6.5. Question 4

The first statement is the user has paid the subscription fee but does not enter a valid password. So you might be wondering what will be the logical connective operator for representing ‘but’. So the but here should be treated as some form of and or conjunction here. That means, you can equivalently pass the statement saying that user has paid the subscription fee and he has not entered a valid password. And I know that the variable r represents the statement user has paid the subscription fee. The variable p represent the user enters a valid password, but I want to represent here, denote here, state here that he has not entered a valid password, that is why it will be ¬ p.

This chunk discusses how to represent the statement about a user who has paid the subscription fee but has not entered a valid password in logical terms. The word 'but' is often used to highlight a contradiction or contrast; however, in logical representation, it functions similarly to 'and'. Here, 'r' denotes the action of paying the subscription, and '¬p' represents not entering a valid password, making the entire proposition two-sided: the user has done one action while negating another.

Imagine a situation where a friend tells you they have a gym membership (they've paid), but they're not exercising (not entering a valid password). In this case, we would depict their commitment to the gym (paying) while highlighting their inaction (not exercising) through logical symbols, similar to how 'but' shows contrast in everyday language.

The second statement is access is granted whenever the user has paid the subscription fee and enters a valid password. So let us parse first this statement. So remember ‘whenever’- is a form of Implication. That means this part is going to be a premise and this is the conclusion. So that means if this is a form of if-then, so whatever is there after whenever that is the if-part, if that thing is ensured then whatever is there before whenever that is the conclusion, that you can conclude. Now in the premise here a conjunction is involved because my premise consists of conjunction of two things.

This chunk breaks down the logical representation of the statement about granting access. The term 'whenever' signals a conditional or implication (if-then format), where the conditions are twofold: both the payment of the subscription fee and the entry of a valid password must happen for access to be granted. The logical form represents these conditions as a conjunction since both conditions must be satisfied for the conclusion to hold true.

Think of a locked building where admittance requires both a key card (representing the subscription fee) and a password (valid password). Only when both conditions are met can the door unlock. This is akin to how the logical representation requires both 'p' (valid password entry) and 'r' (paid subscription) to collectively lead to 'q' (access granted).

The last statement is the user has not entered a valid password but has paid the subscription fee then access is granted. So this is a form of if-then statement. Whatever is there after then that is the conclusion whatever is there before then that is a premise, but in the premise you again have an occurrence of ‘but’ and remember but is nothing but conjunction.

Here, the statement indicates a scenario where the user has not entered a valid password yet has paid. The use of 'but' again indicates a conjunction, emphasizing that despite not fulfilling one condition (valid password), the user’s payment allows access. This sets up a logical implication where the premises must be evaluated to determine if the conclusion can still logically follow.

Consider a situation where a customer pays for a service but forgets their account login details. They may still receive a temporary guest access (granted access) based on their payment despite not fulfilling the standard login requirement. This nuance mirrors the way the logical statement functions, presenting a concession but reaching a conclusion nonetheless.

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