Anyone know Symbolic Logic- proofs and truth trees?

Derivation Extra Credit Assignment: To earn extra credit points for Quiz 5, complete your choice of up to three of the following: Chapter 5 End of Chapter Exercises, Medium Proofs unstarred only, 34-50. Each problem is worth up to 10 points, for a total of up to 30 points or enough points to get you to 100% on Quiz 5, whichever happens first. To earn extra credit points for Exam 3, complete your choice of up to six of the following: Chapter 5 End of Chapter Exercises, Medium Proofs, unstarred only, 51-66; or any of the unstarred Zero-Premise Deductions. Each problem is worth up to 10 points, for a total of up to 60 points or enough points to get you to 100% on Exam 3, whichever happens first. You may do as many or as few of the extra credit problems as you like. To earn full credit, you must:

• Provide proofs that are properly constructed (each line numbered and justified; one proposition in the center column of each line)

• Provide proofs that are properly derived (correct use and application of derivation rules; correct use and discharging of assumptions and subproofs; conclusion derived)

• Email me your completed proofs as a PDF or Word document, with each exercise clearly labeled, no later than 11:30 p.m. Eastern time on Sunday, August 11.

Tips and reminders on how to do derivations: 1) Learn the derivation rules like the back of your hand. Be sure you understand how each rule works, and what it does and does not allow you to do. For example, most of you did something like this on Quiz 5 and/or Exam 3:

1. P∨Q P 2. P 1∨E

------NO! This is wrong, of course, because the rule cited – disjunction elimination – doesn’t work like that. Disjunction elimination says that if you assume each disjunct under a separate subproof and derive a third proposition under each assumption, you can derive that third proposition in your main proof. (If you read that and went, “Huh?” review how to use assumptions and subproofs!) It does not say that you can simply lop off a disjunct and throw it up in your proof! So never, ever do that! To make sure that you never again make a mistake like that, memorize which derivation rules require assumptions and subproofs, and/or make sure you refer to the list of rules at every step of the way to make sure you’re using them correctly. Refer to the appendix in the textbook for a complete list of the rules. Print it out, copy it, carry it with you everywhere you go, sleep with it under your pillow – do whatever it takes to get those

rules firmly in your mind! And even then, refer to the list of rules while you’re doing the exercises. 2) Here’s how to set up a proof:

• Construct a table with three columns. The left-hand columns will provide numbers for each line; the middle column will contain the premises of the argument you are working with; and the right-hand column will contain justifications for each line. The premises are justified with a P, and the conclusion (goal proposition) is placed after the P for the last premise. So if the argument is this (where ⇒ means logical entailment, because I don’t have the symbol used in the book on my computer):

R→S, R ⇒ S

You set up the proof like this:

1. R→S P 2. R P/S

Remember that the “P” in the right-hand column stands for “Premise.” It is a justification, not another proposition. Only justifications, not propositions, go in the right column. Place one premise – the entire premise, not just part of it – on each line in the center column. So if the argument is this:

[R∨¬(P∨T)]→[(F↔D)∨(S∧¬T)], R∨¬(P∨T) ⇒ (F↔D)∨(S∧¬T) The proof is set up like this:

1. [R∨¬(P∨T)]→[(F↔D)∨(S∧¬T)] P 2. R∨¬(P∨T) P/(F↔D)∨(S∧¬T)

The entire first premise – the whole conditional, not just the antecedent or consequent – goes on line 1; the entire second premise – the whole disjunction, not just the right or left disjunct – goes on line 2. The conclusion never goes in the center column! The conclusion is the proposition you are trying to derive from the premises. So it goes after the “P” for ‘Premise’ in the justification column for the last premise, to remind you of the proposition you’re trying to derive.

3) Tips on doing the derivation once your proof is properly set up: • Identify main operators and proposition types to tell you which derivation rules you

can use. For example, take this proposition:

[R∨¬(P∨T)]→[(F↔D)∨(S∧¬T)]

Identifying the main operator, the arrow in the middle, tells you that the proposition is a conditional. Look at the list of derivation rules to see what you can (and cannot) do with a conditional. You cannot, for example, use a rule like DeMorgan’s on a conditional, so it’s crucial to know what kind of proposition you’re working with in order to select the right rule. Conditional elimination tells you that if you have a conditional (P→Q) on one line of your proof and its antecedent (P) on another line, you can derive the consequent (Q). In the proof set up above, note that the premise on line 1 is a conditional and the premise on line 2 is the antecedent of that conditional. That means we can derive the consequent using conditional elimination and citing lines 1 and 2 in the justification column:

1. [R∨¬(P∨T)]→[(F↔D)∨(S∧¬T)] P 2. R∨¬(P∨T) P/(F↔D)∨(S∧¬T) 3. (F↔D)∨(S∧¬T) 1, 2 →E

And just like that, that pretty scary-looking proof is done, because the consequent I just derived happens to be the goal proposition! That leads me to the next tip, a crucial one:

• Look for the goal proposition in the premises. In the above example, looking for the goal proposition in the premises reveals that the goal proposition is the consequent of a conditional. Then look at the derivation rules to see which one(s) allow you to derive the consequent of a conditional. Conditional elimination does that, so that’s another way you can figure out that you should use conditional elimination in the above proof. This works for other types of propositions, too. If the goal proposition shows up as one conjunct in a conjunction, use conjunction elimination to derive it. If the goal proposition shows up as one side of a biconditional, use biconditional elimination to derive it. And so on. But keep in mind this very important point:

o When using introduction and elimination rules, you must work with whole propositions, the entire proposition exactly as it shows up in the proof. You cannot take pieces out of propositions and use an intelim rule on it. So if your proof looks like this:

(S∨P)→(R∧Q), S∨P ⇒ Q 1. (S∨P)→(R∧Q) P 2. S∨P P/Q

You may notice that the goal proposition is the right-hand conjunct in the conjunction on line 1, but you cannot use conjunction elimination right away, because the conjunction in question is itself part of a larger proposition. It’s the consequent of a conditional. You have to derive that conjunction all by itself on its own line of the proof before you can use conjunction elimination on it to derive your goal proposition. For practice, see if you can finish this proof on our own (ask me if you’d like to see the correct answer).

• If you can’t find the goal proposition in the premises, identify the main operator of the goal proposition to determine how to proceed. If the goal proposition is a conjunction, see if you can derive each conjunct and then use conjunction introduction to produce the goal proposition. If the goal proposition is a biconditional, follow the biconditional introduction rule (assume each side of the biconditional and derive the other side under that assumption) to produce the goal proposition. Etc.

• If the goal proposition is an atomic proposition (a single letter with no operators), use

negation elimination: assume the negation of the goal proposition; derive a contradiction. If the goal proposition is a negated atomic proposition, use negation introduction: assume the non-negated proposition; derive a contradiction.

• Review how to use assumptions and subproofs. They can be tricky until you get used

to them, and since some of the derivation rules require their use, you can’t skip learning how to do them right. Go over that section of the text as many times as it takes. Practice the exercises as many times it takes until you can do them easily. Be sure you know how to close subproofs and discharge assumptions. Be sure you are absolutely familiar and comfortable with which lines can be used within a subproof and which cannot (e.g. you can use premises from the main proof within a subproof, but you cannot use premises from a subproof in the main proof).

• Don’t skip steps (e.g. if you want to do conjunction elimination on a conjunct that is the

consequent of a conditional, don’t skip the step of using conditional elimination to get that conjunction derived on its own line first), but also avoid unnecessary steps. For example, if you find that you need a proposition within a subproof and that proposition is already present in the main proof, simply use reiteration; don’t try to re-derive an already derived proposition. Also, if you’re using conditional elimination and you only need the left conjunct to move forward with your proof, just derive the left conjunct; you don’t need to add the unnecessary extra step of deriving the right conjunct, too.

• Use the examples and answered exercises in the textbook as a model. Note how the

book sets up proofs and lists justifications. If the book always cites two lines in a justification for conditional elimination, for example, your justification line when you use that rule should always cite exactly two lines, never one or three or five. Look at examples of proofs that have subproofs within them to help you understand how those work. Look at examples of zero-premise deductions as a model when you’re working on a proof of a zero-premise deduction. If you’re stuck, find an example in the book that’s similar to the problem you’re working on and use it for inspiration: see what steps the book takes and check to see if the same steps will work in the proof you’re deriving.

• If you’re stuck, just start trying things. Go through the rules one by one and see which

ones you can use. Even if you don’t know where you’re going, just try applying the rules that work with the propositions you have, and see where it takes you.

4) One final tip: Do the extra credit problems with your textbook open to pages 241-244. Then you’ll have all the strategic rules and derivation rules at your fingertips. Refer to the rules as often as you need to to make sure you’re doing it right!