Wow! You made it to the end! First, I mistakenly say that at 1:01:00 that (1) and (2) are equivalent to each other. They are not. (2) should be, like you say, (A^~Q) v (B^~C) Second, to your question though: (A ∧ ¬Q) ∨ (B ∧ ¬C) and (¬A v Q) ∨ (B ∧ ¬C) are both in DNF. One of the reasons I made this video was because I kept misunderstanding which wffs were in DNF and (¬A v Q) ∨ (B ∧ ¬C) is just the kind of example I would get confused about. So you are in good company! The reason I would get confused is b/c some sources say that a wff is in DNF provided it is a disjunction of conjunctions, which always made me think it needs to look something like this: (A^B)v(B^C) or like this: A v (B^C). OK, so the reason that (¬A v Q) ∨ (B ∧ ¬C) is in DNF is because it is a disjunction where each of the disjuncts is a literal wff or an elementary conjunction. Let's break this down. So, we have the disjunction (¬A v Q) ∨ (B ∧ ¬C) and the parentheses on (¬A v Q) do not really matter, so we could write ¬A v Q ∨ (B ∧ ¬C). This is a wff of the form: X v Y v Z where X, Y, Z are wffs. Let's check to see if they are either (1) literal wffs or (2) elementary conjunctions: X : ~A, literal wff Y: Q, literal wff Z: B^~C, elementary conjunction Hope this helps! And thanks for watching!
At 1:01:04, the claim is made that formulae #1 and #2 are equivalent to each other. Wouldn't it be the case that ((A -> Q) -> (B ^ ~C)) ((A ^ ~Q) v (B ^ ~C)) instead - that being achieved by the twofold use of (P->Q) ((~P) v Q), by the use of (~(P v Q)) ((~P) ^ (~Q)), and by the use of (~(~P)) P ? Also, what is the best convention for the usage of parentheses? Are other forms of notation more useful? Prefix notation? Postfix notation? Is there one that allows a person to transition seamlessly into model theory? I ask because it seems like there are a host of conventions that are present in logic. Is there perhaps a compendium of all of these various conventions and notations? [Semi-recently, I was able to order your book and am just now getting around to working through it. Cool stuff! Would you please happen to have any recommendations for what book should be read next after yours? Thank you for your consideration of this comment.]
You are right. At that time, I mistakenly say they are equivalent. In reality, the 2nd wff should be (A^~Q) v (B^~C). I'd translate (1) into (2) like this: 1. (A -> Q) -> (B ^ ~C) 2. (~Av Q) -> (B ^ ~C) via Implication. 3. ~(~Av Q) v (B ^ ~C) via Implication 4. (~~A^~Q) -> (B ^ ~C) via DeMorgans 5. (Av~Q)v (B ^ ~C) via Double Negation Removal Sorry about that! There are a bunch of different types of alternative notations. Some remove parentheses altogether and adopt conventions for reading formulas. Others use smaller sets of operators, e.g. a few systems only use a single operator (Sheffer Stroke). The so-called Polish notation prefixes operators. So, instead of ~P, P^Q, PvQ, P->Q, PQ, you have ~P, ^PQ, vPQ, ->PQ, PQ. I don't think I've ever seen a post-fix notation. There are even more graphical (iconic) ways of presenting the same material (the so-called entitative and existential graphs of CS Peirce. This is super fun!). Thanks for picking up my book. I wrote it as a graduate student and it primarily grew out of handouts I used for teaching. In terms of what to read after that book, my recommendation would largely depend on what you are interested in. Here are my favorites: 1. Logic for Philosophy by Ted Sider. It isn't very detailed, but it gives you a really quick review of logic and a broader overview of lots of different topics in logic, particularly as they pertain to philosophy. 2. Logic, Language, and Meaning by L.T.F. Gamut. Two Volumes. I really like this book for how clearly it discusses logic in its relation to language, and I really like its discussion of modal logic in Volume 2. 3. It is sort of an acquired taste, but one widely read book is Graham Priest's Introduction to Non-Classical Logic. I think the book can be hard on people that aren't the most mathematically inclined or the most patient but this book is fun since it systematically considers so many logics. It is fun in that it shows the more creative side of building logical systems. Thanks for watching!
@@LogicPhilosophy Thank you for the extensive reply - especially for the leads on notation types and for the recommendations on further books to read. You have a really nice channel here on TH-cam - it's been very useful to watch your videos: now to read your book!!
Around the 1-hour mark, is the second formula really a DNF? Isn't it supposed to be this: (A ∧ ¬Q) ∨ (B ∧ ¬C) ?
Wow! You made it to the end! First, I mistakenly say that at 1:01:00 that (1) and (2) are equivalent to each other. They are not. (2) should be, like you say, (A^~Q) v (B^~C)
Second, to your question though: (A ∧ ¬Q) ∨ (B ∧ ¬C) and (¬A v Q) ∨ (B ∧ ¬C) are both in DNF. One of the reasons I made this video was because I kept misunderstanding which wffs were in DNF and (¬A v Q) ∨ (B ∧ ¬C) is just the kind of example I would get confused about. So you are in good company! The reason I would get confused is b/c some sources say that a wff is in DNF provided it is a disjunction of conjunctions, which always made me think it needs to look something like this: (A^B)v(B^C) or like this: A v (B^C).
OK, so the reason that (¬A v Q) ∨ (B ∧ ¬C) is in DNF is because it is a disjunction where each of the disjuncts is a literal wff or an elementary conjunction. Let's break this down.
So, we have the disjunction (¬A v Q) ∨ (B ∧ ¬C) and the parentheses on (¬A v Q) do not really matter, so we could write ¬A v Q ∨ (B ∧ ¬C). This is a wff of the form: X v Y v Z where X, Y, Z are wffs. Let's check to see if they are either (1) literal wffs or (2) elementary conjunctions:
X : ~A, literal wff
Y: Q, literal wff
Z: B^~C, elementary conjunction
Hope this helps! And thanks for watching!
At 1:01:04, the claim is made that formulae #1 and #2 are equivalent to each other. Wouldn't it be the case that ((A -> Q) -> (B ^ ~C)) ((A ^ ~Q) v (B ^ ~C)) instead - that being achieved by the twofold use of (P->Q) ((~P) v Q), by the use of (~(P v Q)) ((~P) ^ (~Q)), and by the use of (~(~P)) P ? Also, what is the best convention for the usage of parentheses? Are other forms of notation more useful? Prefix notation? Postfix notation? Is there one that allows a person to transition seamlessly into model theory? I ask because it seems like there are a host of conventions that are present in logic. Is there perhaps a compendium of all of these various conventions and notations? [Semi-recently, I was able to order your book and am just now getting around to working through it. Cool stuff! Would you please happen to have any recommendations for what book should be read next after yours? Thank you for your consideration of this comment.]
You are right. At that time, I mistakenly say they are equivalent. In reality, the 2nd wff should be (A^~Q) v (B^~C).
I'd translate (1) into (2) like this:
1. (A -> Q) -> (B ^ ~C)
2. (~Av Q) -> (B ^ ~C) via Implication.
3. ~(~Av Q) v (B ^ ~C) via Implication
4. (~~A^~Q) -> (B ^ ~C) via DeMorgans
5. (Av~Q)v (B ^ ~C) via Double Negation Removal
Sorry about that!
There are a bunch of different types of alternative notations. Some remove parentheses altogether and adopt conventions for reading formulas. Others use smaller sets of operators, e.g. a few systems only use a single operator (Sheffer Stroke). The so-called Polish notation prefixes operators. So, instead of ~P, P^Q, PvQ, P->Q, PQ, you have ~P, ^PQ, vPQ, ->PQ, PQ. I don't think I've ever seen a post-fix notation. There are even more graphical (iconic) ways of presenting the same material (the so-called entitative and existential graphs of CS Peirce. This is super fun!).
Thanks for picking up my book. I wrote it as a graduate student and it primarily grew out of handouts I used for teaching. In terms of what to read after that book, my recommendation would largely depend on what you are interested in. Here are my favorites:
1. Logic for Philosophy by Ted Sider. It isn't very detailed, but it gives you a really quick review of logic and a broader overview of lots of different topics in logic, particularly as they pertain to philosophy.
2. Logic, Language, and Meaning by L.T.F. Gamut. Two Volumes. I really like this book for how clearly it discusses logic in its relation to language, and I really like its discussion of modal logic in Volume 2.
3. It is sort of an acquired taste, but one widely read book is Graham Priest's Introduction to Non-Classical Logic. I think the book can be hard on people that aren't the most mathematically inclined or the most patient but this book is fun since it systematically considers so many logics. It is fun in that it shows the more creative side of building logical systems.
Thanks for watching!
@@LogicPhilosophy Thank you for the extensive reply - especially for the leads on notation types and for the recommendations on further books to read. You have a really nice channel here on TH-cam - it's been very useful to watch your videos: now to read your book!!
Can you make a video on unanswerable questions and metaphilosophy and Descartes and bertrand Russell