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We've just wrapped up a lengthy discussion on formal logic, so I think one of the next natural questions is "Is there an 'informal' logic? What's the difference?" If you read through the series on formal logic, you may remember clear back in Part 2, that I mentioned that formal logic is primarily concerned with the structure of the argument. For instance: 1. If Max is a Poodle, then Max is a Dog. 2. Max is a Poodle. 3. Therefore, Max is a dog. This argument has the basic structure: 1. If A then B 2. A 3. Therefore B Because the argument has a valid structure (in other words: It takes a valid "form") it is a valid argument. So formal logic concerns itself with the "form" of the argument. (It has nothing to do with whether or not your argument is wearing a bow tie.) That said, as was mentioned early on in the discussion of formal logic, formal logic does not concern itself with the truth of the premises. For purposes of formal logic, we simpl...

When using Identity symbolizations there are a handful of rules all of which (I would argue) are quite intuitive and in line with our normal understanding of the “=” sign. First, we have “Identity Reflexivity” (ID. Ref), which simply states “a=a” at any point in a proof we may introduce “a=a” (or “b=b”, etc.) and simply cite “ID. Ref.” as our justification. Secondly, we have “Identity Symmetry” (ID. Sym), which is a replacement rule stating that “a=b” is identical to “b=a”. Thirdly, “Identity Substitution” (ID. Sub) which is another replacement rule, and states that, from “Ab” and “b=c”, we may conclude “Ac”. Lastly, there is Identity Transitivity (ID. Trans), yet another replacement rule stating that from “a=b” and “b=c” we may conclude “a=c”. Those are the extent of our rules surrounding identity. Again, I expect that none of these rules should be particularly shocking or controversial, and they all fit well with our typical understanding of how the equal sign works. Identity and Ind...

Specifying Numbers: Consider statements such as “there are two senators from Oregon”, or “there are nine planets”, etc. using identity descriptors. For instance, if look at the phrase “there are two senators from Oregon” we could symbolize it as   (∃x)(∃y){[(Sx ^ Ox) ^ (Sy ^ Oy)] ^ x≠y} Or “there is an x and a y, such that x is a senator and is from Oregon, and y is a senator and is from Oregon, and x and y are not the same person.” Similarly, if we wanted to say “there are three representatives from New Mexico” we could symbolize that as (∃x)(∃y)(∃z){[(Rx ^ Nx) ^ (Ry ^ Ny)] ^ (Rz ^ Nz)} ^ [(x≠y ^ x≠z) ^ y≠z] Or “There is an x, a y, and a z, such that x is a Representative and from New Mexico, y is a Representative and from New Mexico, and z is a Representative and from New Mexico, and x is not y, x is not z, and y is not z”. Obviously, if we really wanted to, we could say “there are nine planets” but it is worth noting that numbers above 3 or 4 rapidly start becoming unwieldy. Spe...

Identity: We’ve already learned about statements that involve individuals. For example, if I say “Mark Twain wrote Letters From The Earth” (Wml), Mark Twain is an individual. We’re not merely stating “someone wrote Letters From the Earth” ((∃x)Wxl), we’re stating that it was one specific person. Individuals will often be recognizable because they will be proper names, however they need not be proper names. For instance, consider the phrase “The thirteenth president of the US was a member of the Whig party.” In this “the thirteenth president of the US” denotes one specific person so this person counts as an individual. I stated in a previous post that we can generally identify the subject and predicate of a sentence because the subject is the person, place, or thing being spoken about, and the predicate answers the question “what about them?” As an example: Me: You know Mark Twain? You: Yeah, what about him? Me: He wrote Letters from the Earth. Me: You know the 13th President of the US?...

Sentences with multiple quantifiers Clear back in part 18, we briefly touched on the possibility of sentences with multiple quantifiers Stating that “there is a dog and a cat” could be symbolized as (∃x)(∃y)(Dx ^ Cy). (There is an x and a y such that x is a Dog and y is a Cat.) Today, we’re going to dive a bit deeper into that topic, especially as it relates to the Relational Predicate Logic that we covered last time around. For our discussion, let’s look at the phrase “x loves y”. Using a single quantifier, we could make a handful of sentences. For instance: (∃x)Lxj – someone loves John (x)Lxj – Everyone loves John (∃x)Ljx – John loves someone (x)Ljx – John loves everyone But we can also create sentences using multiple quantifiers such as: (x)(y)Lxy – for all x and all y, x loves y (Everyone loves everyone). In total, there are eight possible configurations that we can form using two quantifiers: (x)(y)Lxy and (y)(x)Lxy – These two are functionally identical and comes out to “everyone...

Multi-place sentences We have previously discussed how certain propositions (such as “Tolstoy was a Russian. All Russians write sad stories. Therefore, Tolstoy writes sad stories.”) simply cannot be usefully symbolized with propositional logic. So we introduced predicate logic. Well boy howdy, guess what: Some sentence can’t be usefully symbolized with the predicate logic we’ve learned so far either! Consider: Some people don’t like any dogs. Puppies are dogs. Therefore some people don’t like puppies. If we try to symbolize this using what we have so far, we end up with something like this: Px (x is a person), Lx (x likes dogs), Ux (x is a puppy), Dx (x likes dogs), Kx (x likes puppies) 1. (∃x)(Px ^ ~ Lx)                P 2. (x)(Ux→Dx)                     P – prove (∃x)(Px ^ ~ Kx) If you look at this, it really leaves us with nowhere to go. The entire second premise is useless, “Kx” is in...

Last time we covered two of our final 4 inference rules (Universal Instantiation and Existential Generalization). Today we’ll be wrapping up the last of our new rules. Beyond this I have a couple posts planned for less-common situations, and then the series will be done. So let’s get to it. Existential Instantiation (EI) If you remember our last past, you might have a guess about how this next rule is going to work. EG let us go from Dm to (∃x)Dx, so EI is probably going to let us go from (∃x)Dx to Dm. If that was your guess, you’re basically right, however, EI is going to have some additional restrictions on how it can be used. To understand the need for these restrictions, I will present the following example where I will INCORRECTLY use EI by treating it exactly the same way as UI. Consider the following premises: Some numbers are odd. Some numbers are even. Let us say that “Nx” is “x is a Number”, “Ox” is “x is Odd”, and “Ex” is “x is Even”. Observe the following argument: 1. (∃x)(...

Universal Instantiation (UI) So we’ve learned how to symbolize statements in predicate logic. Now how do we prove something with them? Let’s consider an argument from way back at the beginning of the course: 1. All poodles are dogs.                P 2. Max is a poodle.                         P 3. Therefore, Max is a dog.          From 1 and 2 Even at a glance this argument is obviously valid when viewed in plain English. That said, let’s look at what happens when we symbolize it. Let’s have Px be “x is a Poodle”, Dx be “x is a dog” and “m” be Max. That gives us: 1. (x)(Px→Dx)                          P 2. Pm                                             P – P...

Last time we covered “All” and “Some”. But what if I want to deny that something exists entirely? Some guy is telling me about the person he read about in the tabloids who was born with working gills and lives entirely underwater, and I get fed up and say “No human has gills!” Now how do we symbolize that? Let’s say Hx is “x is a human” and Gx is “x has gills.” From what we’ve covered so far, there are two major ways that people try to symbolize that: 1) ~(x)(Hx→Gx) 2) ~(∃x)(Hx ^ Gx) (and if you don’t fall into one of these camps, don’t worry, I think I know what you did, and you’re on the right track.) Let’s start with option 1 and translate it back to English. If we start with just (x)(Hx→Gx), it comes to “For whatever you fill in “x” with, if that thing is human, then it has gills.” Or roughly “All humans have gills". Sticking the “Not” in front of it, thus changes the statement to “Not all humans have gills.” While “Not all humans have gills” leaves open the possibility that t...

Last time we discussed the need for predicate logic, as some sentences such as “War and Peace is a novel” simply cannot be usefully symbolized in propositional logic. For predicate logic, we use a capital letter for our predicate, followed by a lower case letter for our subject. So “War and Peace is a novel” becomes “Nw”. “Angela is Happy” becomes “Ha” where “a” is Angela and “H[ ]” is “[ ] is Happy”. As a slightly more complex example: “Max is neither a fish, nor a bird” can be symbolized as ~Fm ^ ~Bm F[ ] = [ ] is a fish B[ ] = [ ] is a bird m = Max Hopefully this is looking similar enough to what we’ve been doing this whole time that denoting the subject and predicate of our propositions isn’t throwing anyone too much, but if you find yourself a bit lost, remember, the comments are your oyster! Now, predicate logic also introduces two other ideas into our logic vocabulary: “Some” and “All”. All (Universal Quantifier) Let’s say that I’m dealing with a sentence where I want to make a ...

As promised, this time we will be beginning our discussion of predicate logic. So what is predicate logic? Why does it exist? Well to answer that first question:  I’m assuming most of you remember back in grade school, your teacher telling you that sentences have a subject and a predicate. I can already hear 95% of you being like “Yeah… I mean I was told that but it’s been… HOW many years now?”, and the other 5% of you saying “Yeah of course I know that, I’m an English major.” So, for the benefit of you Non-English-Majors out there, the subject is the person, place or thing that you’re talking about. If I say “Leo Tolstoy was a Russian”, I’m talking about Leo Tolstoy, he’s the subject. The predicate tells us something about the subject (e.g. “he’s a Russian”). To help separate subjects and predicates, I tend to imagine a conversation like this: Me: Hey, you know Leo Tolstoy? You: Yeah, what about him? Me: He’s a Russian. “The kid threw a ball.” Me: Hey, you see that kid? You: Yeah,...

Our last two replacement rules will be Distribution, and Exportation. I will freely admit that these are the two rules that I’ve found the least use for, but they do exist, and they are occasionally handy. Distribution Distribution is useful with three-proposition sentences that use a mixture of “And” and “Or”. For instance the sentence: Joe Biden is the President, and his eldest son is either named Joseph or Beau. We can symbolize that as “P^(JvB)”. Now looking at it, the part where Biden is president isn’t a point of contention. That’s just stated as fact. The uncertainty comes in regarding what his eldest son’s name is. Distribution allows us to split the sentence up into (P^J) v (P^B) (Either Biden is the President and his son is named Joseph, or Biden is the President and his son is named Beau). Again, since this is a replacement rule, it also works in reverse. The other way that Distribution works is a bit less intuitive (to me at least), and states that P v (J^B) is equivalent t...

Welcome back, welcome back! Glad to have you. Come on in, take a seat, kick your shoes off, and get ready for yet another post in our series in formal logic. We only have three more replacement rules. We’re only going to get to one in this post, because we also have to learn our last operator “If and only if”, so that’ll be the post for today, and we’ll cover the last two next time.  Biconditional Exchange (BE) In our last post, we discussed Conditional Exchange. In this case we will be discussing Biconditional Exchange. Do not let the name fool you, the two are not really all that similar. We’ve previously discussed four operators “If/Then”, “And”, “Or”, and “Not”. To understand Biconditional Exchange, we need to add one more operator “If and Only If”. To start, let’s briefly recap “If/Then” so we can contrast it with “Only If”, and then see how the two behave together. Do note that the “If” does not necessarily have to be at the beginning of the sentence. For instance, “If she Wa...

 Richard Dawkins' book "The God Delusion" is something of a staple text amongst many atheist circles. Not surprisingly, it is held in much lower-regard amongst many Christians, including Christian Philosopher  Alvin Plantinga  who - in 2007 - wrote a 13-page response to the book entitled "The Dawkins Confusion". For most of this post, I will be focusing on responding to the paper itself. Having said that, I know from experience that a good many atheists hear a term like "Christian Philosopher" and are wont to hold their heads in their hands and say something to the general effect of "Oh what pitiful stuff". So I would like to begin by establishing Dr. Plantinga's credentials, and thus why I would bother addressing his paper in the first place, before moving on to the paper itself. Per his  Wikipedia page , Plantinga is an American philosopher who works primarily in the fields of philosophy of religion, epistemology, and logic. He earned h...