Symbolizing Compound Propositions I

Logical System

8/25/12

Not all propositions are atomic propositions. Consider the proposition asserted by 'It is not the case that in 2011 the United States had a female President'. This is a true proposition, yet it is not an atomic one. It is made up of the atomic proposition 'in 2012 the United States had 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.

A remark before the Exercises: Divide and Conquer

As mentioned earlier, there is a powerful technique for symbolization: Divide and Conquer. The Exercises throughout will often illustrate it. The Exercises themselves may be a little awkward, or even clunky, because  Divide and Conquer is not the easiest thing to show. However, what they are showing, the Divide and Conquer technique or heuristic is extremely strong and useful. And it works well with symbolization.


Exercises to accompany Symbolizing Compound Propositions I

Exercise 1 (of 3):

Exercise 2 (of 3):

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Exercise 3 (of 3)

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.