7.5. Question 11

Now, you are given that this argument form is valid where you are given a set of n premises and (n + 1)th premises is q and the conclusion is r. Now you have to show if this is the case then the argument form where only p to p are the premises 1 n and the conclusion is q → r is also valid.

In this chunk, we discuss an argument form involving a set of premises leading to a conclusion. The argument states that if we have premises p1, p2, ..., pn along with an additional premise q, and we conclude with r, this configuration forms a valid argument. To build a similar conclusion with fewer premises, namely p1 to pn and the conclusion q → r, we need to prove this new form is also valid. This relies on the definition of valid arguments, which states that if the combination of premises leads logically to the conclusion, it remains true if represented differently (as in q → r).

Think of a relay race where each runner (premise) must complete their section before passing the baton (leading to the conclusion). If the last runner can finish the race correctly with just the baton (q), the overall team (the argument) is validated. This shows that the team can still win (be valid) even if it doesn’t have all runners (premises).

Since we are given that this argument form is valid as per the definition of valid argument, I can say that conjunction of p to p and q → r is a tautology that means it is never possible that your left hand side is true and RHS is false.

Here, we conclude that if we ascertain that the original argument is valid, we can claim that the conjunction of premises p1 through pn together with q leading to r is always true (a tautology). A tautology implies that there can never be a situation where the premises are true but the conclusion is false. It establishes a reliable logical structure, asserting that the premises guarantee the truth of the conclusion under all circumstances.

Consider a common scenario like a school policy: if students arrive on time (premises), they will be allowed to participate in activities (conclusion). It is a guaranteed scenario as the policy (tautology) ensures participation whenever the arrival condition is met. There cannot be a time when students arrive on time but are turned away.

If my LHS is true, RHS is also true. That means if the conjunction of p to p and q is true then r is also true and as a result I can say that this implication is also true.

In this section, we explore the logical connection between the left-hand side (LHS) being true and the right-hand side (RHS). If we affirm that both the conjunction of the premises and q are factual, then it leads necessarily to r being factual as well. Such a structure reinforces that the implication q → r holds true in every case derived from the initial assumptions.

Imagine a chef (premises) preparing food meticulously (p1 - p_n), followed by serving a delicious dinner (q leading to r). If the chef's preparation is acknowledged, the quality of the dinner cannot be deemed subpar. Thus, we trust the connection between careful preparation and delicious outcomes.

Because if I closely see here, what does exactly this implication means? If I say that this implication is always true then another form of the implication is that you have the premises p to p and the conclusion is q → r.

We analyze the deeper significance of the implications under discussion. This implies that if we establish the logicality of our premises leading toward the conclusion, then we recognize a conditional format (if...then) carries the same validity as our original premises and conclusion. This rearrangement articulates foundational logic principles, ensuring clarity and applicability.

Using a navigation system (premises), we know that following the route (q) leads to our destination (r). This establishes a clear conditional logic path: if following this route is true (we navigate correctly), then surely we will arrive at our destination, mirroring our argument structure.

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