Logic - formula notation

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Logic - formula notation

#1  Postby Dolorosa » Aug 19, 2018 2:03 pm

I decided to do an online Ancient Philosophy course in my spare time, and whilst searching for more information on Heraclitus came across a site that uses formula notation. Now, I've seen propositional logic being expressed by way of formulas before but have never delved into it (formulas scare me), particularly since different sources use different notation.

I was wondering if someone could help me interpret read the formulas in points 3(a),(b) and 4(a),(b) - https://faculty.washington.edu/smcohen/320/heracli.htm

I believe the formula in point 2 reads - for all F, F is identical to its opposite F.

Any advice on learning this form of notation would be greatly appreciated.
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Re: Logic - formula notation

#2  Postby Cito di Pense » Aug 19, 2018 2:15 pm

Dolorosa wrote:I decided to do an online Ancient Philosophy course in my spare time, and whilst searching for more information on Heraclitus came across a site that uses formula notation. Now, I've seen propositional logic being expressed by way of formulas before but have never delved into it (formulas scare me), particularly since different sources use different notation.

I was wondering if someone could help me interpret read the formulas in points 3(a),(b) and 4(a),(b) - https://faculty.washington.edu/smcohen/320/heracli.htm

I believe the formula in point 2 reads - for all F, F is identical to its opposite F.

Any advice on learning this form of notation would be greatly appreciated.


Yeah, I understand your confusion. If Heraclitus says "hot" is identical to "not cold" you have to forget about lukewarm, which is also "not cold". So much for that kind of shit identity.
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Translation by Elbert Hubbard: Do not take life too seriously. You're not going to get out of it alive.
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Re: Logic - formula notation

#3  Postby Keep It Real » Aug 19, 2018 2:17 pm

The similarity to my "broken boiler/washing up water heat" event earlier today is there.
You're only conscious when you're thinking about consciousness.
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Re: Logic - formula notation

#4  Postby Thommo » Aug 19, 2018 2:34 pm

Dolorosa wrote:I was wondering if someone could help me interpret read the formulas in points 3(a),(b) and 4(a),(b) - https://faculty.washington.edu/smcohen/320/heracli.htm


3.(a) ∃x ∃F (Fx & Fbarx)
"There exists x such that there exists F such that Fx is true and Fbarx is true"
3.(b) ∀x ∀F (Fx & Fbarx)
"For all x and For all F Fx is true and Fbarx is true"
4.(a) ∀x ∃F (Fx & Fbarx)
"For all x there exists F such that Fx is true and Fbarx is true"
4.(b) ∀F ∃x (Fx & Fbarx)
"For all F there exists x such that Fx is true and Fbarx is true"

The text is ambiguous about how Fbar relates to F, it's intended to denote the "opposite" of F, which may or may not relate to the general logical concept of negation.

A bit clearer is that he intends the variable x to denote "things" and the variable F to denote "properties", so he's running through the four permutations of the quantifiers on X and F. So the suggestions become:
3.(a) Heraclitus believed that there is at least one object that has at least one property such that both the property and its opposite are true of that object.
3.(b) Heraclitus believed that for every object all its properties and their opposites hold true.
4.(a) Heraclitus believed for every object there is at least one property such that both the property and its opposite are true.
4.(b) Heraclitus believed that for every property there is at least one object such that the property and its opposite both hold true.

In terms of learning to read this kind of symbolic logic you'd need to find a course or primer on "first order logic". The plain text answer is more straightforward, but a bit of practice at what the implications of logic are is probably needed to really understand what's going on.

However, just to read, it's not that complicated. You go from left to right, adhering to the following conventions:
https://en.wikipedia.org/wiki/List_of_logic_symbols
  • ∃ reads "there exists". This is a quantifier, which means there is at least one object in your domain which satisfies the following formulas.
  • ∀ reads "for all". This is a quantifier, which means that every object in your domain (which can be empty - i.e. having zero things in it, in some logics) satisfies the following formulas.
  • Lower case letters like x are just read "x", these typically denote members of the domain.
  • Upper case letters like F are just read "F", these typically denote formulas, which philosophers (although not logicians) will often think of as properties. Note: Thinking of them as properties has no formal implications.
  • ¬ reads "not". It's logical negation and is also sometimes written as ~. So ¬F would mean "not F", which simply reverses the truth value (in classical bivalent logic - bivalent means two truth values, "true" and "false", it can mean more subtle things in some more advanced logics that we should not worry about here) of F, i.e. it's false if F is true and true if F is false.

He also used an overline that I can't quickly replicate here, which I'd just read as "F bar", so I've reformatted it as Fbar here to make it naturally readable in that same way.
Last edited by Thommo on Aug 19, 2018 2:52 pm, edited 3 times in total.
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Re: Logic - formula notation

#5  Postby Dolorosa » Aug 19, 2018 2:44 pm

Thommo wrote:
Dolorosa wrote:I was wondering if someone could help me interpret read the formulas in points 3(a),(b) and 4(a),(b) - https://faculty.washington.edu/smcohen/320/heracli.htm


3.(a) ∃x ∃F (Fx & Fbarx)
"There exists x such that there exists F such that Fx is true and Fbarx is true"
3.(b) ∀x ∀F (Fx & Fbarx)
"For all x and For all F Fx is true and Fbarx is true"
4.(a) ∀x ∃F (Fx & Fbarx)
"For all x there exists F such that Fx is true and Fbarx is true"
4.(b) ∀F ∃x (Fx & Fbarx)
"For all F there exists x such that Fx is true and Fbarx is true"

The text is ambiguous about how Fbar relates to F, it's intended to denote the "opposite" of F, which may or may not relate to the general logical concept of negation.

A bit clearer is that he intends the variable x to denote "things" and the variable F to denote "properties", so he's running through the four permutations of the quantifiers on X and F. So the suggestions become:
3.(a) Heraclitus believed that there is at least one object that has at least one property such that both the property and its opposite are true of that object.
3.(b) Heraclitus believed that for every object all its properties and their opposites hold true.
4.(a) Heraclitus believed for every object there is at least one property such that both the property and its opposite are true.
4.(b) Heraclitus believed that for every property there is at least one object such that the property and its opposite both hold true.


Thank you so much. This makes perfect sense now. The highlighted part is exactly where my confusion arose so your explanation makes this really clear. I just have to wrap my mind about this duality of 'thing' and 'property'. According to Heraclitus, everything is flux; only 'everything', if I understand this correctly, isn't a thing - flux is. I guess, this is what 'coinstantiated' means.

Again, thank you!
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Re: Logic - formula notation

#6  Postby Dolorosa » Aug 19, 2018 2:52 pm

Cito di Pense wrote:Yeah, I understand your confusion. If Heraclitus says "hot" is identical to "not cold" you have to forget about lukewarm, which is also "not cold". So much for that kind of shit identity.


I actually find his idea very interesting, specifically in the philosophy of mind context. I have a number of books on a go, and one of them is The Ethics of Killing by Jeff McMahan. There he begins by trying to describe what constitutes as identity and goes through a number of different accounts, many of which attempt to reconcile the idea of change and how one can be the same as the other. Not sure I fully grasp it yet, or if I ever will for that matter, but it's a great work out for my otherwise atrophied brain.
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Re: Logic - formula notation

#7  Postby scott1328 » Aug 19, 2018 2:57 pm

Remember Mathematics and Logic have progressed quite far since the Bronze Age.
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Re: Logic - formula notation

#8  Postby Calilasseia » Aug 19, 2018 8:59 pm

For a decent primer on symbolic logic, track down Willard Van Orman Quine's Methods Of Logic. Heavy going and terse, but contains everything you need to know about modern symbolic logic, and also contains a discussion on the history of logic.
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Re: Logic - formula notation

#9  Postby newolder » Aug 19, 2018 9:11 pm

I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops. - Stephen J. Gould
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Re: Logic - formula notation

#10  Postby Dolorosa » Aug 19, 2018 10:00 pm

Fantastic. Thank you both. In fact, after Thommo suggested the FOL course, I did a bit of good ol' googling and it turns out that there is an archived Intro to Logic and Philosophy of Language course on the same University of Washington website. The latter even had a lecture on Quine's Quantifiers and Propositional Attitudes, which is basically what I got stuck on. It's a lot of info to get my head around at this point but it's certainly something that I will be dipping into as I get further into my course.

Newolder, it's added to my library :)
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