The recording is started. Let's call the meeting to order at 6.02. We have a quorum. Are there any additions or changes to the agenda? Hearing none. Is there any public comment? Again, hearing none. Janiel, let me pass this over to you so that we can start the meeting with you introducing Olivia, please. Good evening. There you are. There it is. So we have, at CV Fiber, hired our third full-time employee. Olivia Contica is here as our community relations manager, and we are so happy to have Olivia here with us. I will let Olivia talk a bit about herself, her background, and what she hopes to see with CV Fiber. Thanks, Olivia. Thanks, Janiel. Good evening, everyone. My name is Olivia Contica. I live in Berry, Berlin, somewhere on that border. I am originally from New York, so I am a transplant. I moved here about three years ago. In terms of my background, I have a bachelor's in polypsi, and I actually went to law school. I am a lawyer by trade. However, I transitioned over into the world of legal marketing and business development where I worked with a firm who had approximately 80 attorneys. I did professional services for quite some time, and then I transitioned over into working in the world of startups, working with early-stage technology companies all through Fortune 50. I have pretty much seen both sides of the world in terms of marketing and business development, but I am very, very excited to be here. I am really passionate about doing local community work, and I want to be a resource not only for this board, but also moving forward a resource to our communities, which is our first and foremost priority. My virtual door is always open. Feel free to email, call at any time. I will open to any feedback, suggestions, brainstorms. Thank you so much. Thank you so much, Olivia. We are so happy to have you. We are indeed pleased you are here. Thank you, Olivia. Thank you very much. Shall we then move on with the regular business of this meeting? I am going to consider Olivia being here an extraordinary business. Let us move on with the regular business. Jeremy, can we talk about the prior meeting minutes, please? Yes. I sent those out, but only about an hour and a half ago or so ago. I think they are in okay shape for approval, but I do not know that people have had enough of a chance to review. I guess Alan is shaking his head. Why do not we delay this until next meeting? Sorry, I was on a bike ride. I am sorry, I did not get it done sooner. Things have been crazy in Jeremy's world lately, so I apologize. I promise to read it quickly later tonight. All right, then let us move on. I am not sure that we have a treasurer's report today, but Ray, do we? I know we do not have a treasurer with us. Yes, I am surprised that she is not here. She had sent an email saying that she was not going to be able to make it. Oh, I did not see that. I think she said she was going to be later, like 6.30 maybe. I do know that today, just today and this afternoon, I think it was that we received from Bonnie, the reconciled financials, and so we will send them out to the board so that you will have them, and I had no further information for you. No, okay, understood. This is a little problematic with the timing. I guess if our second Tuesdays move back a little bit farther in the month, then those months will have reconciled numbers before the meeting, but when we are short on our number of days for second Tuesday, we will not have it. It is a little bit of a problem, but I think we will have to live with it for now anyway. Okay. Janiel, let me pass this over to you. If I am correct, Lucas is not able to be with us tonight, but if you would please, the next few parts of the meeting, construction through the marketing, if you have run through that, please. Absolutely. I actually like to put the highlight first because I do not want to bury it. The highlight is that we actually are operational, and by that I mean we have six friendly subscribers up and running on the CV Fiber Internet Service. I think we should acknowledge that. We do not want to bury that. That is a huge deal. That is in CLO 1. Is any of them on tonight? No, no. At the meeting? No, I do not believe there are any at the meeting. Janiel, there is a question from Siobhan. Siobhan. I was just wondering, are they happy so far? Have we heard anything? They are happy so far. Actually, Olivia is going to give a marketing update after I talk, and she is going to be delivering the great news. I am sure she can speak to the wonderful things that we have actually heard in terms of that, yes. We are actively working on signing up more subscribers. Those are our first six friendlies. So, we will be signing up general population as soon as we finish testing the network. So, the status we are at now is we are continuing to test the network that involves testing the network itself to make sure the fibers work, and also the payment system, and also the sign up system. So, it is a comprehensive layered system of friendlies that we are testing, and then we will be able to sign up the general subscribers. So, that is where we are with operations. I sort of took it almost in reverse order, but I needed to get that out first. Construction-wise, we are doing great with crews actually. On today's call, we learned we have four aerial crews, one underground crew, and now two splicing crews. Splicing is what has to be done at the end of the construction process. We got about 70 miles of each strand and fiber, so the fiber is catching up to the strand, and we have now 993 passings. That is addresses that we have passed. Possible subscribers. Construction is 192 sign ups for packages as of today. Thank you, David. Yes, 192 sign ups for packages as of today. That is a pretty darn good take rate. I mean, without just not even being technically in service for the general population, we have 192 out of 993. Now, I am not sure if that is exactly the ratio, but those numbers are excellent for where we are at. Materials and warehousing, we have ordered most of 400 miles. We have ordered all of the 400-mile bill of materials that we thought we needed, but every once in a while something new will be needed that we did not realize we needed in the field, and we will need to get it. Some of those are consumables or additional items that just didn't make it onto the bill of materials, which is normal because sometimes you will be able to use an existing piece in the field. Sometimes you can't. There are things that we learn in the field as we go. For the most part, we have the materials we need for 400 miles that should get us through all of next construction season, as well as this construction season, minus the small items, consumables that we will need to purchase. Warehousing is going well. Last month, we had actually, I guess it was back in July, there were some significant floods, but there's been a lot of rain. We have so far been able to escape any damage to the warehouse or materials, so we've been very lucky on that. Our warehouse managers are securing our outdoor storage facility as well, so that our fiber is going to be well protected over the winter. From an operational materials instruction standpoint, things are looking very positive. I do see that Jeremy, you have your hand up. Yeah. I guess my question is we have 400 miles of materials give or take. Do we have funds to have those materials installed, or are we going to have to stop construction for that first? So we do have, we have the funds to continue all of the distribution areas that we have started constructing that we have issued to our construction workers. They tell us that we have about nine weeks of construction left, and then, of course, there's additional work that needs to be done with testing, with splicing. So we're looking to be busy throughout the rest of the year. As far as installing the next distribution areas, that is, Jerry, did you want to speak to that? No, actually, I just want to get to the point for Jeremy's question. We do not have sufficient funds to construct 400 miles. What we have found is that the funds that we have are getting us through 200-some-odd miles, and that'll be a big part of the discussion of the budget and in the second part of our meeting. So everything that Janil has said is absolutely correct, but to cut to the chase on it, no, we don't have sufficient funds to construct all 400 miles. Things were not as they seemed a year ago. Thank you for being very direct, Jerry. Yes, we have seen incredible inflation and markup of costs beyond what we anticipated. They are unprecedented, so we're attacking that as we can. Any other questions? Okay, so that's what I really wanted to share for construction, materials, warehousing, operations, but I know that we have much more exciting stuff that does bleed into the operations scope in marketing. Thanks, Janil. I wanted to start with a couple of testimonials that we've received from our friendlies in CL01. I know sometimes it's difficult taking a step back and seeing all the nitty gritty construction details that are going on, but I want to emphasize in this group that we are really impacting the lives of our customers to date. So I'll start with one testimonial, and this is from one of our friendlies in CL01. The fiber is great. I'm so happy to be finally hooked up. It's all I had hoped for, a far, far cry from the old days of dial-up than satellite internet. Most recently, high-speed internet through a competitor, another ISP, which I have now canceled, from another friendly. I have been so impressed with all the various guys. They haven't been all professional, informed, and able. It's been a pleasure doing business with them. So we've, in general, have been receiving really positive feedback. If there are any ways that we can fine-tune the friendly process, we have been on top of that as well. I would say that in the past couple of weeks we really have solidified the friendly process in terms of having an inventational letter, making sure that we are in constant communication to iron out any kinks throughout the process, and so far to date we've been really successful with CL01. Now, moving forward for CL02, we do have those letters sent out. They were sent out early this morning, and we'll be scheduling those calls, and we'll be moving on to the friendly selection for RSO1 and RSO2. So all in, we are taking our feedback from a friendlies series at this point because we want to make sure that we are continuously improving and integrating based on what we're hearing in the market. Now, as far as what we're hearing, there are areas in terms of our messaging that requires a bit more clarity. As you know, there's a lot of technical nuances in what goes on behind the scenes, and so we are updating our website. So you'll see some homepage updates giving a bit more information, transparency and clarity to our audience as well. So people who are sending us inquiries trying to figure out where they are in the process, everything will be very clear-cut on the homepage, and it is our responsibility to make sure that we establish those trust relationships. So that's one improvement that we will be making to date on the site. Secondly, in terms of the website as well, we're integrating with Smart Hub. So we will be able to take payments, which is a really exciting update for us. Chuck. I want to start by just reiterating the first point that you made for the board.  They don't particularly care about them. For compliance reasons, we have to have them, so we can't get rid of them, but they don't need to be showcased front and center, and what we need is to streamline the conversion subscription funnel of people getting to subscribe. And so Olivia is making great strides on that front, and I thank you, Olivia. Now, to play devil's advocate, Olivia, what's Smart Hub? Smart Hub is actually a piece of technology that is used across the country. So I want to say that it is a well-established piece of technology, a payment system that is used for transactions. Some folks may even be familiar with it here in the Vermont area, paying their own electric bills. So it might not even be something new for some of our customers as they come on board. We are essentially using Smart Hub as a way to securely take payments in a dashboard that is user-friendly and a way for folks to possibly update their plans moving forward. That's something that we've been talking about in the works by the end of the year, but also people can update their credit card information, they can pay by phone. So essentially making the payment flow as frictionless as possible because we would hate for payment to be a friction area. Jeremy? Yeah, I know that WEK uses it and you can report outages and stuff like that. Are there going to be other ways of detecting when we have an outage, or will people be able to also report outages on the app? Yeah, that's an excellent question. I can certainly dig deeper for that. Right now it's just establishing the transactions, which I think is number one for us to receive payments securely. As we become a more mature organization, which is the term that I'd like to use, I'd like to have our touch points with our customers increase so that people are well-informed every step of their journey with us. Jeremy, Watesfield will know when there's an outage. There's no question. I mean, they'll notice the light has stopped. Right? Yeah, so I see Allen and RD and then Olivia, you may not be done, so we'll go back to you, but let's take Allen and RD right now. Here it is. Sorry, I don't want to belabor the outage thing, but WEK's, the problem with reporting online is with WEK's smart meter system. There was a great promise. I don't know if you remember this back in the teens when they got, I don't know, 50 million bucks or whatever it was to install the system and they thought they would be able to use it for online outage reporting. And I don't quite understand what's missing from their system, but it has not worked. And one of the reasons they originally were interested in working with us on fiber was they realized that's really the kind of connection they need in order for their, in order to begin to use some of the aspects of online reporting. So that's sitting in WEK's hands and they've got to figure that out. I've had great luck with the hub in paying my bills to WEK. I've had absolutely no problem. Works great. You can do auto pay if you want. It's very easy to change if you want to pay by check or by bank or by credit card.  So I'm really glad we're using that. I think it'll work great. Although, honestly, for having reporting, that's probably a terrible idea because if the internet is out, they can't use the app. Well, understood. Thank you. That was going to be my point, Jeremy. Oh, okay. Sorry to jump in. I drove to Montpelier today to find out what the heck was going on. Okay, let's give it back to Olivia because I'm not sure that she was finished. Go ahead, Olivia, please. I just have two other items, two categories of items I should say. First and foremost, events. We will indeed be at the Berry Hardage Festival, September 22nd and 23rd. So this event was originally scheduled for, I believe, July. It was rescheduled due to the weather. So we will be there. We have a team of volunteers that have already proactively reached out to me. But if you are interested in being part of the VIP club, please let me know. Please reach out. I do have time slots and two hour increments. So we're really excited for this. Again, it's a brand awareness piece because we don't know quite yet when we'll be in the area. So we really want to start associating our name with the community. So this is really more of an informational 101 presence for us. And I think it'll be a great opportunity. Hardy? Okay. Jerry, two things. First of all, Olivia, there is an event in Cabot that you might wish to have a presence at. Our cheese festival on September 30th. That's the beginning of Fall foliage day in the Northeast Kingdom. And we are right, of course, Cabot is right on the edge between CB fiber and Northeast Kingdom, not to mention Kingdom fiber. So brand awareness would be a very good thing in Cabot. The second thing I wanted to say, Jeremy said it, there's got to be a hotline for reporting averages. If your internet goes out, there's no way to report it online. So a simple hotline is, you know, probably what we should have. That's me. Very good, RD point taken. Janiel, go ahead, please. And just to speak to that RD, we do have our customer service and tech support numbers on our website. So we do have that ability for folks who want to call in. It's essentially our hotline. Yeah. And the second event, which I don't have a date quite yet. However, we have done the majority of prep work behind the scenes. We've been working with Mr. David Healy as well. We are planning a ribbon cutting event in CL01 at the Calistown Hall sometime in October. We are putting together a list for contacts for those who should be invited to the event. Again, this is more of a celebratory awareness piece for us, not necessarily lead generation for people to get signed up, but we really want to get a group together of media contacts, partners, congressional delegates as well. All of you folks as well will receive an invite from us. We want to take the time to celebrate now that we will be officially in CL01. Now, we are waiting on final approval for the Calistown Hall to be greenlit. We want this to be an opportunity for people to actually go and sign in and use our internet while they attend. So we're just waiting to hear back. And as soon as we get the green light that Calistown Hall will be ready to go, we will be sending that out, that invite out at least two weeks in advance. So you'll have that for the month of October. And the last piece in terms of engagement strategy, we are working on a creative campaign. Now that we are actually going to be live, some of you folks may remember that we did have a postcard series that went out in the town of Calist for CL01. We are updating and refining some of that messaging, refreshing the brand to make it a little bit cleaner and consistent. But we also want to make the brand a bit more memorable and human as well. So we are working on those behind the scenes. And again, we're looking to place the CB fiber brand and CL01 in high traffic locations with potentially marketing at satellite hubs, so commercial businesses that are offering free Wi-Fi for people on the go. And we think it's actually a mutual value to add not only for CB fiber, but also for our business customers as well. So we are working on that as we expand. Well, that's fantastic. Thank you, Olivia. Any questions for Olivia, comments? Olivia, anything you want to follow up with before we move on? No, just that everybody should expect an invite for the ribbon cutting ceremony. We will send that out in advance. And if you are interested in Barry Herd, trustable, I'd be happy to send over additional information. Thank you. Thank you. Alan, I'm going to pass this over to you now to give us a brief on the policy update. Again, this will be similar to what you did for the Executive Committee. Please, just a brief on where you're at. We're not voting on anything tonight. Just let us know what you're up to, please. This is the personnel policy, Jerry, you mean? The policy that you've been working on, I've been calling it the HR policy. But if personnel policy is more accurate, then please, as such. Yeah, that's actually what it's been titled. So that's why I'm stuck with that in my mind. But you're right, it is HR. Basically, what's happened is legal counsel looked at our policy. If you remember, once we passed this back in July, we knew we were going to have to have it gone over by legal counsel. And we have ended up with a document that's not 13 pages, but is more like 34 pages. And the number of sections that have been added is about a dozen. So in addition to some changes to what we had already had in the policy, there's a lot of new material in the revised policy that legal counsel has presented us with. So the policy committee dedicated its September regular meeting to reviewing the policy, the new personnel policy. We then had a special meeting on Monday to continue our review. And we're going to have another one next week to continue what's still left of what we haven't done before. It's a lot of stuff. And we hope to get a piece of it before the executive committee when it has its meeting on, is it the 19th, I think next week on the 19th? But we won't present anything to the board, the full board, until we have everything together with the executive committee. I think the only way to do it is to do a piecemeal. So that committee can really take a whack at what we've done. But I think for the board, we really have to have the whole thing together at one time. So it's going to be another, I don't know if we'll have it for October, it might be November, but we're moving as fast as we can. And it's just a lot of work. The committee's been great. I really appreciate the people who have been working on it. John and Sean and Linda and I and Ray, Ray was, Ray's not on the committee anymore, but he's, he's done policy work for us in the past. And we really appreciate everybody's effort. Siobhan, I see that your hand is up, please. I just wanted to add that this is, we're working as a committee.  So any board members who have an interest in this or are concerned about where we might be going with it or something like that are invited to attend. We encourage you to attend and remind you that we do have HR professional attorneys on this as well. So I just, I don't know that we, I mean, we do need to go through it bit by bit, but you might want to make sure you take a look at it before we get to the executive committee because we're already shredding it in the policy committee. And I would like our discussion and the executive committee to be as pertinent and on point as we can make it without going, wafting off if I, because we have a tendency to do that. That's why I'm bringing it up because we have a tendency to do that. So anyway, just get ahead. I'm done. Thank you, Siobhan. Got it. Alan, is there anything else you want to add here? Not unless people have questions. No, I think that's plenty. Are there questions for Alan? Okay. Hearing none. Let's move on then to our 2024 budget discussion. When we did this last week for the executive committee, my preamble took everything out of what, out of Ray's preamble. So I'm going to give Ray an opportunity this time to talk about something other than the numbers. Feel free. Feel free. Go ahead. You can tee it up.  And that is that we have run into a funding gap that we knew we were going to run into. Well, we saw it coming. We didn't know originally we thought we were going to have a continuous flow of funds and we started acting based on the idea that we would have a continuous flow of funds. Then we found out that we wouldn't and now we know we won't. So there are two things that are going on with this budget. One is to be accurate as possible, which is still pretty difficult because we don't have all that much history behind us, but we're trying to forecast 2024 the best we can. And that's on the one hand. And then on the other hand, we're trying to figure out what construction and what level of installations can we maintain through 2024, given that there will not be or likely will not be any grant funding in 2024. So this is the predicament that we're in. And I will stop there. And if we need to go into executive session, I believe we will, Ray, I'll hand it over to you and then we get to the point where we need to stop recording. Please let us know. Okay, I'll start with this. And that is that the process is that in October, the board will approve a budget. That budget will be sent to the towns, the 20 towns, and for them to review and provide feedback if they want to. In November, the board will hold a public hearing on the budget. In December, the board will actually adopt the 2024 budget. The budget's going to change between the time you approve the budget and the time we adopt the budget, because we're going to have a lot more information hopefully by that time. So there will be some changes made to it. The budget is directional and aspirational. It's directional in the sense of something we just talked about. And that is for the last five years, we've been an incubator for a startup company for the CV fiber community network. And that community network is operational. And so we have to change in our governance and in our operation and in our website. There are so many things that we have to change and we have to work out a lot of details in terms of protocols, communications and oversight and what the committees are going to do and not do. The business, they do the operational stuff. The board, it does the governance stuff, policy, budgets, purchases, large purchases. There'll be a number of things that we're still responsible for as a board, but the business has to operate as a business. And they have to be given the kind of flexibility that's necessary. And also the kind of resources that are necessary. And so it's directional in that sense. And it's also aspirational in the sense that we have to make some assumptions with regard to what we can possibly get done in 2024. So in that sense, we're making a bet that we can do one more thing in 2024. And between now and December when we approve a budget, we'll be pretty far along in the knowledge about whether or not we're going to get a funding stream to actually do that. And so we're going to talk about that. Now, many of you have already seen a great deal of this. And it's not short. What we're going to go through is actually foundational for the decision and the approval that we're going to do in October. Knowing you should know that the finance committee has been working on this. The executive committee is overviewing it. The executive committee will be making a recommendation to the governing board with regard to approving a budget. So there's a lot of detail in the background. There's a lot of people that are working on it. So without any further ado, Jerry, unless you have something else you think in a preamble, I'm going to move that we go into executive session. I'll just ask to the group, are there any questions before we go into executive session? Because we need that in order to do the details. Jeremy, you're on mute, Jeremy. I apologize. So we're going to be going into executive session, and then we're going to be coming out of to do additional business. Is that correct? The only additional business I believe will be to adjourn. Oh, perfect. Okay. I didn't realize that the agenda was so short. That's perfect. So I don't know everybody who's on the call because we're going to invite into the call staff, delegates, alternates in the treasurer, but she's not here. If there's, if you are not a delegate, alternate or staff, if you might raise your hand and there's a reason for you to be invited, let us know else that's not going to happen. So here is the motion move. We enter an executive session, discuss records that are confidential pursuant to one VSA section 313 alpha six, specifically budgets, strategic plans and funding that relate to our strategic planning and inviting to executive session all staff, delegates, alternates in the treasurer in accordance with one VSA section 313 Bravo. Second. Second. Seconded by Jeremy. Are there, are there any opposed to the motion? Any abstentions? Hearing none, the motion passes unanimously.


of M1. I recently got, well, it arrived while I was gone, more on that later. I got that M1 Mac mini that I traded that domain for. I told you about and it arrived and all is good. So, you know, that part of it went really well. I replaced the iMac in the office with this and I decided mainly in the interest of time because, you know, I'd just gotten back from a trip. I wanted to get this thing set up and make sure it worked. So, I leveraged my time zone shift. We got back about 10 p.m. on Thursday night, actually about 11 p.m., I guess. And so, I stayed up till about three. I prepped the show, answered all the questions, and then migrated to the or started the migration process to this new Mac mini. And because I knew it was going to take hours for it to slurp all the data across from my clone drive. So, I got that set up and then I would sleep. I am not convinced, in fact, I am convinced that my advice to all of you would be if you are migrating from an Intel Mac to an M1 Mac that you not use migration assistant and that you set it more specifically, that you set it up from scratch. And the reason is I've been going through enough of this process that I know I'm still missing things. And the, I'm not sure I've saved any time. And given that I know that I'm still missing things, I know that at some point I'm going to wipe this thing and start it from scratch and migrate stuff back in manually anyway. So, I highly recommend that if you're going to an M1 Mac that you just do it manually. And that way you're at least starting as close to native as you


Recall that for a conditional statement, p implies q, its contrapositive was the statement not q implies not p, and these statements are logically equivalent to each other. So in terms of logic, these two are the same thing. We would say that they're equal to each other. Remember, that means that if this conditional statement is true, then this conditional statement is true. Likewise, if the contrapositive is true, the original conditional was true. If the conditional was false, then it's contrapositive is false. And lastly, if the contrapositive is false, then the conditional is false. They're logically equivalent to each other. Now in the past, we've talked about how one can prove a conditional statement. We typically use a method which we call direct proof. And the reason we call it direct proof is that there are ways to prove a conditional that are indirect, and the contrapositive proof pattern is exactly one of these first indirect proof methods we can prove here. Because if this statement is true, then its contrapositive is likewise true. And more importantly, if this statement is true, then this statement is true as well. So if we have to prove a conditional statement, what we can do is we can instead look at its contrapositive, we can prove the contrapositive to be true, which then makes the original statement true as well. So remember, with the method of direct proof, what we do is we assume the hypothesis of our conditional is true, and then we argue and then we can conclude that the conclusion is true. Now if you apply the method of direct proof to the contrapositive, what that means for you is you would assume that the negation of the conclusion is true, because that is the premise of the contrapositive. And so then you would make some argument and you would end with the statement that the conclusion is false. That is the conclusion of the contrapositive. But the conclusion of the contrapositive is the negation of the premise of the original conditional there. So if you apply direct proof to the contrapositive, that then proves the contrapositive to be true. And since it's logically equivalent to the original conditional, this would also prove that the conditional is true. But as you're doing it somewhat indirectly, you're assuming the conclusion is false and then arguing that the premise is false. This is what we call the method of contrapositive. So a contrapositive proof. It's an alternative way to prove a conditional statement is true. And there are many times where it can be more helpful to do that in that situation. Now, of course, with direct proof, you assume the premise is true because of the premise is false. The statement is vacuously true. But over here, you assume the conclusion is false because if the conclusion was true, the conditional would be trivially true as we've talked about previously. So let's then compare how does a proof by contrapositive compare to direct proof? So consider the following statement. Suppose x is an integer. If 7x plus 9 is even, then the number x was actually an odd number. I'm going to prove this theorem twice using the method of direct proof and contrapositive proof. Let's start with the direct proof. As we have this conditional statement, if p, then q, I'm going to assume p is true and then work from there to prove that q is true. Well, that would mean I suppose that 7x plus 9 is even. What does it mean to be even? Well, that means there exists some integer k such that 7x plus 9 is equal to 2k. Now, the next step of the problem, well, the next step is not so bad. We want to show that x is odd. So at some point we have to get x equals 2l plus 1, something like that. That's what we're looking to do right here. So how do we do that? So it makes sense to solve for x in this situation. So you can do that by moving the 9 to the other side. You're going to get 7x is equal to 2k minus 9. But then the next natural step would be to divide both sides by 7, for which then you get that x is equal to 2k minus 7. We have a problem here. First of all, how do I know that 2k minus 9 divided by 7 is even an integer? Because when you start dividing by integers, there's no guarantee that you have an integer anymore. The only way that 2k minus 9 divided by 7 is an integer is if 2k minus 9 is divisible by 7. And I don't actually have a good argument available to me to provide why 2k minus 7 is divisible by 7. I would love it to be, but I don't actually know that for a fact. Now it might be tempting to be like, well, 9 minus 2 is 7, maybe. But we don't know what k is. So there could be an argument, perhaps, right? But we don't have it necessarily in front of us. And it might be difficult to find it. And then even if we prove that 2k minus 9 divided by 7 is an integer, even if we know this is an integer, how do we know it's odd? Which is what we're trying to show. It could arguably be done, because x is in fact odd here, because that's what we're trying to prove. But that's a very difficult argument comparatively. Now what we can do instead is we don't necessarily have to solve for x by dividing by 7. What we could do is we could actually subtract 6x from both sides of the equation. So by doing that, we get x equals 2k minus 6x minus 9. Now this right here feels like it's a violation of my mathematical upbringing. I've taught many algebra students, like college algebra, intermediate algebra students over the years, calculus students as well. And oftentimes they're asked to solve an equation of two variables. It's usually an x and y or something. And they try to solve for x. Maybe it was y equals some function, and you get x squared plus x plus 2. And then they might do something like, oh yeah, I'm going to write as x equals y minus x squared minus 2. It's like, oh, I solved for x. But it's like there's still x's on the other side of the equation. You didn't really solve for x right there, because in order to determine x, you have to know y. And you have to know x. That's not very helpful as a formula. Like over here, if you know x, then you know y. But we didn't reverse the direction. We didn't find the inverse function. I've had to tell students so many times that you have to make sure you get all of the variables on the one side of the equation. But I actually did the exact opposite. I did the opposite of what I've been telling students to do for years. x equals 2k minus 6x minus 9. Now I confess in this situation, I'm not trying to come up with a function. So I don't necessarily need to do that. I just have to show that x is odd. But it's kind of odd that I'm going to reference x by using x itself, in addition to k, of course. So there's this awkwardness that's happening here. But nonetheless, 2k is an even number. You can factor out a 2. Negative 6x is an even number. You can factor out the 2 leaving negative 3x. And you could rewrite negative 9 as negative 10 plus 1. You could leave the 1 by itself. And then the negative 10 is an even number. It's 2 times negative 5. So you can rewrite this expression as 2 times k minus 3x minus 5 plus 1. As k minus 3x minus 5 is an integer, 2 times an integer is an even number. Plus 1 is a non-number. We've been able to write x as an odd number. That proves that x is an odd number. So this is a valid proof. But I want to mention that it's sort of an awkward proof. There were some weird things I did that made me uncomfortable as I did them. I was able to get around the problems. But imagine you, if you were trying to write this proof by yourself, and you've never written a proof like this before, you might struggle to have done it in the direct approach because of some of these awkwardness, these obstacles I explained along the way. Now, if we were to instead prove this by the contrapositive, or positive, which instead of assuming the premise, we are going to negate the conclusion. So suppose that x is not odd. Well, if you're an integer and you're not odd, that means you're even. So we're assuming that x is an even number. Now, if x is an even number, that means there is an integer L such that x equals 2L. Now, this proof is actually very much in line with previous proofs we've considered in this lecture series. I need to show that 7x plus 9 is not even. But to show that it's not even, I have to show that it's odd, which means I need to show it looks like some type of 2k plus 1. But the way I approach this is like, okay, 7x plus 9 depends on x. I could plug in this substitution in for x inside of 7x plus 9 and then pull it out. 7x plus 9 is the same thing as 7 times 2L plus 9, which of course, like we did before, 7 times 2L, that's 14L that you can factor out the 2. The 2 is already there, right? And then with 9, you can break it up as 8 plus 1. 8 is 2 times 4. And so this then decomposes as 2 times 7L plus 4 plus 1. So this then shows that 7x plus 9 is an odd number. And if it's odd, then it's not even. And therefore, we've been proven it by contra position. We use the contra positive here because we took not the conclusion and proved not the premise as opposed to assuming the premise and then getting the conclusion. Now, when we proved it via contra positive, we didn't have any of the annoyances, the awkwardness that appeared over here. Now, when you look at the proof side by side, they look like they're the exact same length, okay? They're both involving like five lines of writing here. Yeah, five lines of writing. So it looks like the same length. So in terms of brevity, it's not one that's shorter than the other, which is something nice. But I definitely would argue that the contra positive approach was smoother. Again, it avoided some of the awkwardness. Now, if you're reading it after the fact, maybe you don't notice it. But when you're writing it, those decisions had to be made. And this one was super awkward, super cumbersome. This one was much slicker and smoother. And because of that, I would actually endorse the contra positive as a method here. That because sometimes when you have a conditional statement, sometimes the conditional itself can be awkward to prove, but its contra positive might be much slicker or smoother to prove. So instead, you can prove the conditional by contra positive as opposed to direct proof. Let's see some other examples. Now, in these examples, I'm going to prove it by contra positive, not necessarily because the contra positive is better, but I think many of us could make an argument that the contra positive is the better approach here. So for our next proposition, it's very similar to the one we just did here. Suppose that x is an integer. If x squared minus 6x plus 5 is even, then x is odd. Now, this is sort of actually very similar to the one we just did here. If I assume that x squared minus 6x plus 5 is even, I would know that this number equals 2k, but I have x's and x squared. Again, trying to get x to be odd, so x looks like some 2l plus 1 is going to be very, very, very awkward. The other direction is actually much easier. I'm going to assume the opposite. I'm going to assume that x is not odd, or I might begin with suppose x is even. Now, when you prove something by contra positive, you don't necessarily have to be like, suppose not the following statement, which was the conclusion. As we've practiced previously in this lecture series, we've practiced how one negates a statement. This is exactly that moment to imply that skill, to apply it here. So instead of saying, suppose x is not odd, that is, that is, x is even, you could write something like that. Instead, it's very, it's reasonable to expect the audience knows that if an integer is not odd, it's even. So I'm just going to start with the negation already computed. I don't have to tell anyone that I'm going to get a statement. It's reasonable that my audience can negate statements. This is a fundamental advanced mathematics. So I'm going to do this by contra positive, so I get the negation of the conclusion. So suppose n is even. Well, if n is even, there exists some integer k, so it's at x equals 2k. And much like in the other proof, what I can do is I can take this substitution and plug it into my expression x squared minus 6x plus 5 and then try to argue that that is going to be not even a ka odd. So when we do that, you replace the x with a 2k here and here. So we get 2k quantity squared minus 6 times 2k plus 5. Let's simplify that for a moment. 2k quantity squared, we have 4k squared. Negative 6 times 2k is a negative 12k. Nothing to do with 5 so far. But I need this to look like 2l plus 1.  So 5 breaks up as 4 and 1. Then everything left that's not the 1 is an even number. You can factor out a factor of 2 there. So we end up with 2 times 2k squared minus 6k plus 2 and then you add 1 back in there. Since the 2k squared minus 6k plus 2 is an integer, this then shows that x squared minus 6x plus 5 is an odd number. And of course, if you're odd, that means you're not even. I don't need to specifically say that it's not even, saying that it's odd is good enough. And that then gives us the contrapositive proof. We took the negation of the conclusion and then proved the negation of the premise. That's a contrapositive proof. Let's look at an example using real numbers and inequalities. Suppose that x and y are real numbers. If y cubed plus yx squared is less than or equal to x cubed plus xy squared, then y is less than or equal to x. This one is one that screams to prove it by contrapositive, much for the same reasons as the previous examples. The premise looks awfully complicated while the conclusion looks awfully simple. And it might be a little bit confusing. How do I use this complicated statement to prove this much simpler statement? It's like, I don't even know what to begin with this thing, but I want to get there. So the nice thing about contrapositions, you can go the other way around. I can take the simple conclusion to prove the complicated premise, at least the negations of such. Because if we take y is less than or equal to x, then the negation of that statement would be that we are not less than or equal to. That's the same thing as saying y is greater than x. There's only those possibilities for this. So that's how I'm going to start my statement. Suppose that y is greater than x. Again, I don't need to say not y less than or equal to x. I can just say y is greater than x as my original assumption. The reader would understand that that's the negation of the conclusion. So suppose that y is greater than x. We can move the x to the other side, then y minus x is greater than zero. Again, there is some reasoning that has to go on here, but the thing is, a typical student, if you start with the left-hand side here, you're like, I have no idea what to do with that premise in order to get here. This other one makes a little bit more sense because you have a y minus x.  So if I take this y minus x is greater than zero, if I times both sides by the non-negative value x squared plus y squared, I will then get the following inequality, y minus x times x squared plus y squared. That's greater than zero times x squared plus y squared. Because this is a non-negative value, squared real numbers can never be negative, and so you add together two non-negatives that itself is non-negative. That's important because when you multiply both sides by a non-negative, it doesn't change the direction of the inequality. We have to pay attention to is our quantity positive or negative or zero, right? We know it's non-negative because it's a sum of squares. So if you take the inequality we have right here and then you get this, well, you can simplify things. So the right-hand side's pretty easy. Zero times anything will give us zero. On the left-hand side, I'm just going to foil things out. You get y times x squared. You're going to get y times y squared, which is a y cubed. You're going to get negative x times x squared and negative x cubed.  And then finally, if we move the negative terms to the right-hand side, which was zero, we end up with y cubed plus y x squared is greater than x cubed plus x y squared. Notice these were the terms we wanted, but now we have the negation of our premise, right? Because as this thing is greater than, that means the negation would be y cubed plus y x squared is less than or equal to x cubed plus x y squared, like so. So we have now arrived upon the negation and we don't necessarily need to be like, we've negated the premise. You can say that if you want to. If you think there's any lack of clarity there, put that in there. But sometimes we do ourselves a disservice by saying too much, like we don't necessarily need to say, I will prove this statement by contra position. You can say that if you want to. And again, if you think it would benefit the reader to actually declare the proof technique you're going to use, do it. But in professional mathematical writings, very few people say that. But it's not completely isolated.  Sometimes the proof technique is mentioned specifically, but don't feel like you have to say that. Only say it for the sake of clarity. Like by not saying it, does it make the proof more confusing? And it might be depending upon your audience, right? If you're writing to professional mathematical researchers, you don't need to tell them, this is a proof by contra position. So they've been there before they can do it. But if you're writing to like, transition to advanced mathematics students, then they're not used to proofs by contra position yet. So therefore you might need to say that we will prove this by contra position. That's perfectly fine. And so that judgment has to be made by the author to decide what is necessary for the sake of clarity or not. Let's do one more example here. Let's do one with some divisibility here. Suppose we have integers x and y. And if 5 does not divide the product xy, then 5 does not divide x and 5 does not divide y. Now in this situation, you'll notice that this is a statement that's negated and these statements are also negated. So because there's a bunch of negations, there's a bunch of knots in play, this is also a reason to consider using contra positives because if I negate something that's already been negated, it's a double negative and hence becomes arguably a simple argument in that situation. So if I do this by contra positive, which I'm going to do, I'm going to take my premise here, excuse me, my conclusion, and I have to negate that. So we're going to negate 5 doesn't divide x and 5 doesn't divide y. But as we're negating things, we have to be careful about how negations work. Like how do you negate quantifiers? How do you negate connectives like and and or? The De Morgan law comes into play here. So the negation would look like not 5 doesn't divide x or not 5 doesn't divide y. Now of course these other ones simplify. If 5 does not not divide x, that means 5 divides x, or 5 divides y. So that would be the negation of that. And if you want to put all of those details into your proof, you are welcome to do so, but you have to make this argument between completeness and clarity. Because it's like, wait, how are those things competing with each other? When your proofs start to get longer and longer and longer and longer, because you're providing details upon details upon details, you think you're doing yourself a service for the sake of completeness. It's like, oh, I concluded all the details. But the problem is the longer and longer the proofs get, the more and more cognitive load you are putting onto your reader. There's more things they have to keep track of. There's more things they have to read and understand and remember. And therefore it can make it difficult to understand what's happening because they don't see the trees inside of the forest. I should say they don't see the forest inside the trees. How are the phrase goes, right? There's just too many trees. We'll put it that way. And so they can get confused because of the length of it. And therefore a one way to shorten a proof is to put some of the calculations, and in this case, this is a logical calculation, to put some of the calculations onto the reader. I know that seems strange, but you give them some of the burden, and that actually takes a burden off of themselves. So then when you start with something like suppose that x is five divides x or five divides y, then the reader, when they look at that, it's like, hmm, oh, that's the negation of this. They're proven by contrapositive. Okay, so they make that mental note. They sort of certify that up to this point, I'm good. And then they can keep on going from there. And then they don't have to worry about it anymore. It's actually, believe it or not, is more clear to let your reader participate. It's much like when you're teaching, if I'm teaching students, I need to give them something to do. It's not just me talking over and over and over again. They need to do these calculations themselves. That'll deepen their understanding. By allowing the reader to do some of the calculations, you are doing them a favor. And so while I did this negation calculation, I don't include any of the details in the proof. Instead, the reader will be like, why are we looking at five divide X or five divide Y? That's the negation of the conclusion of contrapositive proof. You're doing them a service by doing that. Now, because our assumption is an OR statement, there are cases that have to be considered. Five could divide X or five could divide Y. I don't know which one it does. But it turns out that there's really no difference between X and Y. You have these two different numbers, they're integers, and I'm going to take their product, but it doesn't matter who's the first factor or the second factor. So this is a great time where we can then be like, without the loss of generality, we assume that X divides a five divides X. Because assuming five divides X is no different than assuming five divides Y. It divides one of them. Let's just assume it was X. If it actually was Y, they will secretly relabel them. Y is now X and X is now Y, problem solved. Now, because five divides X, there exists an integer A such that X equals five times A. Now, if I take this equation, this is an equation, and I times both sides by Y, the left-hand side becomes XY, the right-hand side becomes five AY. Notice how I put the parentheses here. The right-hand side then suggests that XY is divisible by Y. We'll get this statement right here, which of course, if five divides XY, that's then the negation of my premise, and that then finishes the proof by contraposition. The proof is now done. I didn't need this last sentence, but again, I put it in for the sake of clarity. I did want my reader to know the proof is done, in which case they're like, why is the proof done? We just proved that five divides five XY. Oh, that's the negation of the premise. We're doing by contrapositive. I forgot that. Yeah, we're done. So again, you can say things, but we say things for the sake of clarity. And sometimes saying too much can be a problem with that as well. So in addition to some of these writing tips, hopefully we now have a better understanding that we can prove conditionals by proving they're contrapositive instead. And sometimes working with the contrapositive is a simpler approach to them working with the conditional. Not always, but we'll talk about some more of that in the future.


So, so far this semester we've looked at a pretty wide variety of moral theories. Now, at this point we're going to ask the question of, you know, does morality depend upon religion? Now this notion of dependence kind of fluctuates in this chapter. There's different notions of dependence floating around. One way of thinking about whether morality depends upon religion is whether the truth of some ethical theory depends upon the truth of some religion. That's one way to try to understand dependence. Another way is, does our knowledge of morality depend upon our knowledge of a particular religion? That's another way to think about dependence. Another way to think about it is, does the existence of morality depend upon the existence of a religion? Now Rachel's, you know, looks at this question and he provides several different approaches but sometimes it's not always clear which kind of dependence he's talking about. So, we'll think about that as we wrap this up at the end. But the question we're dealing with here is trying to figure out what, if any kind of dependence, morality has upon a religion. So one of the classic arguments in dealing with this kind of dependence is something called the Euthyphro Dilemma. And it's called Euthyphro named after the person with whom Socrates is having a conversation about this. So the Euthyphro Dilemma starts off really simple. He's asking the question, is what is moral, moral because the gods command it or we can just simply say the divine. If you don't want to talk about the existence of God you can simply say it this way. Say, is what is moral, moral because the divine commands it or does the divine command it because it's moral? So if it's the first one, right, so the two possibilities, either it's moral because the divine commands it or the divine commands it because it's moral. So let's try the first one. It's moral because the divine commands it. Okay. So what is this supposed to mean? Well, there's a couple of different ways of trying to understand this. One way that Rachel's looks at is that we have existence, we have the world, we have what's real, we have all these physical properties and then the gods come along, the divine comes along and then kind of adds morality on top of that. So what the divine commands makes this morality. Another way of understanding this is that you have what is natural, you have what is existence, there is just no morality at all and the gods simply just command something or the divine simply commands something. Okay. So this whole idea is that there is some kind of disconnect between what is real, what's true and then what the divine says. Well, now if it's this option it looks like a bad situation here if this is the case that if the divine makes true independently of what's real makes true what is moral. Because it looks like there's just absolutely no reason for deciding one thing or another. Sure, the divine with a lot of faith traditions that the divine commands say something like murder, that murder is immoral. But if there's no reason for that then we say well the divine could have commanded just anything. Could have commanded murder or killing, random killing to be moral. The divine could have commanded a variety of things regarding life. The divine could have commanded that it's absolutely wrong in all situations to take life, any life whatsoever. The divine could have commanded that to take every life. Now you might be thinking well that's just kind of nonsensical because there's good reasons to take life or not take life and this is exactly the point. If there's nothing that existed or is responsible for divine command then it could have just been completely arbitrary. But your gut instinct is that well there's really good reasons for what is moral. There's really good reasons for why this or that is more than moral. We've seen lots of varieties of how to explain that with these different ethical theories. So this is exactly the point. We don't think morality is arbitrary. We think that there is good reason. But if there's already good reason and depending on what the divine commands then it can't be that option. There has to be reason for why the divine commands with the divine commands. Okay, good. But then this is the second option. That the divine commands it because it is good. But if the divine commands it because it is good it looks like it's good regardless of what the divine commands. So randomly killing strangers. There's really good reason not to do them. But that doesn't necessarily mean that some divinity wouldn't command it. You know if divinities are anything like people there are many times when people have commanded killing others for no good reason. But that doesn't mean it's moral. It looks like the standard of what is moral is independent of what the divine command is. So either way you go here I mean either you say that what is moral is moral because the divine commands that then you get this weird notion that morality is arbitrary and we don't want to accept that. The other way that we can go is to say that the divine commands because it is moral. It's like okay, we like to go that way but then it looks like what is moral is independent of divine command. So one approach in saying that and taking this second option with the youth of pro dilemma is what's called natural law theory. Now the reason why Rachel's talks about this is because it's been very popular with some traditions to appeal to natural law theory when trying to explain why what divinity commands is moral. And natural law theory has basically three claims. The first is that there is a natural order to the world or even just order. There is just an order to the world. And another way of saying this is that there are patterns to what's real. Trees sprout from acorns. Trees don't sprout from puppies and vice versa. Puppies don't sprout from acorns. The sky is gaseous the ground is solid and rock. Talk about mathematics, right?  The rules of addition, subtraction, multiplication, division are constant. They don't change when you're dealing with well at least when you're dealing with counting integers. You don't have difference rules. But if there is going to be a difference then there's going to be some way to account for it. If there's going to be a difference in this situation then there's going to be that same difference The laws, what we discovered that the physical sciences discovers a causal order to the universe. The physical sciences are going to tell us things for instance like puppies don't sprout from acorns and acorns don't sprout from puppies. The physical laws are also going to tell us about these causal relationships between material objects. So they're going to describe the physical sciences are going to describe how I behave or at the beginning of account other things that I can do within physical limits. So that's the first claim of natural law theory is that there is a pattern and order of rationality to the universe to all of reality. The second part of natural law theory is that there are laws that describe how things are and there are laws that describe how things should be. So the causal laws of nature are going to describe how these trees are actually going to exist. Trees are these trees well the old trees sprout from acorns the cedar trees sprout from whatever the seed is for cedar trees and this is the causal order of reality. Not just trees but also human beings what we can do what we can't do. The second kind of laws are not how things are how things should be. So we have lots of ideas about how things should be. We talk about this all the time. Human bodies should be healthy. We should strive for health. Human minds should be emotionally and cognitively well balanced. We should be able to have a grasp upon our emotional states and we should strive for believing true things. These are two different kinds of laws. Laws for how things should be and laws for how things actually are. It's not like these things are completely separated. How things should be is in part going to depend upon how things are. So one part of what we are human beings is we in fact have minds. We do have emotions. So the laws that describe how we actually behave are going to have certain descriptions about what we're going to do given certain emotional states. If you're extremely, extremely angry the laws will describe that behavior. Usually it's yelling, sometimes violence, certain facial expressions. These laws will where we're dealing with anger are probably going to explain how we react to people with whom we are angry. That's going to be the descriptive part. The laws that describe what should happen are going to in part depend upon that in the sense that given certain descriptions in how we are we should act in a particular way. But that's not necessarily it's not going to be the same thing as actually what is. So for instance how we actually act when we're angry, it's like we're yelling at somebody and maybe we'll actually hit them or something like that. But the laws describing how we should behave you shouldn't hit people necessarily when you're angry. There are going to be certain limits on when you can use violence. Now you imagine that the descriptive laws are different. Imagine, for instance when you hit somebody it causes damage. It causes bruising, maybe you cause some bleeding or if you hit them hard or something like that. If the descriptive laws were different if no amount of hitting somebody actually caused damage then the laws describing how you should behave would probably be different. If there was no damage that was caused by hitting somebody then it would probably be ok to actually hit somebody when you're angry. So for instance this happens already now when I'm angry at somebody you know my anger doesn't necessarily cause damage. I could just be angry and keep that anger to myself but it wouldn't necessarily hurt another person. So if it doesn't cause that damage at least on the surface there's nothing really wrong with whether or not I should be angry. There are the laws that describe what actually happens what can happen as far as physical descriptions are concerned and there are laws about what should happen. The laws about what should happen aren't completely independent of what actually happens but they're not co-extensive they're not the same thing. There's going to be different sets of laws or as I like to tell people the difference between is and should is called the real world. The third part of natural law theory is that the first part was that there's an order, a pattern, a rationality to existence. The second part is that there are laws that describe how things actually are and laws that describe how they should be. The third part of natural law theory is that all this is all this is discoverable or at least known or understood by reason. So the idea is there is this pattern, there is this rationality to the universe this pattern, this rationality is going to give us two different sets of laws since there is this rationality to the universe we are going to understand it because we are rational creatures we are the kinds of things that use reason we are the kinds of things that can discover patterns and this works really well in every regard because we are really good at discovering the causal laws of nature at least to an extent we understand a lot as far as that is concerned we are really good with for instance mathematics which is another kind of pattern to reality we are really good with logic to an extent we are really good with logic we can discover a lot of patterns and reasoning with logic what we have been doing through this course we have been discovering different ideas of what is moral looking at reality and engaging reality with our minds and trying to figure out what is moral based upon that so these three claims are the real basis for natural law theory the Rachel's provides three objections to natural law theory the first objection is what is natural is not always what is good so you get this kind of popular refrain from people who are endorsing how shall we say certain plants that when either consumed or the smoke of which is inhaled induces certain euphoric states so the claim is well you know this is natural so it's good you also get a lot of claims about this say for instance with food products you say well this food is natural it's naturally made so it's good well as Rachel's points out there are lots of things that are natural like this that are not good arsenic occurs naturally in nature you can literally if you know where to look for it you can literally go into a field and find arsenic poisons are natural the venom of a snake is natural but that doesn't mean it's good that doesn't mean it's good so the Rachel's argument here his objection is that what is natural is not always what is good that's a good argument that's a true claim though what is natural in this sense is not always what is good however that's not the sense of natural that the natural law theorist is dealing with you know it looks like what Rachel's is doing here is he's kind of combining the two different kinds of laws into one he seems to be saying that the causal laws of nature are just the same thing as the laws of what you should do and that's not what the natural law theorist is claiming would agree with Rachel's here that just because it's natural in the sense that it occurs in nature does not mean it's good the natural law theorist would agree with that but the natural law theorist says I'm not talking about just that sense of natural I've got more than one sense of I've got when I'm talking about natural I'm talking about what exists what exists what's part of the nature the essence or definition or meaning of a thing not necessarily what happens in a forest right so this is Rachel's first objection that what is natural in the sense that it occurs independent of human intervention is not always what is good and this is true what is natural in the sense that it occurs independent of human intervention is not always what is good but the natural law theorist doesn't claim this the second objection that Rachel's gives is this kind of famous one that what is the case doesn't imply what ought to be the case so the idea is well there's lots of things that actually are so when we're looking at it to the world we're looking at reality there's lots of things that actually are I mean there is there is the fact that people are dying of disease there is the fact that people are killing each other there is the fact that oh I don't know that people lie to each other all the time but from that fact of is it doesn't mean that you ought to kill that you ought to lie that you ought to steal and this is right there's all kinds of things that actually happen that doesn't mean they ought to happen however the natural law theorist is not claiming this the natural law theorist it actually makes a real distinction between this when with the two different kinds of laws the laws of what actually is the causal laws of what actually happens and the laws of what should be right now you know you might say hold on a second does that mean that they're independent no like I said they're not completely independent some of what should be is going to depend upon what actually is okay well it doesn't mean they're co-extensive right and all that is really meant here when talking about laws that should be with laws that are this kind of laws of existence is that if you know if reality were different the laws of what should be would also be different and that's just I mean that's just going to go here with all the theories that we talked about so far right if there were no rationality we wouldn't have social contract theory or consrespect for persons or even Rachel's bare minimum conception of morality right involving rationality you know if rationality were completely different if we were built differently such that you know right now the way that we're built we have to start with experiences and try to derive conclusions from that right so if I'm going to learn about these trees I got to start by observing the trees and derive the laws of nature about the trees now if our brains were different if our rationality was different such that I didn't have to go through the experiences of looking at that tree instead and say my mind could make direct contact with the tree and be able to infer or be able to just know directly everything there is to know about that tree then I wouldn't have to bother with the scientific method right if there was that change in what was real there would be that change as far as what rationality is but that doesn't mean that what is rational is just co-extensive with what is okay there's still some dependence there there's going to be something more happening than simple inference but but it's not as if the natural law theorist is saying that what is just is what ought to be natural law theorist is not saying that so again it kind of it looks like Rachel's complaint that is does not imply ought that's a good complaint that's true is does not imply ought claims exactly opposite that that there is a difference between what is and what ought to be Rachel's third objection is kind of related to the first two so the third objection deals with the physical sciences now with the physical sciences we can spend all day out here observing nature and we do this a lot and we do it very well so we do study the trees and we look at how the trees are related to the earth and the water and the sky we look at the causal relationships between these things and never not once do we find laws of what ought to be we only ever find of laws of what is so Rachel's is saying look the physical sciences deal with what's natural all the time but we never find these laws of what ought to be we only ever find the laws of what are or what is and you know if that you know that's that complaint again is it's kind of misguided again it's confusing the two kinds of laws there's a lot that science that the physical sciences don't uncover and it's not supposed to for instance you can study nature all you want and you're not going to find any physical evidence for mathematics that's not how math works we don't understand mathematics in virtue of observation and through experiment we understand mathematics through conceptual understanding all the same thing is the same is true for a lot of things there are plenty of properties out there that are not physical properties science and this is what the physical sciences do very good and I'm a fan of science physical sciences can tell you about the physical properties of all these things and about you and about the world and atoms and galaxies from the very tiny to the very big you can describe all those physical properties but natural law theorists say yeah there's physical properties and there are moral properties as well there are laws about what is and there are laws about what should be lots of things do this right lots of other disciplines talk about other properties besides physical properties and you might just start discounting all these different kinds of physical properties but the property of being a good inference is one of those that's not a physical property for instance the whole scientific method does not have physical properties the scientific method is a great method for discovering physical properties and the relationships of the things having those physical properties between them but it itself has no physical properties what an experiment is does not rely upon any given physical thing or any kind of physical thing rather what an experiment is involves all of these physical things and itself doesn't have any physical properties so even truth truth is not a physical property you can't weigh it you can't burn it you can't transfer it from one object to another you can't build it truth itself is not a physical property so you know Rachel's complaint here that the sciences deal with what's natural and they can't find any of these natural laws again it is confusing the two kinds of laws the natural law theorist is very willing to say yes there are causal laws of nature between material objects that's one set of laws and that the physical sciences discovers very well but physical sciences don't discover the laws of what should be because it's not designed for that and for one really simple example of what this is all kinds of physical laws there's going to be all kinds of physical properties about what happens to the body in different situations you give the body different kinds of food you give different kinds of outcomes you have the body do different kinds of activities you give different kinds of outcomes and the physical sciences can give us that complete description but in that description amongst the physical properties you will not find the physical property healthy healthy is what we come along and in a sense add to it and if you don't believe me there have been really constant descriptions of what of what the physical body does but there's been changing definitions of what is healthy so we already engage in this kind of activity of what we should be healthy versus how we actually are and I'm not going to go into my physical descriptions so these three objections that Rachel provides really just confuse the two kinds of laws that the natural theorist is dealing with now this still leaves open the question of what this all has to do with religion now I don't want to give the impression that there's just absolutely nothing wrong or problematic with natural law theory no there's lots of problems even just using the case of what's healthy you're going to get a wide variety of answers of what's healthy when we're talking about physical health we're talking about motion health mental health even in terms of we've been talking a lot about nature obviously lots of people are not happy but trying to figure out what happiness is and answering that question has been at best troublesome because we've had a lot of problems with that really in a lot of ways natural law theory is a very common project when you look at the three claims behind natural law theory that there is an order to the universe that there's a difference between what is and what ought to be and that we can discover through human reason that's real common to all the theories at least to some of them we're dealing with Kant's categorical comparative definitely dealing with rationality definitely dealing with the difference between what is and what ought to be and definitely dealing with the idea that reason can discover this utilitarianism does the same thing utilitarianism is very concerned with what's real out there it demands of us that we look at what's real and especially at the consequences of our actions and we find happiness in all that and we should go after happiness so there's lots of rules about what is and one rule about what we should do namely go after the thing that produces the most happiness and the utilitarianism is really confident that we can do this through human reason I guess what I'm pointing at here is that there is variety of natural law theories in really important ways and there are some really specific ones I mean one there's a version that just really kind of sticks to natural law theory but the three claims behind is one that's just really specifically called natural law theory but the idea behind natural law theory is actually really common across the theories now of course there's going to be problems with natural law theory we've seen that just through the course of the semester we can work out these three claims now what it's really getting to the main point here is what does this have to do with religion and there's even a way you can kind of parse this out you can say what does morality have to do with any given religion and what does morality have to do with with any given if a deity exists the other things are kind of two different questions the fact that there are religions doesn't necessitate that anyone's got it right and if there is no deity it really looks like religions are in trouble but they still make a lot of a lot of claims and the idea is what are we supposed to do with that if there is no divinity so to try to think about this a little bit when we're dealing with something like natural law theory and actually any of these different philosophical theories about what is moral and you see the motivations behind these different theories pop up across time, culture, religion and place the question is what is this relationship between morality and religion and the thing that Rachel's focus is on is a really great question is this various theories the idea is that we can discover what's true about morality independent of what a religion states so that's one way to look at it another way to look at it is to say religions make various claims about what's moral but we have to use reason to understand why they're moral another way to look at it and these are two really important claims to consider as far as I know faiths and different religions they do teach different kinds of moral codes that's right also as far as I know none of them stop there there's always an explanation for why these are the right moral commands of course you can have preachers and maybe even some religious institutions that will say that the reason why these are true is because they're commanded by the divine or by that church but that's pretty few and far between even Rachel's likes to use the catholic church as an example but the catholic church spends a lot of time trying to explain why these moral commands are the right ones and it doesn't always depend upon simply because it pops up in a sacred text as Rachel's points out natural law theory is a favorite of the catholic church in trying to explain these things but natural law theory as we saw with that discussion doesn't rely upon any specific sacred text so yeah these various faiths as far as I know all of them as far as I can tell always offer some kind of explanation some kind of justification for why these theories are the right ones besides simply saying the divine commands it so the different faiths embrace this idea that what is moral is understood and justified by human reason so but this is the crux of the complaint that Rachel's has this is the big question if what is moral does depend or you know is justified but it can't be understood by reason, by human reason why do we need the divine