8/15/06
The computer support for the exercises in Tutorial 2 can either be by 'applet' , as in the original, or by downloadable program. If your browser or computer is not working properly with the applets, you could try the alternative approach of using the downloadable program offered here.
You should now and do Propositional Exercise 2 (Propex2).
You can work on the exercise which is reproduced below.
Help with Tutorial 2 Alternative Exercises
The program knows of four atomic propositions and has itself adopted the following conventions regarding them
The proposition 'Philosophy is hard' is symbolized by 'H'.
The proposition 'Philosophy is interesting' is symbolized by 'I'.
The proposition 'Logic is hard' is symbolized by 'L'.
The proposition 'Logic is interesting' is symbolized by 'M'.
Form a view as to how the program will symbolize the following propositions, then ask it to do so. First check that Propositional Level is chosen under the Semantics Menu. Then symbolize by selecting the proposition, copying your selection (using the Edit Menu on your Web Browser or suitable key combinations) and, in Deriver, clicking To Symbols (under the Semantics Menu). When you have symbolized the propositions, ask the program to translate them back by selecting each symbol and clicking To English (also under the Semantics Menu).
Again, the program will do one step of the translation process. Keep selecting inner symbolic subformulas and clicking To English until the entire formula is translated.
Keep in mind that when you are translating to symbols what you select must be in English, and that when you are translating to English what you select must be a symbolic formula.
When you are half way through the translation process you will have something which is part in English and part in symbols-- be sure that what you select is entirely in English or entirely in symbols (or nothing will happen).
Symbolize
a) It is not the case that philosophy is interesting.
b) Philosophy is hard and philosophy is interesting.
c) Philosophy is hard and it is not the case that logic is hard.
Copy and paste this into the Journal.
In this exercise you must chose how you are going to symbolize the atomic propositions. The program knows of the four propositions above, but that is all. Tell the program how you are going to do it by selecting
remember proposition (<english in here>) <capital in here>
and clicking Do command.
Then ask the machine to translate back and forward for you.
First tell of your conventions (and you can give several commands at once if you put them indivually in brackets and start with 'all')
all
(remember proposition (Capital punishment deters killers) <capital in here>)
(remember proposition (Capital punishment is justified) <capital in here>)
(remember proposition (Capital punishment is inhuman) <capital in here>)
(remember proposition (Capital punishment is cheap) <capital in here>)
then symbolize
a) Capital punishment deters killers and capital punishment is cheap.
b) It is not the case that it is not the case that capital punishment is inhuman.
c) Capital punishment is inhuman and it is not the case that capital punishment is justified.
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.
The program knows of four atomic propositions and has itself adopted the previously mentioned conventions regarding them.
(*If you ever want to know what propositions the program knows, select
write propositions
and click Do Command. *)
Form a view as to how the program will symbolize the following propositions, then ask it to do so. First check that Propositional Level is chosen under the Semantics Menu. Then symbolize by selecting the proposition, copying your selection (using the Edit Menu on your Web Browser or suitable key combinations) and, in Deriver, clicking To Symbols (under the Semantics Menu). When you have symbolized the propositions, ask the program to translate them back by selecting each symbol and clicking To English (also under the Semantics Menu).
Again, the program will do one step of the translation process. Keep selecting inner symbolic subformulas and clicking To English until the entire formula is translated.
Symbolize
a) Philosophy is hard or philosophy is interesting.
b) If philosophy is interesting then philosophy is hard.
c) Philosophy is hard if and only if it is not the case that philosophy is interesting.
d) If philosophy is hard and logic is hard then philosophy is interesting.
e) If philosophy is hard or logic is hard then it is not the case that philosophy is interesting.
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.
The program knows of four atomic propositions and has itself adopted the previously mentioned conventions regarding them.
Symbolize then translate back
a) Both philosophy is hard and logic is hard.
b) Logic is hard if logic is interesting.
c) Logic is hard if either philosophy is interesting or philosophy is hard.
d) Neither logic is hard nor philosophy is interesting.
e) Logic is hard only if logic is interesting.
f) Logic is hard unless logic is interesting.
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.
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).