# Symbolizing Compound Propositions II

Logical System

8/25/12

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 Symbolizing Compound Propositions II

### Exercise 2 (of 4)

What are the main connectives of the following formulas?

a) (W⊃ (X≡Y))

b) (∼(X≡Y) ∨ (X≡Y))

c) ∼(A∨∼(B))

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 3 (of 4):

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 4 (of 4):

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.

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 is 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) HI and this has '∧' as its main connective.

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

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

d) HL 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).