What Is a Valid Argument?
What Is a Valid Argument?
Do you ever listen to someone’s argument and think that it makes sense, and all the ideas are connected, but you don’t know why? Well-crafted arguments need to have certain ingredients, and those ingredients in an argument make sense to us.
Sometimes that “makes sense” feeling is a recognition that an argument is valid. The definition of a valid argument is this: if the premises are true, then it’s impossible for the conclusion to be false. In other words, a valid argument actually proves that its conclusion is true.
If you really want to understand in detail what this definition means, you first have to take the time to understand in detail what an argument is, and that means also understanding what statements are.
What is an argument?
An argument is a series of statements that try to prove a point. The statement that the arguer tries to prove is called the conclusion. The statements that try to prove the conclusion are called premises.
Statements can be either true or false. A statement is true when the world matches the statement. If I were to say, “2 plus 2 is 4,” then this statement is true since it matches how the world is. If I were to say, “2 plus 2 is 5,” then this statement is false since it doesn’t match how the world is.
Statements can be combined using logical operators such as ‘not’, ‘and’, ‘or’, ‘If...then’, and, ‘if and only if.’ When we combine two or more statements using logical operators, the result is a compound statement.
For example, the statements, “Peter is a man,” and, “Quinn is a woman,” can be combined using the logical operator ‘and’ to make a compound statement as follows: “Peter is a man, and Quinn is a woman.” Or they can be combined using ‘if and only if’ as follows: “Peter is a man if and only if Quinn is a woman.”
Here are more examples of statements formed with logical operators: “Ben is not a woman,” “James is tall, or Adam is fast,” “Either you can go straight, or you can make a right,” “If you are tired, then you will make mistakes,” “Shawn can win the race if and only if he enters it.”
What are logical forms?
Validity is a type of logical form. Logical forms are like math formulas. Each comprises variables and operators. For example, the math formula “x + x = 2x” comprises a variable ‘x’ and an operator ‘+’. If we were to plug in the value 1 for x, then we would get “1+1 = 2.” Logical forms are similar. The difference is that instead of mathematical operators, logical forms use logical operators, and instead of variables that are filled in with numbers, the variables of logical forms are filled in with statements.
Let’s use these symbols to represent logical operators:
Further, we can also use symbols to represent statements:
Using these symbols provides a convenient shorthand for representing different kinds of statements. For example, we can represent the statement, “Ben is not a woman,” using the symbol ‘~B’. That is, we use the negation operator ~ together with the variable ‘B’ which represents the statement, “Ben is a woman.”
We can represent the statement, “James is tall, and Adam is fast,” using the symbol ‘J & A’-- that is, using the conjunction operator & together with the variables ‘J’ and ‘A’ which represent the statements, “James is tall,” and, “Adam is fast,” respectively.
We can represent the statement “Either you can go straight, or you can make a right,” using the symbol ‘S V R’-- that is, using the disjunction operator V and the variable ‘S’ and ‘R’ which represent the statements, “You can go straight,” and, “You can make a right.”
These symbolic representations give us the logical forms of the statements. The form of a statement is like a math formula: it comprises variables and operators. For example, the math formula “x + x = 2x” comprises a variable ‘x’ and an operator ‘+’. If we were to plug in the value 1 for x, then we would get “1+1 = 2.”
Likewise, if we were to use the logical form “T => M,” then ‘T’ and ‘M’ are our variables and ‘=>’ is our logical operator. If we were to plug in the value “You are tired” for T, and the value “You will make mistakes” for M, then we would get “If you are tired, then you will make mistakes.”
Compound statements and truth tables
When we combine statements using logical operators, the truth of the compound statement is determined by the truth of its component statements. That determination relation is represented on a truth table. For example, this is the truth table for the conjunction operator:
A truth table is a tool that allows us to determine whether the compound statement is true by looking at its component statements. It shows us how the truth value of the compound statement is determined by the truth values of its component statements.
Let’s look at the first line of the truth table: If we assume P is true and Q is true, then the entire compound statement P & Q is true. Combining two true statements using the conjunction operator results in a true compound statement.
Let’s look at the second line of the truth table: If we assume P is true and Q is false, then P & Q is false. Combining one true statement and one false statement using the conjunction operator results in a false compound statement.
Let’s look at the third line of the truth table: If we assume P is false and Q is true, then P & Q is false. Combining one false statement and one true statement using the conjunction operator results in a false compound statement.
Let’s look at the fourth line of the truth table: If we assume P is false and Q is false, then P & Q is false. Combining two false statements using the conjunction operator results in a false compound statement.
Just like the conjunction operator, all the other logical operators have their own truth tables:
The truth table for negation
The truth table for disjunction
The truth table for conditional
The truth table for biconditional
Now that we understand what the logical form of a statement is and how to use truth tables to determine the truth or falsity of a compound statement, we’re ready to consider what the form of an argument is.
What makes an argument’s form valid?
An argument is a series of statements, so to get at the form of an argument, you just need to get at the form of the statements that compose it. Let’s look at an example. Let’s say we have two statements: A and B.
If we put them in order, then we make an argument.
Premise: A
Conclusion: Therefore, B
Above is an example of something that counts as an argument since it has a premise and conclusion. That’s all it takes for something to be an argument: it needs to have a premise and a conclusion.
When we say an argument is valid, we are talking about an argument’s form. If we plug in true premises, a valid form guarantees a true conclusion. A valid form is similar to an accurate math formula. For example, in mathematics, if you want to get the area of a circle, you will first get the formula to calculate the area of a circle. In this case, the formula will be “A = π (r)^2.” At this point, all you need to do is plug in the radius r of the circle in the formula to get an accurate result. If you get the accurate radius, then you are guaranteed an accurate area.
The values we plug in for the variables in a math formula are numbers. By contrast, the values we plug in for the variables in a logic formula are statements. Let’s look at a deductive argument form that logicians call modus ponens:
If P then Q,
P
Therefore, Q
In the above form, ‘P’ and ‘Q’ are variables, and ‘if...then…’ is the logical operator. Modus ponens is universally regarded as a valid form of argument.
An argument’s form is valid if and only if the truth of the argument’s premises guarantees the truth of its conclusion. A valid form is similar to an accurate math formula. Just as the formula “A = π (r)^2” guarantees an accurate area if you plug in an accurate radius, likewise, a valid form of argument guarantees you a true conclusion if you plug in values for the variables that yield true premises. In other words, if you plug in values for the variables in a valid form of argument, and the resulting premises are true, the conclusion is guaranteed to be true. If an argument is valid and it has true premises, in other words, then it’s impossible for it to have a false conclusion.
Note: there are two categories of invalid argument: inductive arguments and fallacies. If an argument is inductive and its premises are true, then it is possible for it to have a false conclusion; it’s just that it’s unlikely that the conclusion is false if the premises are true. If, on the other hand, an argument commits a fallacy, then even if the premises are true, they still tell you nothing at all about the truth or falsity of the conclusion. (I talk more about these invalid forms of argument in another piece.) Also, an argument with a false premise is like a fallacy: it tells you nothing at all about the truth or falsity of the argument’s conclusion.
Now let’s plug in statements into modus ponens to show that if the premises of a valid argument are true, then the argument is guaranteed a true conclusion.
Let’s suppose P is “It is raining,” and Q is “The street is wet.” The result is the following argument:
If it is raining, then the street is wet.
It is raining.
Therefore, the street is wet.
In our above example, if the premises of the argument are true, then it’s impossible for the conclusion of the argument to be false.
It’s possible to visualize the validity of an argument. To see this, let’s look at another example of a valid argument:
Premise 1: All mammals are animals.
Premise 2: All dogs are mammals.
Conclusion: Therefore, all dogs are animals.
Here’s the form of the argument:
All M are A
All D are M
Therefore, all D are A
The above form is called a categorical syllogism, and it is a valid form. This form is valid because if the two premises are true then the conclusion has to be true.
Now that we understand what a valid argument is, it is worth mentioning what a sound argument is. An argument is sound if and only if it is a valid argument and all the premises are true.
Some people mistakenly use the expression “true argument.” In fact, arguments cannot be true or false. The statements that we plug in the argument can be true or false. The argument has a form: valid or invalid. If the argument’s form is valid, and all the premises are true, then the argument is sound.
Learning forms of argument is an important step in developing your critical thinking skills. Once you understand the argument, the form of an argument, what a valid argument is, you’ll be able to evaluate even complex arguments since those arguments are built up by putting together many simpler ones.