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\)
Section 5.2 Standard arguments
Subsection 5.2.1 Modus ponens
- modus ponens
-
standard argument with form
| \(p \lgccond q\) |
| \(p\) |
| \(q\) |
Worked Example 5.2.1.
Verify the validity of the modus ponens standard argument.
Solution.
Verify the validity by ensuring that each row in the truth table with premises all true also has the conclusion true.
| (pr) |
(c) |
(pr) |
| \(p\) |
\(q\) |
\(p \lgccond q\) |
| \(\lgctrue\) |
\(\lgctrue\) |
\(\lgctrue\) |
\(\correct\) argument is valid |
| \(\lgctrue\) |
\(\lgcfalse\) |
\(\lgcfalse\) |
| \(\lgcfalse\) |
\(\lgctrue\) |
\(\ast\) |
| \(\lgcfalse\) |
\(\lgcfalse\) |
\(\ast\) |
Example 5.2.2.
The argument in
Example 5.1.2 has modus ponens form. So it is valid, even though the first premise and the conclusion are not actually true.
Subsection 5.2.2 Modus tollens
- modus tollens
-
standard argument with form
| \(p \lgccond q\) |
| \(\lgcnot q\) |
| \(\lgcnot p\) |
Worked Example 5.2.3.
Verify the validity of the modus tollens standard argument.
Solution.
Verify the validity by ensuring that each row in the truth table with premises all true also has the conclusion true.
|
|
(pr) |
(pr) |
(c) |
| \(p\) |
\(q\) |
\(p \lgccond q\) |
\(\lgcnot q\) |
\(\lgcnot p\) |
| \(\lgctrue\) |
\(\lgctrue\) |
\(\lgctrue\) |
\(\lgcfalse\) |
\(\ast\) |
| \(\lgctrue\) |
\(\lgcfalse\) |
\(\lgcfalse\) |
\(\ast\) |
\(\ast\) |
| \(\lgcfalse\) |
\(\lgctrue\) |
\(\lgctrue\) |
\(\lgcfalse\) |
\(\ast\) |
| \(\lgcfalse\) |
\(\lgcfalse\) |
\(\lgctrue\) |
\(\lgctrue\) |
\(\lgctrue\) |
\(\correct\) argument is valid |
Example 5.2.4.
Subsection 5.2.3 Law of Syllogism
- Law of Syllogism
-
standard argument with form
| \(p \lgccond q\) |
| \(q \lgccond r\) |
| \(p \lgccond r\) |
The Law of Syllogism may be extended to chains of conditionals of arbitrary (finite) length.
- Extended Law of Syllogism
-
standard argument with form
| \(p_1 \lgccond p_2\) |
| \(p_2 \lgccond p_3\) |
| \(\vdots \phantom{\lgccond p_n}\) |
| \(p_{n-1} \lgccond p_n\) |
| \(p_1 \lgccond p_n\) |
Example 5.2.7. A syllogistic argument in English.
| If I don’t study hard this term, I won’t master the course material. |
| If I don’t master the course material, I will fail the course. |
| If I fail the course, I will have to take it again next year. |
| If I take it again next year, I will have to study harder. |
| Therefore, if I don’t study hard this term, I will have to study harder next year. |
