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Law of Syllogism
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Rule
Law of Syllogism
Given three statements, p, q, and r, the following compound statement is always true.
[(p → q) ∧ (q → r)] →(p → r)
This can also be written as follows.
If p → q and q → r are both true, then p→ r is also true.
The Law of Syllogism is like the Transitive Property of Equality, but for logical statements.
Proof
This law can be proven by making a truth table. The goal is to see that the compound statement is always true.
[( p → q) ∧ ( q → r)] → (p → r)
The proof consists of three steps.
1
Determine the Number of Columns
expand_more To determine the number of columns in the truth table, the compound statement can be 
broken downinto simpler statements until the simplest statements p, q, and r are obtained.
The columns are formed by ordering each unique step of simplification from the simplest to the most complex. The last column of a truth table is always the original statement.
| p | q | r | p→ q | q → r | p → r | (p → q) ∧ (q → r) | [(p → q) ∧ (q → r)] → (p→ r) |
|---|---|---|---|---|---|---|---|
In this case, the truth table has eight columns.
2
Determine the Number of Rows
expand_more A truth table has as many rows as there are combinations of truth values of the variables. In this case, there are three simplest statements, p, q, and r.
Simplest Statements p, q, andr
Each of these statements can be either true (T) or false (F). There are eight possible combinations for these truth values, so the table will have eight rows.
| p | q | r | p→ q | q → r | p → r | (p → q) ∧ (q → r) | [(p → q) ∧ (q → r)] → (p→ r) |
|---|---|---|---|---|---|---|---|
| T | T | T | |||||
| T | T | F | |||||
| T | F | T | |||||
| T | F | F | |||||
| F | T | T | |||||
| F | T | F | |||||
| F | F | T | |||||
| F | F | F |
3
Fill in the Cells
expand_more The last step in making a truth table is to fill in the remaining empty cells, column by column. The first columns that will be filled in are the conditional statements p→ q, q → r, and p → r. A conditional statement is false only when the hypothesis is true and the conclusion false.
| p | q | r | p→ q | q → r | p → r | (p → q) ∧ (q → r) | [(p → q) ∧ (q → r)] → (p→ r) |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | ||
| T | T | F | T | F | F | ||
| T | F | T | F | T | T | ||
| T | F | F | F | F | F | ||
| F | T | T | T | T | T | ||
| F | T | F | T | F | T | ||
| F | F | T | T | T | T | ||
| F | F | F | T | T | T |
The column containing the conjunction (p→ q) ∧ (p → r) is next to fill. A conjunction is false unless both statements are true.
| p | q | r | p→ q | q → r | p → r | (p → q) ∧ (q → r) | [(p → q) ∧ (q → r)] → (p→ r) |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | |
| T | T | F | T | F | F | F | |
| T | F | T | F | T | T | F | |
| T | F | F | F | F | F | F | |
| F | T | T | T | T | T | T | |
| F | T | F | T | F | T | F | |
| F | F | T | T | T | T | T | |
| F | F | F | T | T | T | T |
The last column is another conditional statement, which is only false only when the hypothesis is true and the conclusion false.
| p | q | r | p→ q | q → r | p → r | (p → q) ∧ (q → r) | [(p → q) ∧ (q → r)] → (p→ r) |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T |
| T | T | F | T | F | F | F | T |
| T | F | T | F | T | T | F | T |
| T | F | F | F | F | F | F | T |
| F | T | T | T | T | T | T | T |
| F | T | F | T | F | T | F | T |
| F | F | T | T | T | T | T | T |
| F | F | F | T | T | T | T | T |
As the truth table describes, the compound statement is always true. Therefore, the law is proven.
Extra
ExampleThe Law of Syllogism is usually better understood with an example. Suppose the following two statements are true.
- If it is Wednesday, there is meatloaf for dinner.
- If there is meatloaf for dinner, there is pudding for dessert.
According to the Law of Syllogism, a third true statement can be derived.
- If it is Wednesday, there is pudding for dessert.
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