Read this tutorial on formalization, which means turning statements and arguments in ordinary language into their symbolic counterparts; we might just as well call it translation. Notice that ordinary language contains hint words, letting us know when we are likely to need one of the logical connectives.
Complete the exercises for this tutorial and check your answers.
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 valid, since it is an instance of modus ponens. To use the methods of sentential logic (SL) to show it is indeed valid, we need to translate it from English into the language of sentential logic (SL). This process of translation is called formalization.
First, 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 sentential logic (SL) is simply a pairing of sentence letters of sentential logic (SL) with statements in natural language. In carrying out formalization you should always write down the translation scheme first.
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)
In sentential logic (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:
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:
Take this valid argument for example:
This argument can be formalized as a modus ponens argument:
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|
Suppose “P” translates the sentence “Santa exists.” Then “~P” can be used to translate these sentences:
(P&Q) can be used to translate:
(P ∨ Q)
Source: Joe Lau and Jonathan Chan, https://philosophy.hku.hk/think/sl/formal1.php
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