6 ms·

Paradoxes of Material Implication (1997)

32 points by olooney 8y ago

gus_massa 8y ago
Original title: "Paradoxes of Material Implication"
- pierrebai 8y ago
  About the example given with teh number 3, I prefer this even more absurd form: if the number 3 is not the number 3 then the number 3 is the number 3. It's true!
ninegunpi 8y ago
Descendants of Aristotle still find limitations of the system amusing, that’s amusing itself. I hope to live to the day when philosophical advancements of 20th century (or re-discovery of 2500-old Indian logic, if you like), formalized in accessible forms, get widespread acceptance, could leave plenty of people who’se job it to juggle limited abstractions with the need to pick more useful jobs. No pun intended, these are terribly useful abstractions we’ve built our world on, but they barely hold up against thorough reality check and leave out a lot as ‘paradoxes’.
- nerdponx 8y ago
  This article reminds me of how undergrad economics is taught. 1. Assume humans have a known, unchanging utility function that can be globally maximized, and assume they maximize it at all times 2. Lay out a whole bunch of reasons why this makes no sense 3. Ignore #2 and proceed to build a whole theory on #1
- deleted 8y ago
  [deleted]
- rudolfwinestock 8y ago
  Keep in mind that Aristotelian logic did not stop with Aristotle. It kept developing through the Middle Ages and even into today. Now, it's called term logic. Fred Sommers made tremendous advances in expanding syllogistic logic into something more versatile than what Aristotle worked on.
  - ninegunpi 8y ago
    Indeed. Yet, it is still based on True/False pair, which does not reflect neither reality or human experience in most cases. Where it is applicable - it perfectly works. But the scope is limited.
    - foldr 8y ago
      Many-valued logics have been studied for quite a while. As far as I know, their applications are fairly limited. https://en.wikipedia.org/wiki/Many-valued_logic https://en.wikipedia.org/wiki/Many-valued_logic
      - ninegunpi 8y ago
        1. A "quite a while" is less than a hundred years after Godel and in math? Compared to 2000+ years of Aristotlean logic dominance in hard sciences just because Romans inherited most of their scientific views from Greeks, not from Indians/Chinese? 2. It depends on domains of applicability, if you think of it. In pure CS and math? Yes, the visible value is limited, because most problems we choose to try to solve can be solved with math apparatus we're armed with. Value I know is mostly limited to optimizing problems that have poor solutions with binary logic. In practical engineering? ATPG, to my understanding, requires multi-valued logic. Analysis of large phenomena and automated decision making becomes an order of magnitude simpler problems, with better efficiency over chosen metric. Temperature controllers, decisions based on photo-metering (autofocus, exposure adjustment), etc. Somehow, even with lack of readily-available building blocks and tooling, it turns out that there are problems people are motivated to solve from scratch, and MVL/FL comes handy. It's only stuff I overheard of through my life among bright engineers. 3. Moreover, the biggest impact is not in CS (as original presented paradox isn't, as well), it's on human judgment, decision-making and general assessment of reality, where "neither true nor false" (I don't know) is the first stepping stone to make the world much easier place to live in.
- myWindoonn 8y ago
  Please add some substance, or I will make your argument for you. Yes! We all need to get on board with constructivist mathematics [0][1] already. Construction is very similar to computation, and it is not inconsistent to take "all reals are computable" or "all functions are continuous", the same rules Turing discovered, as axioms if we like. We can therefore move computer science fully onto a foundation that is more rigorous than typical maths. [0] https://plato.stanford.edu/entries/mathematics-constructive/ https://plato.stanford.edu/entries/mathematics-constructive/ [1] https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016...
  - ninegunpi 8y ago
    You've made far better one than me below. Hats off.
foldr 8y ago
The claim that the material implication analysis preserves the validity of valid arguments is pretty questionable. Consider the following argument: No student will succeed if he goofs off Every student will succeed |- No student will goof off Analyzing the 'if' in the first premise of the argument above as material implication, we get: For every student x, it's not the case that [x won't goof off or x will succeed]. = For every student x, x will goof off and x won't succeed. The following argument is valid only trivially, as the premises contradict each other (assuming the existence of at least one student): For every student x, x will goof off and x won't succeed. Every student will succeed |- No student will goof off I suspect it's quite easy to construct other similar examples involving quantifiers where validity is not even trivially preserved. (It's easier to find examples of the material implication analysis failing to preserve the invalidity of invalid arguments.)
- a-nikolaev 8y ago
  No student will succeed if he goofs off = Forall x: (GoofOff(x) -> not Succeed(x)) = not Exist x : (GoofOff(x) and Succeed(x))
  - foldr 8y ago
    Sure, but then you're doing violence to the structure of the original sentence. (How did the negation get into the consequent?) If you're allowed free reign to paraphrase, then you can always get the right result. "No student will succeed if he goofs off" is a standard example discussed in the semantics literature, by the way. What you're pointing out, in effect, is that this sentence seems to mean "No student who goofs off will succeed". The problem is that it's unclear how to get to that interpretation given the actual syntactic structure of the sentence. In other words, you can't get there just by following a direction to interpret if...then... as material implication.
    - a-nikolaev 8y ago
      English works that way, this is not logic's fault.
dwheeler 8y ago
I created the "allsome" quantifier to reduce the risk of some of these confusions. Details here: https://dwheeler.com/essays/allsome.html https://dwheeler.com/essays/allsome.html
tr352 8y ago
Let me copy/paste my reply from the same discussion yesterday (https://news.ycombinator.com/item?id=18531650 https://news.ycombinator.com/item?id=18531650): There's another "solution" to this paradox: if we assert something we are guided by a set of "conversational principles". For example, asserting "X implies Y" if we know that X is false is inappropriate. If X is false, "not-X" would be the appropriate assertion. According to this theory, there's nothing wrong with the truth-functional meaning of "X implies Y". We just need to take into account what is implied by asserting "X implies Y", rather than e.g. "not-X", or "X and Y". Same with disjunction: "X or Y" is true if we know that X is true. However, if we assert "X or Y", it is implied that we're not certain that X is true, otherwise we would have used "X", which is the simplest way to convey what that fact. This is known as Grice's Pragmatic Defence of Truth-Functionality.
leoc 8y ago
Graham Priest's book is great: https://www.cambridge.org/ie/academic/subjects/philosophy/philosophy-science/introduction-non-classical-logic-if-2nd-edition https://www.cambridge.org/ie/academic/subjects/philosophy/ph...

32 points by olooney 8y ago

gus_massa 8y ago

Original title: "Paradoxes of Material Implication"

pierrebai 8y ago
About the example given with teh number 3, I prefer this even more absurd form: if the number 3 is not the number 3 then the number 3 is the number 3. It's true!

ninegunpi 8y ago

Descendants of Aristotle still find limitations of the system amusing, that’s amusing itself. I hope to live to the day when philosophical advancements of 20th century (or re-discovery of 2500-old Indian logic, if you like), formalized in accessible forms, get widespread acceptance, could leave plenty of people who’se job it to juggle limited abstractions with the need to pick more useful jobs. No pun intended, these are terribly useful abstractions we’ve built our world on, but they barely hold up against thorough reality check and leave out a lot as ‘paradoxes’.

nerdponx 8y ago
This article reminds me of how undergrad economics is taught. 1. Assume humans have a known, unchanging utility function that can be globally maximized, and assume they maximize it at all times 2. Lay out a whole bunch of reasons why this makes no sense 3. Ignore #2 and proceed to build a whole theory on #1
deleted 8y ago
[deleted]
rudolfwinestock 8y ago
Keep in mind that Aristotelian logic did not stop with Aristotle. It kept developing through the Middle Ages and even into today. Now, it's called term logic. Fred Sommers made tremendous advances in expanding syllogistic logic into something more versatile than what Aristotle worked on.
- ninegunpi 8y ago
  Indeed. Yet, it is still based on True/False pair, which does not reflect neither reality or human experience in most cases. Where it is applicable - it perfectly works. But the scope is limited.
  - foldr 8y ago
    Many-valued logics have been studied for quite a while. As far as I know, their applications are fairly limited. https://en.wikipedia.org/wiki/Many-valued_logic https://en.wikipedia.org/wiki/Many-valued_logic
    - ninegunpi 8y ago
      1. A "quite a while" is less than a hundred years after Godel and in math? Compared to 2000+ years of Aristotlean logic dominance in hard sciences just because Romans inherited most of their scientific views from Greeks, not from Indians/Chinese? 2. It depends on domains of applicability, if you think of it. In pure CS and math? Yes, the visible value is limited, because most problems we choose to try to solve can be solved with math apparatus we're armed with. Value I know is mostly limited to optimizing problems that have poor solutions with binary logic. In practical engineering? ATPG, to my understanding, requires multi-valued logic. Analysis of large phenomena and automated decision making becomes an order of magnitude simpler problems, with better efficiency over chosen metric. Temperature controllers, decisions based on photo-metering (autofocus, exposure adjustment), etc. Somehow, even with lack of readily-available building blocks and tooling, it turns out that there are problems people are motivated to solve from scratch, and MVL/FL comes handy. It's only stuff I overheard of through my life among bright engineers. 3. Moreover, the biggest impact is not in CS (as original presented paradox isn't, as well), it's on human judgment, decision-making and general assessment of reality, where "neither true nor false" (I don't know) is the first stepping stone to make the world much easier place to live in.
myWindoonn 8y ago
Please add some substance, or I will make your argument for you. Yes! We all need to get on board with constructivist mathematics [0][1] already. Construction is very similar to computation, and it is not inconsistent to take "all reals are computable" or "all functions are continuous", the same rules Turing discovered, as axioms if we like. We can therefore move computer science fully onto a foundation that is more rigorous than typical maths. [0] https://plato.stanford.edu/entries/mathematics-constructive/ https://plato.stanford.edu/entries/mathematics-constructive/ [1] https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016...
- ninegunpi 8y ago
  You've made far better one than me below. Hats off.

foldr 8y ago

The claim that the material implication analysis preserves the validity of valid arguments is pretty questionable. Consider the following argument: No student will succeed if he goofs off Every student will succeed |- No student will goof off Analyzing the 'if' in the first premise of the argument above as material implication, we get: For every student x, it's not the case that [x won't goof off or x will succeed]. = For every student x, x will goof off and x won't succeed. The following argument is valid only trivially, as the premises contradict each other (assuming the existence of at least one student): For every student x, x will goof off and x won't succeed. Every student will succeed |- No student will goof off I suspect it's quite easy to construct other similar examples involving quantifiers where validity is not even trivially preserved. (It's easier to find examples of the material implication analysis failing to preserve the invalidity of invalid arguments.)

a-nikolaev 8y ago
No student will succeed if he goofs off = Forall x: (GoofOff(x) -> not Succeed(x)) = not Exist x : (GoofOff(x) and Succeed(x))
- foldr 8y ago
  Sure, but then you're doing violence to the structure of the original sentence. (How did the negation get into the consequent?) If you're allowed free reign to paraphrase, then you can always get the right result. "No student will succeed if he goofs off" is a standard example discussed in the semantics literature, by the way. What you're pointing out, in effect, is that this sentence seems to mean "No student who goofs off will succeed". The problem is that it's unclear how to get to that interpretation given the actual syntactic structure of the sentence. In other words, you can't get there just by following a direction to interpret if...then... as material implication.
  - a-nikolaev 8y ago
    English works that way, this is not logic's fault.

dwheeler 8y ago

I created the "allsome" quantifier to reduce the risk of some of these confusions. Details here: https://dwheeler.com/essays/allsome.html https://dwheeler.com/essays/allsome.html

tr352 8y ago

Let me copy/paste my reply from the same discussion yesterday (https://news.ycombinator.com/item?id=18531650 https://news.ycombinator.com/item?id=18531650): There's another "solution" to this paradox: if we assert something we are guided by a set of "conversational principles". For example, asserting "X implies Y" if we know that X is false is inappropriate. If X is false, "not-X" would be the appropriate assertion. According to this theory, there's nothing wrong with the truth-functional meaning of "X implies Y". We just need to take into account what is implied by asserting "X implies Y", rather than e.g. "not-X", or "X and Y". Same with disjunction: "X or Y" is true if we know that X is true. However, if we assert "X or Y", it is implied that we're not certain that X is true, otherwise we would have used "X", which is the simplest way to convey what that fact. This is known as Grice's Pragmatic Defence of Truth-Functionality.

leoc 8y ago

Graham Priest's book is great: https://www.cambridge.org/ie/academic/subjects/philosophy/philosophy-science/introduction-non-classical-logic-if-2nd-edition https://www.cambridge.org/ie/academic/subjects/philosophy/ph...