Predicate (mathematical logic) Puns

I've done my due dilligence and tried to answer this question using every resource I could get. KhanAcademy, NesoAcademy, and Rosen's Discrete Mathematics book. I still can't wrap my head around it. Here is how I answered it:

Let x be things you did to me.

Let y be you making me happy.

Let P(x,y) be the thing you did to me that made me happy.

∀x∃!yP(x,y)

According to Rosen (p. 62), the statement with the form ∀x∃yP(x,y) is false when “there is an x such that P(x,y) is false for every y.” Since “there exists one thing that makes me happy,” there is all minus one thing that did not “make me happy.” Therefore, the statement is FALSE because there is an x (a thing you did to me) such that P(x,y) is false (you did not make me happy) for every y (the thing(s) you did to me that make me happy).”

Is this right?

Long time ago when I did my undergraduate in mathematics I was just an average student (not the best and certainly not the worst) and simply liked math. But after my first year, I began to ask myself: Is there a theory or a topic in math that allows me to talk about different fields of mathematics?. I did my research and found out (and realised) that everything (in the contemporary higher math education) is built from ZFC Set Theory and Predicate Logic. I also found out Bourbaki and their view/paradigm about thinking about pure math, where all mathematical structures (e.g. Posets, Groups, Topology, Metric Space, etc.) are all built axiomatically using Set theory and first-order logic.

With this framework of thinking, I started to become more and more formalist over the years and did pretty well in writing proofs, to the point that in order to prove a mathematical statement I dont even need to intuitively understand some mathematical concept(s) all I needed was the relevant formal definitions, theorems, and axioms. And with Logic & their applications in proofs, I did manage to solve a lot of trivial and non-trivial statements in higher mathematics.

Do you think that in order for students to do extremely well in higher mathematics, they need to have a very deep understanding of ZFC Set Theory and Logic?

P.S. I'm not suggesting that this is the only way to build math or the only paradigm of thinking about math (There are other theories about math foundation such as Categories and Types).

Title basically says it all. My programming teacher says that over the years I have developped excellent knowledge of Python, but often start choosing the incorrect methods and focusing on coding rather than solving the problem at its root in my head, with a piece of paper and a pencil, and developing an algorhithm, before moving on to actually coding.

Did anyone else actually struggle with that, if yes, what helped?

I have tried Edabit and found it useful, but it is way out of my price range right now.

This is the field I’m conducting research in, and it seems quite rare (at least at my uni). I’m curious what people’s impressions are of this field and the people who pursue it :)

Hi everyone, I have a math question from college I would like to clarify. It goes like this:

Using only the following predicates and no others:

- Likes(x, y): x likes y

- Dog(x): x is a dog

Convert the following english sentences into predicate logic statements. You may leave out the domain in your statements.

a) Harry likes Daphne, and nobody else likes Daphne

My own solution:

∃!x (Likes(x, Daphne)) ∧ Likes(Harry, Daphne)

b) Anyone who likes dogs likes at most one dog (answer this question WITHOUT using the ∃! quantifier)

My own solution:

∀x( ∃y (Dog(y) ∧ Likes(x, y) → ~Likes(x, ~y)))

(I feel that there is something wrong with this solution because it does not specify that ~y is also a dog, but I have no idea how to include it in my solution)

Thank you!!

I've got two degrees in Mathematics and Physics, and I'm passionate about Logic. Of course, a big part of Logic is Mathematical, but I feel like the really important or vital parts of it for me are its epistemic status and its relationships with knowledge, the mind, language or rationality. That is, topics relating to Philosophical Logic, Philosophy of Mind, Semantics, Decision Theory or Artifical Intelligence, which aren't as purely mathematical.

Given how difficult it is to actually get a job in the Philosophical academia, I'm much more inclined to take graduate studies in Mathematical Logic, which I also love. So my question is, how hard will it be later to be able to treat or write about those other, more philosophical topics? How hard will it be to get in touch with experts in those fields? To attend seminars? To publish on these issues? Etc.

I'm talking either about a complete switch to Philosophical Logic, or at least partaking in research areas more directly related with Epistemology, like issues regarding our reasoning or machine reasoning.

I've also posted this to AskAcademia. Any further advice on my future career is very welcome.

Thank you all in advance :)

Throughout this post, I might be (probably am) using the terms "semantics", "syntax", "geometric", and "combinatorial" in non-standard ways, which is why I have them all in quotes.

It seems to me that in general, problems (or systems) tend to exist on two separate levels. The "semantic" level is intuitive, approximate, not well-defined, vague, lacks form, etc. On the other hand, the syntactic level is well-defined, precise, and "mechanical" (though of course, one still develops intuition for the syntax).

Formalization involves "collapsing" semantics into syntax. You could also argue that it's "translating", rather than "collapsing", but the reason I say "collapsing" is that I don't think the syntax can ever correctly capture the semantics (i.e., there's always loss of information rather than an injective mapping).

Is there a field of math that studies formalization itself? (i.e., the process of turning semantics into syntax)? I imagine this is probably one of {model theory, proof theory, category theory}, but I'm a noob at all 3, so I wouldn't know.

This question isn't as pure-mathy as it seems. For example:

Consider traffic laws. Traffic laws are a syntactic manifestation of the more vague "semantic" system of: "don't get into collisions". Traffic laws are more "computable" (or verifiable) than the "don't get into collisions" semantic system. For example, being forced to stop at a stop sign is (for argument's sake) a sufficient condition on not hitting other cars going across the intersection, but it's not necessary (eg, if there's no other cars anywhere nearby, you don't actually need to stop to avoid hitting any cars). In this case, you could "relax" the traffic laws (i.e., bring the syntax closer to the semantics), but you may risk making the computability/verifiability harder. For example, suppose someone runs a stop sign and claims "well, I looked carefully both ways and there was no cars", that is harder to verify/"compute" compared to checking whether a vehicle stopped at a stop sign or not.

Edit: specifically, I'm wondering if there's a field that studies properties of the "mapping" from semantics to syntax. So, given some semantic system, let's conceive of the set of all possible attempts at formalizations of that system. Is there a field that studies the properties of these "attempts"? For example, one formalization is "stronger" than another if its syntax is able to prove more statements than the other.

Using only the following predicate: equal(x, y), which denotes that x equals y, and using the function p(x) which returns the predecessor of the input x, formalise the following statement: Every natural number except 1 in the set of natural numbers {1,2,…,10} has one and only one predecessor.

I'm assuming it's something along the lines

equal(x,y)

S(x)

Entails x equal(x, s(y)) ^ EntailsAll z(equal(z, S(y)) -> equal(z,x)

https://imgur.com/kU499w8

Any letter here is used to represent a binary input (1 or 0) and then you just follow the basic of principles of maths, oh and work in DENARY. The idea is that you could punch one these formula into a standard school calculator without changing any settings and have it working.

Heres the basic ones:

1. NOT = 1-x
2. AND = xy
3. NAND 1-xy
4. OR x+y-xy
5. NOR 1-x-y+xy
6. XOR x+y-2xy
7. XNOR 1-x-y+2xy

Heres an adder that takes a carry and 2 bits and outputs the binary in:

• x+y+z+8xy+8xz+8yz-16xyz

Sorry I didn't create a half adder as my main source for logic gates was a video series by Sebastian Lague so I just followed that along .

Anyway just for fun and to prove how utterly useless this whole thing is here is an untested 4 bit adder:

z+a+b-2ab-2az-2bz+4abz+10((ab+az+bz-2abz)+c+d-2c(ab+az+bz-2abz)-2d(ab+az+bz-2abz)-2cd+4cd(ab+az+bz-2abz))+100(e+f+(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2ef-2e(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz)-2f(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz)+4ef(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+1000(g+h+(ef+e(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+f(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2ef(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2gh-2g(ef+e(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+f(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2ef(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2h(ef+e(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+f(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2ef(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+4gh(ef+e(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+f(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2ef(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz)))+10000(gh+g(ef+e(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+f(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2ef(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+h(ef+e(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+f(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2ef(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2gh(ef+e(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))+f(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz))-2ef(cd+c(ab+az+bz-2abz)+d(ab+az+bz-2abz)-2cd(ab+az+bz-2abz)))

z is a carry value (if you want to string together multiple adders) and

Urgh! I think I will get attacked by including science under "non-mathematical pursuits" but you sort of get what I meant, I hope.

Hello, this post is to look for modern bibliography on mathematical logic. I mean, Enderton, Mendelson, Kleene, are great references with their books including methamatematics but is there any new bibliography good as the forementioned?.

Thanks in advance for all the excellent recommendations that I know will be posted.

Hi! I'd really appreciate any opinions or experiences regarding these Masters, since I want to take one and then a PhD in Mathematical Logic (probably Model or Set Theory):

Master in Pure and Applied Logic, Barcelona
Logic, Master's Programme, Gothenburg
Master of Logic, ILLC, Amsterdam
M2 in Mathematical Logic and Foundations of Computer Science, Paris

Any other possibilites are welcomed!

I'm also interested in good Mathematical Logic PhDs. I've been recommended Münster and Lyon for Model Theory. Any other considerations or recommendations?

Thanks in advance :)

We all know the physical stuff and how you need to be mentally and physically tough to make it, but what about written tests and logical tests? How much grey matter do you need to have to make it through? I don’t hear this talked about a lot so I’m curious what will be said.

Hey everyone, After finally deciding to do something with myself, I find myself "stuck" on the first problem set of the class I chose to take. I think I'm on the right track, but without any available answers, I'm feeling hesitant.

Question: Problem 1. [24 points] Translate the following sentences from English to predicate logic. The domain that you are working over is X, the set of people. You may use the functions S(x), meaning that “x has been a student of 6.042,” A(x), meaning that “x has gotten an ‘A’ in 6.042,” T(x), meaning that “x is a TA of 6.042,” and E(x,y), meaning that “x and y are the same person.” (a) [6 pts] There are people who have taken 6.042 and have gotten A’s in 6.042

Is the answer literally as simple as it seems? ie. S(x)˄A(x) ?

Thank you for your help in advance! :)

If anyone has taken this class before, is it a hard class? I could only find the syllabus from 2007 so what should I expect from this class?

I love to read and I just began looking at logic as related to mathematics/philosophy. Please suggest a book that will enhance my understanding of this subject matter. I'm a layman. Thanks

Hi,

I've been trying to solve this puzzle for hours now and I'm (more or less) sure that something is missing and it can't be solved in this form. Can anyone confirm this before I go insane?

First the riddle:

The inhabitants of an island always either tell the truth or always lie. There are eight inhabitants.
-H says, "If S is lying, then O is telling the truth.",
-B says, "If O is telling the truth, then Z is telling the truth.",
-S says, "If B is telling the truth, then H is lying.",
-Z says, "If T is lying, then B is lying.",
-S says, "If M is lying, then G is telling the truth."

I have several problems with the puzzle. First, it is unclear what exactly "lie" means. Let's say H is lying, is his statement then just false or is the opposite true? In the literal sense, "lie" actually just means that the statement is false - but then it seems to be too little information!? Another problem is that T and M only appear once, so even if I know that Z's statement "if T is lying, then B is lying" is true, I can't conclude from there whether T is actually lying or not (right?). I googled the author and found more of his similar puzzles - usually, there is an additional assumption, e.g. that the ratio of liars to non-liars is 1:1. Would that help?

I'm lost - thanks for any help, and sorry if this is in the wrong sub for this kind of thing (the r/LogicPuzzels seemed pretty voided)

In math there's a proof technique called mathematical induction that proceeds as follows.

Suppose you have propositions

Suppose one proposition is true, and all of the propositions can be ordered so that the first one is the known true proposition, the first implies the second the second implies the third etc etc.

Then all of the propositions are true.

Is there something similar to this taught in formal logic in philosophy? How is this concept used in philosophy if so?

Edit:I said possibly infinitely many previously but that was not entirely true. It is true for any number of propositions.

EDIT:

The reason I'm asking is because I'm starting my master's soon and I need to choose a 2nd general area of focus. My first is logic. The other areas are Algebra, Optimization, Stochastics, Numerical Analysis, and Analysis.

Will pay for it

I've done my due dilligence and tried to answer this question using every resource I could get. KhanAcademy, NesoAcademy, and Rosen's Discrete Mathematics book. I still can't wrap my head around it. Here is how I answered it:

Let x be things you did to me.

Let y be you making me happy.

Let P(x,y) be the thing you did to me that made me happy.

∀x∃!yP(x,y)

According to Rosen (p. 62), the statement with the form ∀x∃yP(x,y) is false when “there is an x such that P(x,y) is false for every y.” Since “there exists one thing that makes me happy,” there is all minus one thing that did not “make me happy.” Therefore, the statement is FALSE because there is an x (a thing you did to me) such that P(x,y) is false (you did not make me happy) for every y (the thing(s) you did to me that make me happy).”

Is this right?

I've done my due dilligence and tried to answer this question using every resource I could get. KhanAcademy, NesoAcademy, and Rosen's Discrete Mathematics book. I still can't wrap my head around it. Here is how I answered it:

Let x be things you did to me.

Let y be you making me happy.

Let P(x,y) be the thing you did to me that made me happy.

∀x∃!yP(x,y)

According to Rosen (p. 62), the statement with the form ∀x∃yP(x,y) is false when “there is an x such that P(x,y) is false for every y.” Since “there exists one thing that makes me happy,” there is all minus one thing that did not “make me happy.” Therefore, the statement is FALSE because there is an x (a thing you did to me) such that P(x,y) is false (you did not make me happy) for every y (the thing(s) you did to me that make me happy).”

Is this right?

I've done my due dilligence and tried to answer this question using every resource I could get. KhanAcademy, NesoAcademy, and Rosen's Discrete Mathematics book. I still can't wrap my head around it. Here is how I answered it:

Let x be things you did to me.

Let y be you making me happy.

Let P(x,y) be the thing you did to me that made me happy.

∀x∃!yP(x,y)

According to Rosen (p. 62), the statement with the form ∀x∃yP(x,y) is false when “there is an x such that P(x,y) is false for every y.” Since “there exists one thing that makes me happy,” there is all minus one thing that did not “make me happy.” Therefore, the statement is FALSE because there is an x (a thing you did to me) such that P(x,y) is false (you did not make me happy) for every y (the thing(s) you did to me that make me happy).”

Is this right?