Learning Objectives
- What is a logical argument?
- Standard Forms of Arguments
- Law of Detachment
- Law of Contraposition
- Law of Syllogism
- Disjunctive Syllogism
- Fallacy of the Converse
- Fallacy of the Inverse
All of the work we have done so far in building the basics of logic has prepared us for the real point which is analyzing logical arguments logically. We want to use logic to evaluate an argument or situation and decide what is and is not reasonable. Setting up our truth tables with letters representing the simple statements allows us to check our opinions and emotions at the door and just focus on the operations.
A logical argument consists of two parts: a set of premises and a conclusion based on the premises. Our goal is to decide whether an argument is valid or invalid. The premises are supporting evidence for the conclusion. Logic is not about deciding if something is true. Logic is about deciding if the conclusion can be deduced by the given premises.
Valid Argument
An argument is valid if the conclusion follows from the premises. Otherwise, an argument is invalid. To be valid, the conclusion of an argument has to follow from the premises. Reasoning that leads to an invalid argument is called a fallacy.
The method that we will use in this class to determine the validity of an argument is by using a truth table.
Analyzing arguments using truth tables
To analyze an argument with a truth table:
- Represent each of the premises symbolically
- Create a conditional statement, joining all the premises with and to form the antecedent, and using the conclusion as the consequent.
- Create a truth table for that statement. If it is always true, then the argument is valid.
Example
Consider the argument:
| Premise: | If you bought bread, then you went to the store |
| Premise: | You bought bread |
| Conclusion: | You went to the store |
Example
| Premise: | If I go to the mall, then I’ll buy new jeans. |
| Premise: | If I buy new jeans, I’ll buy a shirt to go with it. |
| Conclusion: | If I got to the mall, I’ll buy a shirt. |
Let M = I go to the mall, J = I buy jeans, and S = I buy a shirt.
The premises and conclusion can be stated as:
| Premise: | [latex]M{\rightarrow}J[/latex] |
| Premise: | [latex]J{\rightarrow}S[/latex] |
| Conclusion: | [latex]M{\rightarrow}S[/latex] |
We can construct a truth table for [latex]\left[\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)\right]{\rightarrow}\left(M{\rightarrow}S\right)[/latex]
| [latex]M[/latex] | [latex]J[/latex] | [latex]S[/latex] | [latex]M{\rightarrow}J[/latex] | [latex]J{\rightarrow}S[/latex] | [latex]\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)[/latex] | [latex]M{\rightarrow}S[/latex] | [latex]\left[\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)\right]{\rightarrow}\left(M{\rightarrow}S\right)[/latex] |
| T | T | T | T | T | T | T | T |
| T | T | F | T | F | F | F | T |
| T | F | T | F | T | F | T | T |
| T | F | F | F | T | F | F | T |
| F | T | T | T | T | T | T | T |
| F | T | F | T | F | F | T | T |
| F | F | T | T | T | T | T | T |
| F | F | F | T | T | T | T | T |
From the truth table, we can see this is a valid argument.
Forms of Valid Arguments
Rather than making a truth table for every argument, we may be able to recognize certain common forms of arguments that are valid (or invalid). If we can determine that an argument fits one of the common forms, we can immediately state whether it is valid or invalid.
Law of Syllogism
Law of Detachment
Law of Contraposition
Law of Disjunctive Syllogism
The previous problem is an example of a syllogism.
Law of Syllogism
A syllogism is an implication derived from two others, where the consequence of one is the antecedent to the other. The general form of a syllogism is:
| Premise: | [latex]p{\rightarrow}q[/latex] |
| Premise: | [latex]q{\rightarrow}r[/latex] |
| Conclusion: | [latex]p{\rightarrow}r[/latex] |
This is sometimes called the transitive property for implication.
more on the transitive property
The transitive property appears regularly in the various branches of mathematical study. For example, the transitive property of equality states
if [latex]a = b[/latex] and [latex]b = c[/latex] then [latex]a = c[/latex].
Example
| Premise: | If I work hard, I’ll get a raise. |
| Premise: | If I get a raise, I’ll buy a boat. |
| Conclusion: | If I don’t buy a boat, I must not have worked hard. |
Try It
Is this argument valid?
| Premise: | If I go to the party, I’ll be really tired tomorrow. |
| Premise: | If I go to the party, I’ll get to see friends. |
| Conclusion: | If I don’t see friends, I won’t be tired tomorrow. |
Lewis Carroll, author of Alice in Wonderland, was a math and logic teacher, and wrote two books on logic. In them, he would propose premises as a puzzle, to be connected using syllogisms.
Example
Solve the puzzle. In other words, find a logical conclusion from these premises.
All babies are illogical.
Nobody who can manage a crocodile is despised.
Illogical persons are despised.
Law of Detachment (Modus Ponens)
The law of detachment applies when a conditional and its antecedent are given as
premises, and the consequent is the conclusion. The general form is:
Premise: p → q
Premise: p
Conclusion: q
The Latin name, modus ponens, translates to “mode that affirms”.
Law of Contraposition (Modus Tollens)
The law of contraposition applies when a conditional and the negation of its consequent
are given as premises, and the negation of its antecedent is the conclusion. The general
form is:
Premise: p → q
Premise: ~q
Conclusion: ~p
The Latin name, modus tollens, translates to “mode that denies”
Law of Disjunctive Syllogism
Premise: p V q
Premise: ~p
Conclusion: q
Forms of Invalid Arguments
Fallacy of the Converse
Premise: p → q
Premise: q
Conclusion: p
Fallacy of the Inverse
Premise: p → q
Premise: ~p
Conclusion: ~q
Logical Inference
Suppose we know that a statement of form [latex]P{\rightarrow}Q[/latex] is true. This tells us that whenever P is true, Q will also be true. By itself, [latex]P{\rightarrow}Q[/latex] being true does not tell us that either P or Q is true (they could both be false, or P could be false and Q true). However if in addition we happen to know that P is true then it must be that Q is true. This is called a logical inference: Given two true statements we can infer that a third statement is true. In this instance true statements [latex]P{\rightarrow}Q[/latex] and P are “added together” to get Q. This is described below with [latex]P{\rightarrow}Q[/latex] stacked one atop the other with a line separating them from Q. The intended meaning is that [latex]P{\rightarrow}Q[/latex] combined with P produces Q.
| [latex]P{\rightarrow}Q\\\underline{P\,\,\,\,\,\,\,\,\,\,\,\,}\\Q[/latex] | [latex]\,\,P{\rightarrow}Q\\\underline{{\sim}Q\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sim}P[/latex] | [latex]\,\,P{\vee}Q\\\underline{{\sim}P\,\,\,\,\,\,\,\,\,\,\,\,}\\Q[/latex] |
Two other logical inferences are listed above. In each case you should convince yourself (based on your knowledge of the relevant truth tables) that the truth of the statements above the line forces the statement below the line to be true.
Following are some additional useful logical inferences. The first expresses the obvious fact that if P and Q are both true then the statement [latex]P{\wedge}Q[/latex] will be true. On the other hand, [latex]P{\wedge}Q[/latex] being true forces P (also Q) to be true. Finally, if P is true, then [latex]P{\vee}Q[/latex] must be true, no matter what statement Q is.
| [latex]\,\,P\\\underline{\,\,Q\,\,\,\,\,}\\P{\wedge}Q[/latex] | [latex]\underline{P{\wedge}Q}\\P[/latex] | [latex]\underline{\,P\,\,\,\,\,\,\,\,\,}\\\,P{\vee}Q[/latex] |
An Important Note
It is important to be aware of the reasons that we study logic. There are three very significant reasons. First, the truth tables we studied tell us the exact meanings of the words such as “and,” “or,” “not,” and so on. For instance, whenever we use or read the “If…, then” construction in a mathematical context, logic tells us exactly what is meant. Second, the rules of inference provide a system in which we can produce new information (statements) from known information. Finally, logical rules such as DeMorgan’s laws help us correctly change certain statements into (potentially more useful) statements with the same meaning. Thus logic helps us understand the meanings of statements and it also produces new meaningful statements.
Logic is the glue that holds strings of statements together and pins down the exact meaning of certain key phrases such as the “If…, then” or “For all” constructions. Logic is the common language that all mathematicians use, so we must have a firm grip on it in order to write and understand mathematics.
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