Add logical argument, premise, conclusion and rule of inference

_proof-writing.qmd

@@ -177,6 +177,30 @@ Let's save $\rightarrow$ (`\rightarrow`) for [convergence](algebra.qmd#def-conve
 
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+{{< include mds/_subfiles/proof-writing/_def-logical-argument.qmd >}}
+
+{{< slidebreak >}}
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+{{< include mds/_subfiles/proof-writing/_def-premise.qmd >}}
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+{{< include mds/_subfiles/proof-writing/_def-conclusion.qmd >}}
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+{{< slidebreak >}}
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+{{< include mds/_subfiles/proof-writing/_exm-logical-argument.qmd >}}
+
+{{< slidebreak >}}
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+{{< include mds/_subfiles/proof-writing/_def-rule-of-inference.qmd >}}
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+{{< slidebreak >}}
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+{{< include mds/_subfiles/proof-writing/_exm-rule-of-inference.qmd >}}
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_subfiles/proof-writing/_def-conclusion.qmd

@@ -0,0 +1,8 @@
+:::{#def-conclusion}
+#### Conclusion
+
+The **conclusion** of a [logical argument](#def-logical-argument)
+$P_1, \ldots, P_n \therefore Q$
+is the [proposition](#def-proposition) $Q$.
+
+:::

_subfiles/proof-writing/_def-logical-argument.qmd

@@ -0,0 +1,9 @@
+:::{#def-logical-argument}
+#### Logical argument
+
+A **logical argument** is a list of [propositions](#def-proposition)
+$P_1, \ldots, P_n$
+together with one more [proposition](#def-proposition) $Q$.
+It is written with "so" or $\therefore$ before $Q$.
+
+:::

_subfiles/proof-writing/_def-premise.qmd

@@ -0,0 +1,8 @@
+:::{#def-premise}
+#### Premise
+
+The **premises** of a [logical argument](#def-logical-argument)
+$P_1, \ldots, P_n \therefore Q$
+are the [propositions](#def-proposition) $P_1, \ldots, P_n$.
+
+:::

_subfiles/proof-writing/_def-rule-of-inference.qmd

@@ -0,0 +1,14 @@
+:::{#def-rule-of-inference}
+#### Rule of inference
+
+A **rule of inference** is a pattern of [logical argument](#def-logical-argument),
+written with [propositions](#def-proposition) $P, Q, \ldots$ as placeholders,
+whose [premises](#def-premise) [logically entail](#def-logical-entailment) its [conclusion](#def-conclusion)
+whatever propositions are put in place of $P, Q, \ldots$.
+It is written with its premises above a line and its conclusion below:
+
+$$
+\frac{P_1, \ldots, P_n}{Q}.
+$$
+
+:::

_subfiles/proof-writing/_exm-logical-argument.qmd

@@ -0,0 +1,14 @@
+:::{#exm-logical-argument}
+#### A logical argument
+<!-- no-figure: three short sentences; nothing to plot -->
+
+"If $n$ is [even](number-systems.qmd#def-even-odd), then $n^2$ is even.
+$n$ is even.
+So $n^2$ is even."
+is a [logical argument](#def-logical-argument).
+Its [premises](#def-premise) are
+"if $n$ is even, then $n^2$ is even"
+and "$n$ is even".
+Its [conclusion](#def-conclusion) is "$n^2$ is even".
+
+:::

_subfiles/proof-writing/_exm-rule-of-inference.qmd

@@ -0,0 +1,20 @@
+:::{#exm-rule-of-inference}
+#### A pattern that is not a rule of inference
+<!-- no-figure: the truth table row is the display -->
+
+The pattern
+
+$$
+\frac{P \Rightarrow Q, \quad Q}{P}
+$$
+
+is not a [rule of inference](#def-rule-of-inference).
+In the row of the [truth table](#def-truth-table) where $P$ is false and $Q$ is true,
+both [premises](#def-premise) are true,
+by the definition of [implies](#def-implies),
+but the [conclusion](#def-conclusion) $P$ is false.
+For example, "if $n = 2$, then $n$ is [even](number-systems.qmd#def-even-odd)" and "$n$ is even"
+are both true for $n = 4$,
+but "$n = 2$" is false.
+
+:::