Module: Sentential logic
Quote of the page
The heart of man is made to reconcile the most glaring contradictions.
- David Hume
Consider this particular argument :
[Premise 1] The pollution index is high.
[Premise 2] If the pollution index is high, we should stay indoors.
[Conclusion] We should stay indoors.
This argument is of course valid, as it is an instance of modus ponens. To use the methods of SL to show that it is indeed valid, we need to translate it from English into the language of SL. This process of translation is called formalization.
First of all we need to find sentence letters to translate the different parts of the argument. Let us use the following translation scheme. A translation scheme in SL is simply a pairing of sentence letters of SL with statements in natural language. In carrying out formalization you should always write down the translation scheme first.
Translation scheme :
P : The pollution index is high.
Q : We should stay indoors.
Remember that → is used to translate if ... then ___. So using the above translation scheme we can formalize the argument as follows :
Premise #1 : P
Premise #2 : (P→Q)
Conclusion : Q
In SL we can rewrite this argument as a one line sequent, with the premises separated by comma :
P, (P→Q) ⊧ Q
There are a few things to bear in mind regarding formalization. First, we usually try to discern as much structure as we need in the original sentences. For example, consider this argument :
Lychees are sweet and lemons are sour. Lychees are sweet.
To show that the argument is valid, we need to formalize the premise as (L&W) rather than just L. Whereas we can just use L to formalize both the premise and the conclusion in the following argument :
Lychees are sweet and lemons are sour. So, lychees are sweet and lemons are sour.
Another point to remember is that in formalization we are in effect translating from a natural language into an artificial language. It is often not possible to find a translation that has exactly the same meaning as the original sentence. In such a case we should aim to find a wff that is closest in meaning, or which is logically equivalent. Take this valid argument for instance :
Cinta will grow up whatever her parents think. But when Cinta grows up she will argue with her parents. So Cinta will argue with her parents.
This argument can be formalized as a modus ponens argument :
Translation scheme :
C : Cinta will grow up.
A : Cinta will argue with her parents.
C, (C→A) ⊧ A
However, notice the following features about our translation :
This argument is valid : If there is a stock market crash tomorrow Paul will be poor. There is a stock market crash tomorrow. So Paul will be poor. But if we change the conclusion to Paul is poor it will no longer be valid, and it would be a mistake to use the same sentence letter to translate both Paul is poor and Paul will be poor.
So far we have said that the connectives can be used to translate their natural language counterparts :
|~||It is not the case that|
|→||if ... then ___|
|↔||if and only if|
But in fact many other locutions can be translated using these five sentential connectives :
Suppose P translates the sentence Santa exists. Then ~P can be used to translate these sentences :
Translate these sentences into SL, preserving as much structure as possible. Provide the translation scheme in each case. Use the negation sign wherever appropriate.
Formalize these two statements in sentential logic and use truth-tables to check whether they are logically equivalent:
For more exercises, please go through Exercise 1.3 at http://logic.tamu.edu/cgi-bin/quizmaster