Logical Inferencing

We continue our discussion of logical forms by exploring logical inferencing. Recall, that one of the main goals of studying logic is to be able to derive new statements from existing statements. This process is known as logical inferencing. We can assess the validity of arguments using various rules of inference.

Arguments and Validity

An argument is a sequence of propositions, where the last proposition is the conclusion. An argument form is a sequence of logical forms, called premises, where the last form is the conclusion. An argument can be valid or invalid. Please note, that the validity of an argument does not depend on the propositions involved, but rather on the strucutre of the argument itself.

We can use truth tables and rules of inference to decude the validy of an argument.

Truth Tables for Validity

The first step in deducing the validy oif an argument is determing which propostionss are premises and conclusions. Once we have identified the premises and conclusion, we can construct a truth table that includes all the premises and the conclusion. An argument is valid if, in every row of the truth table where all the premises are true, the conclusion is also true. These rows are called critical rows. All critical rows must have a true conclusion for the argument to be valid.

Rules of Inference

A 8rule of inference* is a logical form that describes how new statements can be derived from existing statements using valid logical steps.

Generalization

A common rule of inference is Generalization. It states that if we have a statement $p$ that is true, then we can conclude that $p \lor q$ is also true, for any proposition $q$. The form of Generalization is as follows: $p$ therefore $p \lor q$. We can also inference $q$ therfore $p \lor q$. The inclusive or allows us to infer the disjunction from either of its components, as the statement will be true as long as at least one of the components is true.

Specialization

Another common rule of inference is Specialization. It states that if we have a statement $p \land q$ that is true, then we can conclude that $p$ is also true, and we can also conclude that $q$ is also true. The form of Specialization is as follows: $p \land q$ therefore $p$, and $p \land q$ therefore $q$. This rule allows us to infer the individual components of a conjunction from the conjunction itself. As we know that for a conjunction to be true, both components must be true.

Elimination

If have a disjunction $p \lor q$ and we know that one of the components is false, we can conclude that the other component must be true. This rule is known as Elimination. The form of Elimination is as follows: $p \lor q$, not $p$ therefore $q$, and $p \lor q$, not $q$ therefore $p$. This rule allows us to infer the truth of one component of a disjunction when we know that the other component is false, as for the disjunction to be true, at least one of the components must be true.

Transposition

If $p \to q$ and $q \to r$, we can conclude that $p \to r$. This rule is known as Transposition. The form of Transposition is as follows: If $p$ then $q$, if $q$ then $r$ therefore if $p$ then $r$. This rule allows us to chain conditionals together to derive new conditionals infinitely.

Division by Cases

If we know that $p \lor q$ is true, and we also know that if $p$ is true then $r$ is true, and if $q$ is true then $r$ is true, we can conclude that $r$ is true. This rule is known as Division by Cases. The form of Division by Cases is as follows: $p \lor q$, if $p$ then $r$, if $q$ then $r$ therefore $r$. This rule allows us to infer the truth of a statement based on the truth of a disjunction and the implications of its components. As we know that either $p$ or $q$ is true, and both lead to $r$ being true, we can conclude that $r$ must be true.

Modus Ponens

One of the most common rules of inference is Modus Ponens. It states that if we have a conditional statement $p \to q$ and we know that $p$ is true, then we can conclude that $q$ is also true. The form of Modus Ponens is as follows: If $p$ then $q$, $p$ therefore $q$.

In this truth table, we can see that in the first row, where both $p$ and $q$ are true, the conclusion is also true. In the second row, where $p$ is true and $q$ is false, the conclusion is false. In the third and fourth rows, where $p$ is false, the conclusion is not applicable. Therefore, Modus Ponens is a valid rule of inference.

Modus Tollens

Another important rule of inference is Modus Tollens. It states that if we have a conditional statement $p \to q$ and we know that $q$ is false, then we can conclude that $p$ is also false. The form of Modus Tollens is as follows: If $p$ then $q$, not $q$ therefore not $p$. This follows from the contrapositive of the conditional statement.

Contradiction Rules

Another useful rule of inference is the Contradiction Rule. If we can show that $p$ is false logically leads to a contradiction $c$, we can conclude that $p$ must be true. The form of the Contradiction Rule is as follows: Not $p$ leads to contradiction $c$ therefore $p$. This rule allows us to infer the truth of a statement by demonstrating that its negation leads to an inconsistency.

Beware of Fallacy

Although rules of inference are powerful tools for deducing new statements, it is important to be aware of common logical fallacies that can lead to invalid conclusions, even when using the rules. Ensure that the premises you begin with are well defined.

The Converse Error

One very common fallacy is the Converse Error. This occurs when we mistakenly assume that if $p \to q$ is true, then $q \to p$ must also be true. However, this is not necessarily the case. The truth of the converse depends on the specific propositions involved and cannot be assumed based solely on the original conditional statement.

The converse is only true if you have a biconditional statement, $p \leftrightarrow q$, which states that both $p \to q$ and $q \to p$ are true. This does not hold for simple implications.

The Inverse Error

Another common fallacy is the Inverse Error. This occurs when we mistakenly assume that if $p \to q$ is true, then $\neg p \to \neg q$ must also be true. However, this is not necessarily the case. The truth of the inverse depends on the specific propositions involved and cannot be assumed based solely on the original conditional statement.