5. Logically correct arguments

Logically correct arguments

Argument

An argument is a sequence of [[Proposition|propositions]].

$$\begin{array}{rlrl}& \underline{\begin{array}{l}{p}_1\\ p_2\\ ..\\ p_n\end{array}\}}& & n\text{ premises}\\ \therefore & q& & \text{one conclusion}\end{array}$$

You can read ”$p_1,p_2,...,p_n$, therefore $q$”.

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Logically correct

An argument is logically correct if in every situation that makes all premises true, the conclusion is true as well.

$$\boxed{\begin{array}{r}\underline{\begin{array}{l}{p}_1\\ p_2\\ ..\\ p_n\end{array}}\\ q\end{array}}\text{ is logically correct}⟺\text{the formula }(p_1\wedge p_2\wedge p_3\wedge ..\wedge p_n)\to q\text{ is a tautology}$$ Link to original

Situation

A situation is a row in the truth table of $p_1, p_2, .., p_n, q$ or otherwise a possible assignment for the propositional variables in $p_1, p_2, .., p_n, q$.

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Examples

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Rules of inference

Inference rule

An inference rule is a representation of a valid step in a formal proof, indicating the conditions that must be satisfied before a conclusion can be obtained in the proof.

A rule will be presented in the form: $R:\frac{A_1,A_2,...,A_n}{B}$

$A_1,A_2,...,A_n$ are the premises or conditions of the rule, and $B$ is the conclusion.

Informally, $R:$ “if $A_1,A_2,...,A_n$ are true, then $B$ is also true.”

Alternatively, we use the syntax:

$$\begin{array}{c}{A}_1\\ A_2\\ ⋮\\ A_n\\ B\end{array}$$Link to original

We can always use a truth table to decide whether an argument is correct. This can be very tedious.

If we consider $10$ different variables, the truth table requires $2^{10}=1024$ rows.

Instead, we can establish the correctness of simple argument forms using rules of inference, and then use them to construct more complicated correct arguments.

For example, Modus ponens (the rule of detachment).

$$\begin{array}{llc}& p\to q& \text{If it snows today, then we will go skiing.}\\ & p& \text{It snows today.}\\ & & \\ \therefore & q& \text{We will go skiing.}\end{array}$$

Examples

Should practice trying to justify these!
There will be no exam Q in this module for building these, no need to memorise.

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