Understanding soundness/completeness as "validity classification"?

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brendan.hill
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Joined: Thu Mar 09, 2017 5:28 am

Understanding soundness/completeness as "validity classification"?

I want to test my understanding of soundness and completeness. In particular, I will attempt to describe it in terms of *validity classification* where "positive" relates to "valid", and "negative" relates to "invalid", hence we can judge a classification to be one of (*true positive/false positive/true negative/false negative*). Finally I will consider some example scenarios where the notions of soundness and completeness are applied to draw conclusions (in particular with a specific proof method - proof trees).

#Definitions

Def. An **argument** is a statement of the form "(A and B and C and ... ) implies Z", where (A,B,C,...) is the set of premises and Z is the conclusion.

Def. **Validity** is a property of *arguments*, ie. a given argument is either valid or invalid. Roughly speaking, validity is the property that whenever the premises are true, the conclusion is also true.

Def. A **proof method** is a formal process for *classifying* a given argument as either valid or invalid. Examples of proof methods include proof trees, clausal resolution, an algorithm which examines every line of a truth table, etc.

Def. **Soundness** is a property of *proof methods*, ie. a given proof method is either sound or unsound. If a proof method is "sound", then whenever the method indicates that an argument is valid, we can be sure it is indeed valid. In other words, the proof method **never yields a "false positive"** (ie. incorrectly classifying an argument is valid when in fact it isn't).

Def. **Completeness** is a property of *proof methods*, ie. a given proof method is either complete or incomplete. If a proof method is "complete", then whenever the method indicates that an argument is invalid, we can be sure that it is indeed invalid. In other words, the proof method **never yields a "false negative"** (ie. incorrectly classifying an argument as invalid when it actually is valid).

Observations

- If a proof method is **sound but incomplete**, then I can trust its "valid" classifications but I should mistrust it's "invalid" classifications.

- If a proof method is **complete but unsound**, then I can trust its "invalid" classifications, but I should mistrust its "valid" classifications.

- If a proof method is **both unsound and incomplete**, then I should mistrust all of its classifications!

- However if a proof method is **sound and complete**, then I can always trust its classifications, as it classifies both valid and invalid arguments infallibly.

Example scenarios

Example 1

Suppose a Faulty Logician friend gave me a novel proof method. I use it on an argument which I already know is **actually invalid**, and yet the proof method **classifies it as valid**! This is a **false positive**, which never happens with *sound* proof method. Hence I can conclude that the proof method given to me is **unsound** (but know nothing about its completeness yet).

Example 2

Suppose this Faulty Logician handed me another novel proof method. I use it on an argument which I already know is **actually valid**, and yet the proof method **classifies it as invalid**! This is a **false negative**, which never happens in a *complete* proof method. Hence I can conclude that the proof method given to me is **incomplete** (but know nothing about its soundness yet).

(12-second recap of proof trees as a particular proof method)
Using proof trees as a proof method involves:

- Starting with the list of premises and the *negated* conclusion, ie. the set of statements (A,B,C,...,~Z)

- Using various (sound and complete) branching rules to grow the tree and close branches off (I will not attempt to describe here)

- If all branches of the tree for (A,B,C,...,~Z) close, then the original argument "A and B and C and ... implies Z" is classified as **valid**

- However if one or more branches of the tree for (A,B,C,...,~Z) are completely finished and yet remain open, then the original argument "A and B and C and ... implies Z" is classified as **invalid**

Example 3

Suppose my Faulty Logician friend turned his attention to proof trees, and invented a novel proof tree method with faulty rules such that the trees for (A,B,C...~Z) actually **stayed open more often** than they should (but never closed when they shouldn't)! This leads us to classify arguments as invalid more often than we should, hence the method yields **false negatives** (but never false positives). Of course, this doesn't happen with a *complete* proof method, so I conclude that the proof method is **incomplete** (but still sound).

Example 4

Suppose the Faulty Logician, a bit flustered now, invented yet another proof tree method with faulty rules such that the trees resulting from various roots (A,B,C...~Z) actually **closed more often** than they should (but never remained open when they shouldn't)! This leads us to classify some arguments as valid when we shouldn't, hence the method yields **false positives** (but never false negatives). Of course, this doesn't happen with a *sound* proof method, so I conclude that the proof method is **unsound** (but still complete).

Example 5

Suppose in a final fit of rage the Faulty Logician invented a proof tree method which faulty rules such that the trees resulting from various roots (A,B,C...~Z) sometimes **closed when they should remain open**, and sometimes **remained open when they should close**!
We sometimes incorrectly label arguments as valid, and sometimes incorrectly label arguments as invalid as well, yielding both **false positives** and **false negatives**. Thus the proof method lacks both soundness and completeness, hence I conclude that the proof method is **unsound and incomplete**.

Eodnhoj7
Posts: 1102
Joined: Mon Mar 13, 2017 3:18 am

Re: Understanding soundness/completeness as "validity classification"?

brendan.hill wrote:I want to test my understanding of soundness and completeness. In particular, I will attempt to describe it in terms of *validity classification* where "positive" relates to "valid", and "negative" relates to "invalid", hence we can judge a classification to be one of (*true positive/false positive/true negative/false negative*). Finally I will consider some example scenarios where the notions of soundness and completeness are applied to draw conclusions (in particular with a specific proof method - proof trees).

#Definitions

Def. An **argument** is a statement of the form "(A and B and C and ... ) implies Z", where (A,B,C,...) is the set of premises and Z is the conclusion.

Def. **Validity** is a property of *arguments*, ie. a given argument is either valid or invalid. Roughly speaking, validity is the property that whenever the premises are true, the conclusion is also true.

Def. A **proof method** is a formal process for *classifying* a given argument as either valid or invalid. Examples of proof methods include proof trees, clausal resolution, an algorithm which examines every line of a truth table, etc.

Def. **Soundness** is a property of *proof methods*, ie. a given proof method is either sound or unsound. If a proof method is "sound", then whenever the method indicates that an argument is valid, we can be sure it is indeed valid. In other words, the proof method **never yields a "false positive"** (ie. incorrectly classifying an argument is valid when in fact it isn't).

Def. **Completeness** is a property of *proof methods*, ie. a given proof method is either complete or incomplete. If a proof method is "complete", then whenever the method indicates that an argument is invalid, we can be sure that it is indeed invalid. In other words, the proof method **never yields a "false negative"** (ie. incorrectly classifying an argument as invalid when it actually is valid).

Observations

- If a proof method is **sound but incomplete**, then I can trust its "valid" classifications but I should mistrust it's "invalid" classifications.

- If a proof method is **complete but unsound**, then I can trust its "invalid" classifications, but I should mistrust its "valid" classifications.

- If a proof method is **both unsound and incomplete**, then I should mistrust all of its classifications!

- However if a proof method is **sound and complete**, then I can always trust its classifications, as it classifies both valid and invalid arguments infallibly.

Example scenarios

Example 1

Suppose a Faulty Logician friend gave me a novel proof method. I use it on an argument which I already know is **actually invalid**, and yet the proof method **classifies it as valid**! This is a **false positive**, which never happens with *sound* proof method. Hence I can conclude that the proof method given to me is **unsound** (but know nothing about its completeness yet).

The example is base upon an assumption and an undefinable variable (novel proof method) therefore the unification of approximates could theoretically lead to various answers.

Example 2

Suppose this Faulty Logician handed me another novel proof method. I use it on an argument which I already know is **actually valid**, and yet the proof method **classifies it as invalid**! This is a **false negative**, which never happens in a *complete* proof method. Hence I can conclude that the proof method given to me is **incomplete** (but know nothing about its soundness yet).

The same principle applies above. If you could give an example without assumption and an undefinable variable then your rationality would give to further definition. The whole basis of your rational statement is dependent on variables of irrationality to prove it.

(12-second recap of proof trees as a particular proof method)
Using proof trees as a proof method involves:

- Starting with the list of premises and the *negated* conclusion, ie. the set of statements (A,B,C,...,~Z)

- Using various (sound and complete) branching rules to grow the tree and close branches off (I will not attempt to describe here)

- If all branches of the tree for (A,B,C,...,~Z) close, then the original argument "A and B and C and ... implies Z" is classified as **valid**

- However if one or more branches of the tree for (A,B,C,...,~Z) are completely finished and yet remain open, then the original argument "A and B and C and ... implies Z" is classified as **invalid**

Example 3

Suppose my Faulty Logician friend turned his attention to proof trees, and invented a novel proof tree method with faulty rules such that the trees for (A,B,C...~Z) actually **stayed open more often** than they should (but never closed when they shouldn't)! This leads us to classify arguments as invalid more often than we should, hence the method yields **false negatives** (but never false positives). Of course, this doesn't happen with a *complete* proof method, so I conclude that the proof method is **incomplete** (but still sound).

Example 4

Suppose the Faulty Logician, a bit flustered now, invented yet another proof tree method with faulty rules such that the trees resulting from various roots (A,B,C...~Z) actually **closed more often** than they should (but never remained open when they shouldn't)! This leads us to classify some arguments as valid when we shouldn't, hence the method yields **false positives** (but never false negatives). Of course, this doesn't happen with a *sound* proof method, so I conclude that the proof method is **unsound** (but still complete).

Example 5

Suppose in a final fit of rage the Faulty Logician invented a proof tree method which faulty rules such that the trees resulting from various roots (A,B,C...~Z) sometimes **closed when they should remain open**, and sometimes **remained open when they should close**!
We sometimes incorrectly label arguments as valid, and sometimes incorrectly label arguments as invalid as well, yielding both **false positives** and **false negatives**. Thus the proof method lacks both soundness and completeness, hence I conclude that the proof method is **unsound and incomplete**.

In regards to the above example the "proof tree methods" are never really defined and because of this the arguments based around them would automatically prove themselves as true because they are essentially reflecting only against themselves, therefore propagative. What I am saying is the test you are using have no credibility, because of a deficiency in definition as to what "method" is being used. Because of this, they automatically propagate themselves because they are reflecting against a blank variable.

wtf
Posts: 405
Joined: Tue Sep 08, 2015 11:36 pm

Re: Understanding soundness/completeness as "validity classification"?

brendan.hill wrote:]
Def. **Completeness** is a property of *proof methods*, ie. a given proof method is either complete or incomplete. If a proof method is "complete", then whenever the method indicates that an argument is invalid, we can be sure that it is indeed invalid. In other words, the proof method **never yields a "false negative"** (ie. incorrectly classifying an argument as invalid when it actually is valid).
This seems a little murky to me. If a system is inconsistent, it will yield a proof or disproof of every proposition. Both a proof AND a disproof in fact. That makes it complete by the definition as I understand it, as in the sense of Gödel's theorems.

Whether you regard a proof of both P and not-P as a false negative or invalid, you'll have to say, because I don't really understand your terminology.

But an inconsistent system is always complete in the definition of the word as I understand it. Whether that conflicts with your ideas I can't say.

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