MODULE: Sentential logic

TUTORIAL SL06: Formalization in SL

SL06.1 An example

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

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SL06.2 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 :

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”.

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 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.

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SL06.3 Translating natural language connectives

So far we have said that the connectives can be used to translate their natural language counterparts as follows :

~	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 “God exists”. Then “~P” can be used to translate these sentences :

God does not exist. It is not the case that God exists. It is false that God exists.

Conjunction

“(P&Q)” can be used to translate the following :

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

Disjunction

“(PQ)” can be used to translate the following :

P or Q. Either P or Q. P unless Q. [Comments] Unless Q, P.

Conditional

“(P→Q)” can be used to translate the following :

If P then Q. P only if Q. [Comments] Q if P. Whenever P, Q. Q provided that P. P is sufficient for Q. Q is necessary for P.

Biconditional

“(P↔Q)” can be used to translate the following :

P if and only if Q. P iff (if and only if) Q. P when and only when Q. P is equivalent to Q. P is both necessary and sufficient for Q.

SL06.4 Exercises

? Question 1 - Translate these sentences into SL, preserving as much structure as possible. Provide the translation scheme in each case. Use the negation sign wherever appropriate.

1. The school has collapsed, but it is not true that five students died. [Show answer]
   S : The school has collapsed. F : Five students died.

   (S&~F)

2. This footprint comes from a man or a woman. [Show answer]
   M : This footprint comes from a man.
   W : This footprint comes from a woman.

   (M∨W)

3. It is not the case that May is in Oxford or Hong Kong. [Show answer]
   O : May is in Oxford
   H : May is in Hong Kong.

   ~(O∨H)

4. If you want to go, then I shall go with you if it is sunny. [Show answer]
   G : You want to go.
   I : I shall go with you
   S : It is sunny.

   (G→(S→I))

5. If the weather is bad or I am sick, then I will not go with you. [Show answer]
   B : The weather is bad.
   S : I am sick
   G : I will go with you.

   ((B∨S)→~G)

6. Whether Peter is coming or not, Mary is not going to come. [Show answer]
   P : Peter is coming.
   M : Mary is going to come.

   You can formalize the sentence as either "((P→~M)&(~P→~M))" or simply "~M". [But why not "((P→~M)∨(~P→~M))"?]

7. Unless you try to improve yourself, and unless you improve your attitude, you are not going to succeed. [Show answer]
   Y : You try to improve yourself.
   A : You improve your attitude.
   S : You are going to succeed.

   ((Y&A)∨~S). Note that this is equivalent to : (S→(Y&A)).

8. I will go to the concert only if Christina and Ricky do not sing together. [Show answer]
   G : I will go to the concert.
   C : Christina and Ricky sing together.

   (G→~C)

9. A man and a woman ate the whole cake. [Show answer]
   Use a single sentence letter to formalize the whole statement. Note that the statement is NOT equivalent to "A man ate the whole cake and a woman ate the whole cake." Why?
10. Jane and Matthew are fond of each other. [Show answer]
    You may use a single sentence letter, but sometimes it might be better to analyse the sentence as a conjunction - "Jane is fond of Matthew and Matthew is fond of Jane."

? Question 2 - This is an old final exam question - Formalize these two statements in sentential logic and use truth-tables to check whether they are logically equivalent:

(a) If there are intelligent aliens, then either they are already here, or the earth is not an interesting place to visit.

(b) If the earth is an interesting place to visit, then intelligent aliens are already here if they exist.

[Show answer]

Translation scheme:

A : Intelligent aliens exist.
E : Intelligent aliens are already on the earth.
I : The earth is an interesting place to visit.

Statements (a) and (b) can then be formalized as (A→(Ev~I)) and (I→(A→E)) respectively.

A

E

I

(A

→

(E

∨

~

I))

(I

→

(A

→

E))

T

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Since for any assignment of truth-values, (A→(Ev~I)) and (I→(A→E)) have the same truth-value under that assignment, the two statements are logically equivalent.

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