6/8/07 10Software

### Skills to be acquired in this tutorial:

To learn how compound sentences are true or false depending on the truth or falsity of their component sentences.

### Why this is useful:

One of our goals is to assess arguments-- to say whether an individual argument is valid or whether it is invalid. An argument is invalid if and only if it is possible for all its premises to be true and its conclusion false at one and the same time. And an argument is valid if and only if it is not invalid. Assessing arguments is easier if you have a good grasp of what is possible and what is not possible by way of the truth of sentences.

### Reading

Bergmann[2004] The Logic Book Section 3.1 .

### The Tutorial:

Atomic sentences are either true or false.

We will not be looking here at what conditions are required for an atomic sentence to be true-- we will just take it that each of the atomic sentences either is true or is false.

For example, we take it that the sentence expressed by 'Hills are rare' either is true or is false. Notice here that from the point of view of logic we are not concerned whether anyone knows it to be true or knows it to be false, or believes it to be true, or believes it to be false. We just want there to be the two possibilities-- that it is true, that it is false-- and that the sentence is definitely one of them.

The truth value of compound sentences is fixed entirely by the truth value of their (immediate) constituents.

Consider the formula ∼F... If the atomic formula F is true, then the compound formula ∼F is false. If the atomic formula F is false, then the compound formula ∼F is true. Summing this up in a table

F ∼F True False False True

And this extends to more complicated cases.

We just work our way out from the atomic sentences to the ever more complicated compound sentences. Consider the formula ∼(∼(∼F)). There are two possibilities for F itself, that it is true, and that it is false. Let us follow them through in turn.If F is true, ∼(∼(∼F)) amounts to ∼(∼(∼True)) which amounts to ∼(∼(False)) which amounts to ∼(True) which amounts to False. If F is false, ∼(∼(∼F)) amounts to ∼(∼(∼False)) which amounts to ∼(∼(True)) which amounts to ∼(False) which amounts to True. Summing this up in a table

F ∼(∼(∼F)) True False False True

Similar considerations apply to the other connectives.

If F is true and G is true, then (F&G) is true. If F is true and G is false, then (F&G) is false. If F is false and G is true, then (F&G) is false. If F is false and G is false, then (F&G) is false. Summing this up in a table

F G (F&G) True True True True False False False True False False False False

If F is true and G is true, then (F∨G) is true. If F is true and G is false, then (F∨G) is true. If F is false and G is true, then (F∨G) is true. If F is false and G is false, then (F∨G) is false. Summing this up in a table

F G (F∨G) True True True True False True False True True False False False

If F is true and G is true, then (F⊃G) is true. If F is true and G is false, then (F⊃G) is false. If F is false and G is true, then (F⊃G) is true. If F is false and G is false, then (F⊃G) is true. Summing this up in a table

F G (F⊃G) True True True True False False False True True False False True

If F is true and G is true, then (F≡G) is true. If F is true and G is false, then (F≡G) is false. If F is false and G is true, then (F≡G) is false. If F is false and G is false, then (F≡G) is true. Summing this up in a table

F G (F≡G) True True True True False False False True False False False True

You should learn these tables.

## Exercises to accompany Tutorial 3

### Exercise 1 (of 4)

The notion of the 'main connective' is important in many of the areas to follow. Make sure that you can get 10 out of 10 right in 30 seconds.

### Exercise 2 (of 4)

Say we use True as shorthand for a true proposition, and False as shorthand for a false proposition, form a view as to the truth-value of the following propositions. Select the formula and click *True?* to find out what the program thinks. The program will write over your selection, showing the value of what you have selected (ie whether your view is right or wrong).

### Exercise 3 (of 4)

Form a view as to the truth-value of the following propositions. Each time select a subformula, with no more than two components, and click *True?* to find out what the program thinks. Work your way out from the inside -- keep doing this until the formula as a whole is exhausted. For example,

(True ∨ False) ⊃ (False & True)

Select a subformula, say

(True ∨ False) ⊃ (False & True) and click True? and the widget will write

True ⊃ (False & True) which is the truth value of the subformula you selected

Select another subformula, say

True ⊃ (False & True) and click True? and the widget will write

True ⊃ False which is the truth value of the subformula you selected

Select this and click True? and the widget will write

False

which is the truth value of the whole original formula.

We just work outward from subformula to subformula (this is just like evaluating (3 x 4) + (2+1) by means of the steps 12 +( 2+1) then 12 + 3 and finally = 15.)

## Exercise 4 (of 4)

You need to be reasonably quick with Truth Tables. Make sure that you can get 10 out of 10 right in 5 minutes. [The game here uses 'T' and 'F' for 'True' and 'False' (and, in computer science, you often see '1' and '0' for 'True' and 'False' ).]