PHIL102: Introduction to Critical Thinking and Logic

§1. An Example

Consider this particular argument:

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.

Translation scheme:

Remember that “→” is used to translate “if... then ___.”

So using the above translation scheme we can formalize the argument as follows:

In sentential logic (SL) we can rewrite this argument as a one line sequent, with the premises separated by comma:

§2. Things to Bear in Mind

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:

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 well-formed formula (WFF) that is closest in meaning, or which is logically equivalent.

Take this valid argument for example:

This argument can be formalized as a modus ponens argument:

Translation scheme:

However, notice the following features about our translation:

1. We are ignoring the phrase “whatever her parents think.” This is legitimate as the shorter version should be equivalent to the original one.
2. We are ignoring the word “but” in the second premise. The word adds contrast and emphasis in natural language, but it is irrelevant to the validity of the argument.
3. We are ignoring the difference in tense in ”Cinta will grow up” in the first premise, and “Cinta grows up” in the second premise. There are no symbols in sentential logic (SL) to indicate tense, but here ignoring tense is acceptable because it does not affect the assessment of validity. But sometimes tense cannot be ignored.

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.” 

§3. Translating Natural Language Connectives

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
&	and
∨	or

But in fact many other locutions can be translated using these five sentential connectives:

• Negation

Suppose “P” translates the sentence “Santa exists.” Then “~P” can be used to translate these sentences:

• Santa does not exist.
• It is not the case that Santa exists.
• It is false that Santa exists.

• Conjunction
  

(P&Q) can be used to translate:

• P and Q.
• P but Q.
• Although P, Q.
• P, also Q.
• P as well as Q.

• Disjunction
  

(P ∨ Q)

• P or Q.
• Either P or Q.
• P unless Q.
• Unless Q, P.

• Conditional
  

(P→Q)

• If P then Q.
• P only if Q.
• Q if P.
• Whenever P, Q.
• Q provided that P.
• P is sufficient for Q.
• Q is necessary for P.

• Biconditional
  

(P↔Q)

• P if and only if Q.
• P if (if and only if) Q.
• P when and only when Q.
• P is equivalent to Q.
• P is both necessaryand sufficient for Q.

Exercise #1

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Source: Joe Lau and Jonathan Chan, https://philosophy.hku.hk/think/sl/formal1.php
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