(related to Part: Methods of Mathematical Proving)
Following the notation of rules of inference, logical arguments can be written using a schema, in which all propositions are listed above a line, below which the conclusion is formulated.
$$\begin{array}{rl} \text{All humans are mortal.}&p_1\\ \text{Socrates is human.}&p_2\\ \hline \text{Socrates is mortal.}&q\\ \end{array}$$
This is a valid argument since the conclusion $q$ is true whenever the premises $p_1$ and $p_2$ are both true.
$$\begin{array}{rl} n+m+k=4&p_1\\ n=1&p_2\\ m=-1&p_3\\ \hline k=4&q\\ \end{array}$$
This is a valid argument since the conclusion $q$ is true whenever the premises $p_1,p_2,$ and $p_3$ are all true.
$$\begin{array}{rl} \text{It is raining.}&p\\ \hline \text{It is raining.}&q\\ \end{array}$$
This is a valid argument since the conclusion $q$ is true whenever the premise $p$ is true.
$$\begin{array}{rl} \text{If Bob asks the question, then he will receive the answer.}&p_1\\ \text{Bob receives the answer.}&p_2\\ \hline \text{Therefore, Bob have asked the question.}&q\\ \end{array}$$
This is a fallacy since the conclusion $q$ is can be false, even though the premises $p_1$ $p_2$ are true (Bob could have received the answer, even though he has never asked the question).
$$\begin{array}{rl} \text{Adam always thinks what he says.}&p_1\\ \text{Adam does not say anything.}&p_2\\ \hline \text{Therefore, Adam does not think it.}&q\\ \end{array}$$
This is a fallacy since the conclusion $q$ can be false, even though the premises $p_1$ $p_2$ are true (Adam could think it, even though he does not say anything).