Hidden Premises

Can we prove that the following argument is valid?

1. Ottawa is north of Toronto
2. Toronto is north of Waterloo
3. Therefore, Ottawa is north of Waterloo

No, we're missing the obvious statement that the predicate "northOf" is transitive:

Now we can show that the argument is valid.

An enthymeme is an argument that contains a hidden premise (i.e an unstated premise that is obviously true)

• Billy the kid was in the jail at 6pm.
• Therefore, Billy the kid did not commit the crime at the general store at 6pm

Formalize these sentences and prove that it is a valid argument.

solution

• The crime was committed by someone at the general store at 6pm
• Billy the kid was in the jail at 6pm
• Therefore, Billy the kid did not commit the crime at the general store at 6pm

where:

• means committed the crime at at time
• means was at location at time
• is the constant Billy
• is the constant jail
• is the constant general store

There are 3 hidden premises:

1. If something commits a crime at a location at a time, then they are at that location at that timewhich relates crime and location in all interpretations
2. Something cannot be at two locations at the same timewhich limits the interpretations of the location predicate
3. The general store is not the jail

All together,

proof