6/23/07
and do the 4 exercises of Sentential Exercise 5 (Propex5).
Each of the derivations in the following Exercise can be done using only the Rule of Elimination of Conjunction ∼&.
If the formula in a line in a derivation has the form
<formula1>&<formula2>
the Rule of Elimination of Conjunction &E allows you to remove the conjunction and add either
<formula1>
or
<formula2>
as an extra line. You have a choice here (whether to take the left conjunct or the right conjunct)-- you will be asked to choose.
Show each of the following arguments to be valid. If you are stuck, ask the program to do one line for you (choose Next Line under the Wizard Menu); then, having seen what the next line is, go to the Edit Menu click Undo Next Line and try to do it yourself.
a) F&G ∴ F
b) F&G ∴ G
c) (F&G)&(F&H) ∴ F&G
d) F,(F&G)&(F&H),H ∴ G
Each of the derivations in the following Exercise can be done using only two rules, the Rule of Introduction of Conjunction &I, and &I (learned previously).
If the formula in a line of a derivation has the form
<formula1>
and a formula in a line of a derivation has the form
<formula2>
the Rule of Introduction of Conjunction &I allows you to add
<formula1>&<formula2>
To use this Rule, select two formulas <firstformula> and <secondformula> (*note that the way you select a second formula is to press the command key (clover leaf on a Macintosh) and then click on the line*) and click &I. You choose whether the first formula is to go on the left or on the right.
Show each of the following arguments to be valid.
If you are stuck, ask the program to do one line for you (choose Next Line under the Wizard Menu); then, having seen what the next line is, go to the Edit Menu click Undo Next Line and try to do it yourself.
a) F,G ∴ G&F
b) F&G,F,H ∴ (F&G)&(F&H)
c) F,G,H ∴ (F&G)&(F&H)
d) F ∴ F&F
e) F&(G&H) ∴ (F&G)&H
f) (F&G)&(F&H) ∴ F&(G&H)
g) F,G&H,(A&B)&C ∴ (A&G)&F
Each of the derivations in the following Exercise can be done using only three rules, the Rule of Introduction of Disjunction ∨I, and two earlier rules &E and &I.
If the formula in a line of a derivation has the form
<formula>
the Rule of Introduction of Disjunction ∨I allows you to add
<formula>∨<newformula>
or to add
<newformula>∨<formula>
To use this Rule, select a formula <formula> and click ∨I. The program will ask you what you want as the <newformula>. The program will also ask you whether the <newformula> is to go on the left or on the right.
Show each of the following arguments to be valid.
If you are stuck, ask the program to do one line for you (choose Next Line under the Wizard Menu); then, having seen what the next line is, go to the Edit Menu click Undo Next Line and try to do it yourself.
a) F ∴ F∨G
b) F ∴ G∨F
c) F&G ∴ G∨H
d) F&G ∴ F&(G∨H)