When 'all' does not mean 'all'
The more I study language as a living instrument, the more I feel that a kind of veil is being removed from my eyes after many years of being 'blinded by computer science'.
Anyone who has been trained in mathematics, formal logic or computer science will tell you that 'all' in English translates to universal quantification. "All robots are artifacts" can be represented by
∀x P(x) ⇒ Q(x)
where P(x) denotes "x is a robot" and Q(x) denotes "x is an artifact".
I still remember taking an introductory Artificial Intelligence class at Stanford, taught by Michael Genesereth. He gave us the assignment of translating a bunch of (English) sentences into predicate logic. When pressed as to exactly how one did that (the students in the audience had no trouble coming up with weird cases), he finally burst out with something like "I can't explain it, just figure it out!" A hilarious moment in my AI education.
In the 1990's, Steven Sloman of Brown University got interested in how people used categories and applied categories in reasoning, and did a series of experiments on Brown students to examine their reasoning with what computer scientists call inheritance - and what the rest of academia apparently calls subordinate and superordinate categories. Setting aside the question of whether "Cognitive Psychology" should properly be called American Undergraduate Psychology, Prof. Sloman found that his subjects consistently undervalued the rules of logic, specifically reasoning of this form:
All x's are y's.
for all y's, P(y) is true
for all x's, P(x) is true
In Categorical inference is not a tree: The myth of inheritance hierarchies,(punning off Christopher Alexander's classic paper?) he presented subjects with facts and conclusions, and asked them how convincing they found the conclusions, given the facts. Example:
(G) Fact: All bodies of water have a high number of seiches.
Concl: All lakes have a high number of seiches.
The reasoning by inclusion in the example would be: All lakes are bodies of water therefore if something is true of all bodies of water, it is true of all lakes, therefore the conclusion is true given the premise.
As a good computer scientist, Sloman expected (G) to be assigned a confidence of 10 (out of 10). Instead, his subjects consistently gave confidences that averaged less than a perfect 10. And no matter how he manipulated the material, clarified the questions, and emphasized the syllogism, his subjects declined to assign absolute confidence to the conclusions.
Sloman's interpretation: People don't use the correct logical reasoning, even when it is highly accessible - there is a defect in human reasoning.
I have a different interpretation: Sloman was blinded by computer science, so he could not see the correct naturalistic reasoning his subjects were using. There are four key points:
1. 'All' is not understood by English speakers as universal quantification
2. 'inclusion' in categories (such as lakes being bodies of water) is not grounds for 100% confident deduction.
3. Unknown terms reduce certainty in reasoning
4. Different rules apply to real-world reasoning versus mathematical/logical reasoning.
Point 1. 'all' does not mean all.
Example: I return from shopping, place the grocery bags on the counter and ask my daughter to "please put all the eggs in the refrigerator." I make a phone call and return to the kitchen. My daughter says "One of the eggs was cracked so I threw it away, and I hid that chocolate Easter egg in the rice cooker."
Q: Am I surprised, disappointed, or angry that she didn't do as I asked? She did not put 'all' the eggs in the refrigerator!
A: No, because she did as I asked. She understood my phrase "all the eggs" as meaning "all the eggs that need to be refrigerated and are useable." In fact, that's how we commonly use and understand "all". It means something like "all members of the specified class that are applicable in context, except for surprises and cases where it doesn't make sense in context." 'All' comes very close to meaning 'all the ones you think it makes sense for me to mean.' In casual speech, 'all' has very much this sense. Joke: "Did you eat all the cookies?" "No! I left one..." Why is this funny? I think it's partly because the word 'all' in the question means something like "all the ones I didn't mean for you to eat." Note that the joke doesn't strike us as nonsensical, even though both parties know that 'all' the cookies weren't eaten, because 'all' doesn't mean all.
Point 2. A lake is not always a lake.
Contraty to Sloman's unquestioned assumption, all (in the mathematical sense) lakes are not bodies of water. Take a look at Wikipedia: Dry lake. Also Alvord Lake (Oregon) - a seasonal lake. Finally: Lakes of titan. In all cases, these are lakes (hence the use of the noun 'lake'...), but they are not bodies of water. Titan's lakes are bodies - but of ethane/methane. A dry lake has no water in it, at least most of the time. A seasonal lake has water only some parts of the year. For humans, native speakers of English, these are all 'lakes' - subcategories of the cognitive and linguistic category lake, but not subcategories of the Aristotelian category. The prototypical lake is a body of water, and Prof. Sloman makes a classic error of human reasoning in treating the protoype of a category as the category, and then treating the category as Aristotelian (defined by attributes) instead of cognitive.
Point 3. Traditional logical syllogisms don't extend to arbitrary predicates
Sloman's question (G) uses the phrase "a high number of seiches". The word 'seiche' and other words in his experiment were chosen specifically because they were expected to be meaningless to most of his subjects, so the experimental results would measure syllogistic reasoning with categories, instead of e.g. commonsense or domain-specific knowledge.
But because of the tremendous flexibility of language, especially the reflexive and meta-discursive aspects of language, the introduction of an unknown noun like 'seiches' introduces more uncertainty than you might expect (if you were a computer scientist.) Let's consider "All bodies of water have a high number of seiches" again.
What if "has a high number of seiches" means "has another, basic-level category label."
(G') Fact: All bodies of water have another, basic-level category label.
Concl: All lakes have another, basic-level category label
Whatever it means, I'm pretty sure the Conclusion doesn't follow from the Fact, even if we assume that all lakes are bodies of water. That's because the Fact is both self-referential and meta-discursive. In the absence of any knowledge of what 'seiches' are, a reader is correct to be cautious in extending a logical syllogism to an unknown predicate.
Point 4. People use different reasoning processes in the context of real-world facts versus mathematical/logical facts.
If Sloman's questions had used mathematical or logically-defined concepts and if some substantial fraction of his subjects had mathematical training, I believe his results would have been different. When we ask about the integers, or set theory, or the rules of chess, I think most college students understand that we are talking about formal systems with 'perfect' definitions and reasoning systems. In that universe, 'all' can mean all, categories are Aristotelian, and logical syllogism can produce conclusions that are 100% convincing.
If all real numbers have additive inverses, then all integers have additive inverses.
On the other hand, when we talk about lakes and hammers and Snickers bars, people know we are talking about the real world and (presumably) real-world concepts and categories. And in this universe, 'all' does not mean all, definitions are not perfect, properties are not necessarily shared by all members of a category nor inherited by all subordinate categories, unknown words and phrases can be ontological land mines, and logical syllogisms must be used with caution.
Anyone who has been trained in mathematics, formal logic or computer science will tell you that 'all' in English translates to universal quantification. "All robots are artifacts" can be represented by
∀x P(x) ⇒ Q(x)
where P(x) denotes "x is a robot" and Q(x) denotes "x is an artifact".
I still remember taking an introductory Artificial Intelligence class at Stanford, taught by Michael Genesereth. He gave us the assignment of translating a bunch of (English) sentences into predicate logic. When pressed as to exactly how one did that (the students in the audience had no trouble coming up with weird cases), he finally burst out with something like "I can't explain it, just figure it out!" A hilarious moment in my AI education.
In the 1990's, Steven Sloman of Brown University got interested in how people used categories and applied categories in reasoning, and did a series of experiments on Brown students to examine their reasoning with what computer scientists call inheritance - and what the rest of academia apparently calls subordinate and superordinate categories. Setting aside the question of whether "Cognitive Psychology" should properly be called American Undergraduate Psychology, Prof. Sloman found that his subjects consistently undervalued the rules of logic, specifically reasoning of this form:
All x's are y's.
for all y's, P(y) is true
for all x's, P(x) is true
In Categorical inference is not a tree: The myth of inheritance hierarchies,(punning off Christopher Alexander's classic paper?) he presented subjects with facts and conclusions, and asked them how convincing they found the conclusions, given the facts. Example:
(G) Fact: All bodies of water have a high number of seiches.
Concl: All lakes have a high number of seiches.
The reasoning by inclusion in the example would be: All lakes are bodies of water therefore if something is true of all bodies of water, it is true of all lakes, therefore the conclusion is true given the premise.
As a good computer scientist, Sloman expected (G) to be assigned a confidence of 10 (out of 10). Instead, his subjects consistently gave confidences that averaged less than a perfect 10. And no matter how he manipulated the material, clarified the questions, and emphasized the syllogism, his subjects declined to assign absolute confidence to the conclusions.
Sloman's interpretation: People don't use the correct logical reasoning, even when it is highly accessible - there is a defect in human reasoning.
I have a different interpretation: Sloman was blinded by computer science, so he could not see the correct naturalistic reasoning his subjects were using. There are four key points:
1. 'All' is not understood by English speakers as universal quantification
2. 'inclusion' in categories (such as lakes being bodies of water) is not grounds for 100% confident deduction.
3. Unknown terms reduce certainty in reasoning
4. Different rules apply to real-world reasoning versus mathematical/logical reasoning.
Point 1. 'all' does not mean all.
Example: I return from shopping, place the grocery bags on the counter and ask my daughter to "please put all the eggs in the refrigerator." I make a phone call and return to the kitchen. My daughter says "One of the eggs was cracked so I threw it away, and I hid that chocolate Easter egg in the rice cooker."
Q: Am I surprised, disappointed, or angry that she didn't do as I asked? She did not put 'all' the eggs in the refrigerator!
A: No, because she did as I asked. She understood my phrase "all the eggs" as meaning "all the eggs that need to be refrigerated and are useable." In fact, that's how we commonly use and understand "all". It means something like "all members of the specified class that are applicable in context, except for surprises and cases where it doesn't make sense in context." 'All' comes very close to meaning 'all the ones you think it makes sense for me to mean.' In casual speech, 'all' has very much this sense. Joke: "Did you eat all the cookies?" "No! I left one..." Why is this funny? I think it's partly because the word 'all' in the question means something like "all the ones I didn't mean for you to eat." Note that the joke doesn't strike us as nonsensical, even though both parties know that 'all' the cookies weren't eaten, because 'all' doesn't mean all.
Point 2. A lake is not always a lake.
Contraty to Sloman's unquestioned assumption, all (in the mathematical sense) lakes are not bodies of water. Take a look at Wikipedia: Dry lake. Also Alvord Lake (Oregon) - a seasonal lake. Finally: Lakes of titan. In all cases, these are lakes (hence the use of the noun 'lake'...), but they are not bodies of water. Titan's lakes are bodies - but of ethane/methane. A dry lake has no water in it, at least most of the time. A seasonal lake has water only some parts of the year. For humans, native speakers of English, these are all 'lakes' - subcategories of the cognitive and linguistic category lake, but not subcategories of the Aristotelian category. The prototypical lake is a body of water, and Prof. Sloman makes a classic error of human reasoning in treating the protoype of a category as the category, and then treating the category as Aristotelian (defined by attributes) instead of cognitive.
Point 3. Traditional logical syllogisms don't extend to arbitrary predicates
Sloman's question (G) uses the phrase "a high number of seiches". The word 'seiche' and other words in his experiment were chosen specifically because they were expected to be meaningless to most of his subjects, so the experimental results would measure syllogistic reasoning with categories, instead of e.g. commonsense or domain-specific knowledge.
But because of the tremendous flexibility of language, especially the reflexive and meta-discursive aspects of language, the introduction of an unknown noun like 'seiches' introduces more uncertainty than you might expect (if you were a computer scientist.) Let's consider "All bodies of water have a high number of seiches" again.
What if "has a high number of seiches" means "has another, basic-level category label."
(G') Fact: All bodies of water have another, basic-level category label.
Concl: All lakes have another, basic-level category label
Whatever it means, I'm pretty sure the Conclusion doesn't follow from the Fact, even if we assume that all lakes are bodies of water. That's because the Fact is both self-referential and meta-discursive. In the absence of any knowledge of what 'seiches' are, a reader is correct to be cautious in extending a logical syllogism to an unknown predicate.
Point 4. People use different reasoning processes in the context of real-world facts versus mathematical/logical facts.
If Sloman's questions had used mathematical or logically-defined concepts and if some substantial fraction of his subjects had mathematical training, I believe his results would have been different. When we ask about the integers, or set theory, or the rules of chess, I think most college students understand that we are talking about formal systems with 'perfect' definitions and reasoning systems. In that universe, 'all' can mean all, categories are Aristotelian, and logical syllogism can produce conclusions that are 100% convincing.
If all real numbers have additive inverses, then all integers have additive inverses.
On the other hand, when we talk about lakes and hammers and Snickers bars, people know we are talking about the real world and (presumably) real-world concepts and categories. And in this universe, 'all' does not mean all, definitions are not perfect, properties are not necessarily shared by all members of a category nor inherited by all subordinate categories, unknown words and phrases can be ontological land mines, and logical syllogisms must be used with caution.