to say, “You brought both. You can’t come in. I said to bring either one or the other”. The presence of “either” does not specify whether the “or” is exclusive or inclusive. Use common sense to determine this. A good rule of thumb is to assume the “or” is inclusive if unsure.
Chapter Attribution
Content adapted, with editorial changes, from:
How to think critically (2024) by Stephanie Gibbons and Justine Kingsbury, University of Waikato, is used under a CC BY-NC licence.
