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### Skills to be acquired in this tutorial:

Symbolizing compound propositions. Learning about logical connectives, and the notion of the main connective. Recognizing different constructions in English which have the same underlying logical form. Paraphrasing the English into a standard form.

### Why this is useful:

It is the next step in learning how to symbolize. Main connectives are very important-- they are central to symbolization, they are central to the semantics, and they are central to derivations.

### Tutorial:

Not all propositions are atomic propositions. Consider the proposition asserted by 'It is not the case that the United States has a female President'. This is a true proposition, yet it is not an atomic one. It is made up of the atomic proposition 'the United States has a female President' (which is false) and negation (expressed by 'It is not the case that...'), and the resulting compound proposition, which is the negation of a false proposition, is true.

There are several types of compound proposition.

Negation is one. To symbolize negation, the symbol '∼' is used to express the 'It is not the case that ...' and then the remaining simpler proposition is symbolized in the standard way. For example, to symbolize 'It is not the case that forests are widespread.' first symbolize the 'It is not the case that ... to get '∼(FORESTS ARE WIDESPREAD)' and then symbolize the 'Forests are widespread' (which is F, under the conventions of the exercises) to get ∼(F) as the symbolization.

Notice that some brackets appeared around the F -- brackets or parantheses are used to avoid ambiguity. They are actually not needed in this case, for there is no ambiguity. However, it is better to use brackets all the time to start with until you learn what is ambiguous and what is not. More remarks will appear on this as we go along.

Conjunction is another type of compound proposition. The proposition asserted by 'Forests are widespread and grass is plentiful. ' is a compound proposition composed of the two atomic propositions 'Forests are widespread' and 'Grass is plentiful' . To symbolize conjunction, the symbol '∧' is used to express the '...and ...' and then the remaining two simpler propositions are symbolized in the standard way. With the example, the result is (F∧G).

The symbols '∼' and '∧' are examples of logical connectives. Each compound proposition has a main connective which links up its immediate components. For example,the main connective of (A∧∼(B)) is '∧', and it connects up A and ∼(B); and in turn the compound formula ∼(B) has '∼' as its main connective. Atomic propositions do not have main connectives (they are atomic and have no parts that need connecting up).

If you can tell whether an English sentence expresses a compound proposition and can recognize what that proposition's main connective is, then you have the skills to symbolize any proposition. (All you do is use your skills on the sentence as a whole, perhaps dividing it up into a main connective, and some parts; then use your skills on the parts; and keep doing this until there are no parts left).

Equally, if you can recognize the main connective in a logical formula, and can translate it into English, you should be able to translate any symbolic formula back to English by repeated use of a similar 'divide-and-conquer' tactic.

There are a few more types of compound proposition.

Disjunction is one. The proposition asserted by 'Forests are widespread or grass is plentiful. ' is a compound proposition composed of the two atomic propositions 'Forests are widespread' and 'Grass is plentiful' . To symbolize disjunction, the symbol '∨' is used to express the '...or ...' and then the remaining two simpler propositions are symbolized in the standard way. With the example, the result is (F∨G).

Conditional is another. The proposition asserted by 'If forests are widespread then grass is plentiful.' is a compound proposition composed of the two atomic propositions 'Forests are widespread' and 'Grass is plentiful' . To symbolize a conditional, the symbol '⊃' is used to express the 'If ...then ...', it is placed between the two propositions, and then the remaining two simpler propositions are symbolized in the standard way. With the example, the result is (F⊃G).

And biconditional is the final one. The proposition asserted by 'Forests are widespread if and only if grass is plentiful. ' is a compound proposition composed of the two atomic propositions 'Forests are widespread' and 'Grass is plentiful' . To symbolize a biconditional, the symbol '≡' is used to express the ' ...if and only if ...' , and then the remaining two simpler propositions are symbolized in the standard way. With the example, the result is (F≡G).

The symbols '∨ , '⊃' and '≡' are further examples of logical connectives.

In English there is usually more than one way to say the same thing. For example, the sentences 'Forests are widespread or grass is plentiful.' and 'Either forests are widespread or grass is plentiful.' assert the same compound proposition-- the new word 'either' at the beginning of the second sentence does not alter the underlying logical structure. Both these sentences should be symbolized to (F∨G).

One symbolic formula can represent the logical structure of a proposition asserted by several different English sentences (this is one reason why we symbolize).

You can discover the common cases for yourself in the Exercise (and there are some further remarks after the Exercise).

Of course, when we translate a symbolization back into English we might not get exactly the same English sentence that we started with-- but we will get an English sentence which accurately depicts the underlying logical structure of the starting sentence. For example, 'Either forests are widespread or grass is plentiful. ' should be symbolized (F ∨ G), and if (F ∨ G) is translated back into English we get 'Forests are widespread or grass is plentiful.' which is not what we started with (the 'either' is missing) but the retranslation conveys the entire logical force of the original.

## Exercises to accompany Tutorial 2

### Exercise 1 (of 7):

### Exercise 2 (of 7):

### Help Video:

*[This is a Video, click the Play button to view it..] *

### Exercise 3 (of 7)

What are the main connectives of the following formulas?

(The answers are below).

a) (A∧ (B∧C))

b) (~(A) ∧ B)

c) ∼(A∧B)

Answers

a) The main connective of (A∧ (B∧C)) is the '∧' which occurs between A and (B∧C).

b) The main connective of (∼(A) ∧ B) is the '∧' which occurs between ∼(A) and B.

c) The main connective of ∼(A∧B) is the '∼' which occurs at the beginning.

### Exercise 4 (of 7):

### Exercise 5 (of 7)

What are the main connectives of the following formulas?

(The answers are below).

a) (W⊃ (X≡Y))

b) (∼(X≡Y) ∨ (X≡Y))

c) ∼(A∨∼(B))

Answers

a) The main connective of (W⊃ (X≡Y)) is the '⊃' which occurs between W and (X≡Y).

b) The main connective of (∼(X≡Y) ∨ (X≡Y)) is the '∨' which occurs between ∼(X≡Y) and (X≡Y).

c) The main connective of ∼(A∨∼(B)) is the '∼' which occurs at the beginning.

Exercise 6 (of 7):

Some further remarks on the more difficult cases.

From the point of view of logic.....

'Neither A nor B' amounts to 'It is not the case that either A or B'.

'A if B' amounts to 'If B then A' . For example, 'The bomb explodes if the red button is pushed' amounts to 'If the red button is pushed the bomb explodes'.

'A only if B' amounts to 'If A then B'. For example, 'Plants flourish only if there is sunlight' amounts to 'If plants flourish there is sunlight'.

'A unless B' amounts to 'A or B'. For example, 'Plants flourish unless there is no sunlight' amounts to 'Plants flourish or there is no sunlight'.

A suggestion regarding how to solve translation problems: If you do not recognize the English as an example of a standard form, try to paraphrase it into a standard form. For example, 'Taxes are unpopular, but revenue is needed' is not in a form that we have met, but paraphrasing it to 'Taxes are unpopular and revenue is needed' takes it to a form that we know and which has the same logical structure as the original.

### Exercise 7 (of 7):

Symbolize each of the following, and for each one identify its main connective. None of these can be done by the program (with the program working at this level of analysis), but answers and paraphrases are given below:

a) Philosophy is hard but interesting.

b) Philosophy isn't both hard and interesting.

c) Although philosophy is hard, it is interesting.

d) Not only is philosophy hard but so too is logic.

e) Logic is interesting, yet not hard.

Answers. Paraphrases

a) Philosophy is hard and philosophy is interesting.

b) It is not the case that philosophy is hard and philosophy is interesting.

c) Philosophy is hard and philosophy is interesting.

d) Philosophy hard and logic is hard.

e) Logic is interesting and it is not the case that logic is hard.

Let

H = philosophy is hard

I = philosophy is interesting

L = logic is hard

M = logic is interesting

a) H∧I and this has '∧' as its main connective.

b) ∼(H∧I) and this has '∼' as its main connective.

c) H∧I and this has '∧' as its main connective.

d) H∧L and this has '∧' as its main connective.

e) M∧∼L and this has '∧' as its main connective.

Notice here that we have started to leave out brackets (when there is no ambiguity).