1/7/06
and do Propositional Exercise 3 (Propex3).
Help with Tutorial 3 Alternative Exercises
Deriver will read what you have selected, and read it literally. So, for example, if you have the formula (A ⊃B) and inadvertently select (A ⊃ B) and issue a command, chances are that Deriver will write an error message to you (because the right bracket or paranthesis is missing from your selection and so your selection is 'ill formed'). This can happen in more subtle ways. First you may select 'invisible' characters (such as tabs, returns, enters, etc.). So, it looks to you as though your selection is good, when it is not. Of course, Deriver is smart enough to know that it should ignore tabs, returns, etc. But, second, there is html 'noise'. This system uses html, the language of the Web, for its display. And when you, the User, are selecting, copying, etc., you are selecting html. In turn, html can accomodate many thousands of different characters, sixty thousand upward, and many thousands of these are invisible. Some of these can creep in from time to time and cause error messages. In sum, be careful with your selecting, and if you receive apparently mysterious error messages maybe there are some invisible characters. (Somewhat later in the piece you will be able to inspect the html and see what is happening.)
The notion of the 'main connective' is important in many of the areas to follow. Try the Main Connective Drill (in Deriver 'Games'->Propositional->Main Connective). Make sure that you can get 10 out of 10 right in 30 seconds.
(As in previous exercises, copy and paste the relevant formulas into the Journal or just copy them.)
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? under the Semantics Menu to find out what the program thinks. Notice that when the program writes to you what it writes is selected-- just press the delete key and the message will go.
a) True ∧ False
b) True ⊃ False
c) ∼ True
d) False ∨ True
e) False ≡ False
f) True ∧True
g) True ⊃ True
h) ∼ True
i) True ∨ True
j) False ≡ True
k) True ∧ False
l) True ⊃ False
m) True ∨ False
n) True ≡ True
o) True ∧ False
p) True ⊃ False
q) False ∨ False
r) True ≡ False
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. If your view is different from the programs try working your way out from the inside. Select one of the sub-formulas and click True?, then remove the sub-formula leaving the answer that you get instead (that will be True or False)-- 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 Deriver will write
(True ∨ False)True ⊃ (False ∧ True) which is the truth value of the subformula you selected
Remove the original subformula, but leave its truth value
True ⊃ (False ∧ True)
Select another subformula, say
True ⊃ (False ∧ True) and click True? and Deriver will write
True ⊃ (False ∧ True)False which is the truth value of the subformula you selected
Remove the original subformula, but leave its truth value
True ⊃ False
Select this and click True? and Deriver will write
True ⊃ False 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.)
a) (True ∨ False) ⊃ (False ∧ True)
a) (True ∨ False) ⊃ (False ∧ True)
b) (True ≡ False) ∧ (False ∧ True)
b) (True ≡ False) ∧ (False ∧ True)
c) (True ∨ True) ⊃ (False ∧ False)
c) (True ∨ True) ⊃ (False ∧ False)
d) (∼True) ⊃ (True ∧ True)
d) (∼True) ⊃ (True ∧ True)
e) (True ∨ False) ∨ False
e) (True ∨ False) ∨ False
The program knows about the truth and falsity of atomic propositions. When it starts, it assumes that all atomic propositions are false. You can ask it which propositions are true under the Semantics Menu. Note that when the program writes to you it will do so to the place it considers most appropriate and this will usually be immediately after any selection or the insertion point. You may be wise to decide where you want the answer to come and click on the text there (immediately before an exercise would be a good place). Now try it asking about true propositions under the Semantics Menu.
Assume for this exercise that all atomic propositions are false.
Form a view on the truth of the following compound propositions. The program will go through it step by step for you, if you select the subformula you are interested in and click True?. Work your way out from the atomic to the more and more complex, as in the previous exercises.
a) ∼G
a) ∼G
b) ∼∼G
b) ∼∼G
c) A ∨ G
c) A ∨ G
d) A ∧ ∼G
d) A ∧ ∼G
e) (∼G)⊃F
e) (∼G)⊃F
f) B ≡ K
f) B ≡ K
You can ask the program to consider certain atomic propositions to be true by selecting
assign true <atomic propositions>
and clicking Do Command.
Try selecting
assign true A C D
and clicking Do Command, and then ask for Current True Propositions under the Semantics Menu. Then select
assign false C
and click Do Command, and then ask for Current True Propositions under the Semantics Menu.
You should now have A and D as true atomic propositions, and all other atomic propositions are taken to be false.
Form a view on the truth of the following compound propositions. The program will go through it step by step for you, if you select the subformula you are interested in and click True?. Work your way out from the atomic to the more and more complex, as in the previous exercises.
a) ∼A
b) ∼∼D
c) A ∨ G
d) A ∧ ∼G
e) (∼G)⊃A
f) A ≡ D
g) ∼(A∧T)
h) D∧(∼∼D)
i) (A≡A) ∧(S≡S)
j) A ∨ (B∨C)
k) (A ∧ B) ∨C
l) (A ⊃ A)⊃(B⊃B)
You need to be reasonably quick with Truth Tables. Try Truth Table (in Deriver 'Games'->Propositional->Truth Table). 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' ).]