An extract from the book “Language Proof and Logic – J Barwise, J Etchemendy”,
It was thought that by avoiding the ambiguities and complexities of ordinary language, we would be able to recognize the consequences of our claims more easily. This is, to a certain extent, true; but it is also true that we should be able to recognize the consequences of our claims whether or not they are expressed in FOL.
To understand the validity or soundness of an argument, one needs to understand what “logical consequence” means or when an argument is “logically valid”.
Whereas we may assume that logical has everything to do with mathematics or something unrelated to the English language, it is actually not all that true.
Firstly, before we start with any argument, let explain what an argument consist of: a conclusion, and a premises. An argument, in FOL sense, is not about two people fighting, but rather about one person trying to convince another person of some conclusion on the basis of mutually accepted premises.
Take for example the argument:
All sportsmen are physically fit. Mark Fish is a sportsman. So he must be physically fit.
Let’s analyze this argument, sentence by sentence.
All sportsmen are physically fit.
True, as men who wish to participate in any kind of sport has to be physically fit for that kind of sport.
Mark Fish is a sportsman.
Whereas Mark Fish may be a sportsman, but over the years he could have developed some medical condition, that may not completely stop him from participating anymore but could slow him down.
So he must be physically fit.
Guess, I’ve explained a different circumstance with the sentence above. Given, that he has to be physically fit to be a sportsman, a person cannot be physically all his life.
So, how would you conclude this argument? Firstly, what is the conclusion and what are the premises?
Note, a conclusion can be at the beginning or at the end of an argument. In the argument above, the conclusion is at the end of the argument.
How we figure that?
Words like hence, thus, so, and consequently are used to indicate that what follows is the conclusion of an argument.
The words because, since, after all, and the like are generally used to indicate premises.
Next, is this argument logically valid?
“All sportsmen are physically fit. Mark Fish is a sportsman.” Is true, therefore the conclusion must also be true, that “Mark Fish is physically fit.”
Therefore, our argument is logically valid and a logical consequence.
Next up, Fitch format:
Let’s look at another argument:
All rich actors are good actors. Brad Pitt is a rich actor. So he must be a good actor.
Fitch format is a special format to display arguments, and was named after the logician Frederic Fitch.
**For the benefit to understand this part, I've attached the full word document for the line drawing that is not visible here.
In Fitch format, we would display an unsound argument like this:
All rich actors are good actors.
Brad Pitt is a rich actor.
Brad Pitt is a good actor.
The sentences above the short, horizontal line are the premises, and the sentence below the line is the conclusion. The horizontal line is called the Fitch bar. You will notice the words “So … must be …” was omitted because they are not necessary in the Fitch format.
Next tutorial, Methods of proof…
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