Here are a few examples of mnemonic devices that could be used to remember symbolic representations:
For the symbol ∀ (for all), you might use the phrase "For All Xs." This helps you remember that the symbol represents the idea of "for every" or "for all" in a statement.
For the symbol ∃ (there exists), you might use the phrase "Exists at least one." This helps you remember that the symbol represents the idea of "there is at least one" or "there exists" in a statement
For the symbol → (implies), you might use the phrase "If, then." This helps you remember that the symbol represents the idea of "if this is true, then that must also be true" in a statement.
For the symbol ∈ (belongs to), you might use the phrase "In set." This helps you remember that the symbol represents the idea of "belongs to" or "is a member of" a particular set.
Remember, these are just a few examples, and you may find that other mnemonic devices work better for you. The important thing is to find a device that helps you remember the meaning of the symbols and use it regularly as you practice and learn.
Saturday, December 31, 2022
Monday, December 26, 2022
A game theoretic representation of the three little pigs
It’s interesting having technology represent physically what one may have as a passing thought. Here's chatGPT giving an example of a bayesian game using the three little pigs.
Sure, we can represent the story of the three little pigs as a Bayesian game. In this game, the players are the wolf and the little pigs, and the payoffs to the players depend on the strategies chosen by the players and the state of the game.
To represent the story of the three little pigs as a Bayesian game, we can use a tuple (S,A,P,u), where S is the set of possible states of the game, A is the set of possible actions (or strategies) for each player, P is the probability distribution over the possible states of the game, and u is the utility function that assigns a payoff to each player for each combination of strategies and states.
Here is an example of the three little pigs represented as a Bayesian game:
S = {house of straw, house of sticks, house of bricks}
A = {blow, hide}
P = {0.4, 0.3, 0.3}
u(wolf,blow,house of straw) = 1
u(wolf,hide,house of straw) = 0
u(pig,blow,house of straw) = -1
u(pig,hide,house of straw) = 0
u(wolf,blow,house of sticks) = 1
u(wolf,hide,house of sticks) = 0
u(pig,blow,house of sticks) = -1
u(pig,hide,house of sticks) = 0
u(wolf,blow,house of bricks) = 0
u(wolf,hide,house of bricks) = 0
u(pig,blow,house of bricks) = 1
u(pig,hide,house of bricks) = 0
In this example, the set S represents the possible states of the game (the house of straw, the house of sticks, or the house of bricks), the set A represents the possible actions (or strategies) for each player (blow or hide), the probability distribution P represents the likelihood of each state occurring, and the utility function u assigns a payoff to each player for each combination of strategies and states.
For example, the utility function u(wolf,blow,house of straw) = 1 indicates that if the wolf chooses to blow and the pig is in the house of straw, the wolf gets a payoff of 1. Similarly, the utility function u(pig,hide,house of bricks) = 0 indicates that if the pig chooses to hide and is in the house of bricks, the pig gets a payoff of 0.
I hope this helps! Let me know if you have any other questions.
Sure, we can represent the story of the three little pigs as a Bayesian game. In this game, the players are the wolf and the little pigs, and the payoffs to the players depend on the strategies chosen by the players and the state of the game.
To represent the story of the three little pigs as a Bayesian game, we can use a tuple (S,A,P,u), where S is the set of possible states of the game, A is the set of possible actions (or strategies) for each player, P is the probability distribution over the possible states of the game, and u is the utility function that assigns a payoff to each player for each combination of strategies and states.
Here is an example of the three little pigs represented as a Bayesian game:
S = {house of straw, house of sticks, house of bricks}
A = {blow, hide}
P = {0.4, 0.3, 0.3}
u(wolf,blow,house of straw) = 1
u(wolf,hide,house of straw) = 0
u(pig,blow,house of straw) = -1
u(pig,hide,house of straw) = 0
u(wolf,blow,house of sticks) = 1
u(wolf,hide,house of sticks) = 0
u(pig,blow,house of sticks) = -1
u(pig,hide,house of sticks) = 0
u(wolf,blow,house of bricks) = 0
u(wolf,hide,house of bricks) = 0
u(pig,blow,house of bricks) = 1
u(pig,hide,house of bricks) = 0
In this example, the set S represents the possible states of the game (the house of straw, the house of sticks, or the house of bricks), the set A represents the possible actions (or strategies) for each player (blow or hide), the probability distribution P represents the likelihood of each state occurring, and the utility function u assigns a payoff to each player for each combination of strategies and states.
For example, the utility function u(wolf,blow,house of straw) = 1 indicates that if the wolf chooses to blow and the pig is in the house of straw, the wolf gets a payoff of 1. Similarly, the utility function u(pig,hide,house of bricks) = 0 indicates that if the pig chooses to hide and is in the house of bricks, the pig gets a payoff of 0.
I hope this helps! Let me know if you have any other questions.
Thursday, December 22, 2022
How definite descriptions improve critical thinking
Testing chatGPT and thought this collection of ideas was helpful. There are many questions to explore with this technology.
In symbolic logic, a definite description is a phrase of the form "the X" where X is a property or a set of properties. For example, "the tallest building" or "the prime numbers" are definite descriptions. The use of definite descriptions allows us to talk about unique objects or sets of objects, such as a specific building or a particular set of numbers, in a formal and precise way.
Here is an example of a definite description being used in symbolic logic:
Suppose we want to say that the capital city of France is Paris. We could express this statement using the definite description "the capital city of France" as follows:
Let C be the capital city of France. Therefore, C = Paris.
In this example, "the capital city of France" is a definite description that refers to a unique object, namely the city that is the capital of France. The statement "C = Paris" asserts that this unique object is in fact Paris. This allows us to make a precise and formal statement about the capital city of France using the symbolic language of logic.
Definite descriptions are often used in logical arguments to make precise and formal statements about unique objects or sets of objects. This can help improve critical thinking in several ways.
First, using definite descriptions allows us to be more precise and clear in our reasoning and arguments. This can help us avoid confusion or misunderstandings, and make it easier for others to follow our line of thinking.
Second, using definite descriptions forces us to be more precise and explicit in our assumptions and premises. In order to use a definite description, we need to clearly specify the properties that define the unique object or set of objects we are talking about. This can help us identify any assumptions or implicit premises in our arguments, and make sure they are stated explicitly and clearly.
Third, using definite descriptions can help us evaluate the validity of an argument by examining the relationship between the premises and the conclusion. In a logical argument that uses definite descriptions, the premises must be sufficient to support the conclusion. If the premises do not logically entail the conclusion, then the argument is not valid. This can help us improve our critical thinking skills by encouraging us to carefully evaluate the logical structure of an argument.
Here are some examples of how using definite descriptions can improve critical thinking in the ways described above:
1. Precise and clear reasoning:
Suppose we are having a discussion about the tallest building in the world. Without using definite descriptions, our conversation might go something like this:
Person 1: I heard that the tallest building in the world is in Dubai.
Person 2: No, I heard it's in Shanghai.
Person 1: But I read an article that said it's in Dubai.
Person 2: Well, I saw a documentary that said it's in Shanghai.
In this conversation, there is a lot of confusion and disagreement because it is not clear which building we are talking about. By using a definite description, we can be more precise and clear in our reasoning:
Person 1: I heard that the building with the highest architectural height is in Dubai.
Person 2: No, I heard that the building with the highest occupied floor is in Shanghai.
Person 1: Ah, I see what you're saying. So they are different buildings with different records.
Person 2: That's right.
In this revised conversation, the use of definite descriptions ("the building with the highest architectural height" and "the building with the highest occupied floor") helps us avoid confusion and have a more productive conversation.
2. Precise and explicit assumptions and premises:
Suppose we are trying to prove that all prime numbers are odd. Without using definite descriptions, our proof might go something like this:
Proof:
Let P be a prime number.
P is not even, because if it were, it could be divided by 2.
Therefore, P is odd.
In this proof, the second step is not explicitly stated, so it is not clear what the proof is based on. By using a definite description, we can be more precise and explicit in our assumptions and premises:
Proof:
Let P be a prime number.
Let E be the set of even numbers.
P is not an element of E, because if it were, it could be divided by 2.
Therefore, P is odd.
In this revised proof, the use of definite descriptions ("the set of even numbers" and "an element of E") helps us make our assumptions and premises explicit and clear.
3. Valid logical structure:
Suppose we are trying to prove that all triangles have three sides. Without using definite descriptions, our proof might go something like this:
Proof:
Let T be a triangle.
T has three sides.
Therefore, all triangles have three sides.
In this proof, the second step does not logically follow from the first step, so the argument is not valid. By using a definite description, we can make the logical structure of the argument more clear:
Proof:
Let T be a triangle.
Let S be the set of polygons with three sides.
T is an element of S.
Therefore, all triangles have three sides.
In this revised proof, the use of definite descriptions ("the set of polygons with three sides" and "an element of S") helps us make the logical structure of the argument more clear and valid.
In symbolic logic, a definite description is a phrase of the form "the X" where X is a property or a set of properties. For example, "the tallest building" or "the prime numbers" are definite descriptions. The use of definite descriptions allows us to talk about unique objects or sets of objects, such as a specific building or a particular set of numbers, in a formal and precise way.
Here is an example of a definite description being used in symbolic logic:
Suppose we want to say that the capital city of France is Paris. We could express this statement using the definite description "the capital city of France" as follows:
Let C be the capital city of France. Therefore, C = Paris.
In this example, "the capital city of France" is a definite description that refers to a unique object, namely the city that is the capital of France. The statement "C = Paris" asserts that this unique object is in fact Paris. This allows us to make a precise and formal statement about the capital city of France using the symbolic language of logic.
Definite descriptions are often used in logical arguments to make precise and formal statements about unique objects or sets of objects. This can help improve critical thinking in several ways.
First, using definite descriptions allows us to be more precise and clear in our reasoning and arguments. This can help us avoid confusion or misunderstandings, and make it easier for others to follow our line of thinking.
Second, using definite descriptions forces us to be more precise and explicit in our assumptions and premises. In order to use a definite description, we need to clearly specify the properties that define the unique object or set of objects we are talking about. This can help us identify any assumptions or implicit premises in our arguments, and make sure they are stated explicitly and clearly.
Third, using definite descriptions can help us evaluate the validity of an argument by examining the relationship between the premises and the conclusion. In a logical argument that uses definite descriptions, the premises must be sufficient to support the conclusion. If the premises do not logically entail the conclusion, then the argument is not valid. This can help us improve our critical thinking skills by encouraging us to carefully evaluate the logical structure of an argument.
Here are some examples of how using definite descriptions can improve critical thinking in the ways described above:
1. Precise and clear reasoning:
Suppose we are having a discussion about the tallest building in the world. Without using definite descriptions, our conversation might go something like this:
Person 1: I heard that the tallest building in the world is in Dubai.
Person 2: No, I heard it's in Shanghai.
Person 1: But I read an article that said it's in Dubai.
Person 2: Well, I saw a documentary that said it's in Shanghai.
In this conversation, there is a lot of confusion and disagreement because it is not clear which building we are talking about. By using a definite description, we can be more precise and clear in our reasoning:
Person 1: I heard that the building with the highest architectural height is in Dubai.
Person 2: No, I heard that the building with the highest occupied floor is in Shanghai.
Person 1: Ah, I see what you're saying. So they are different buildings with different records.
Person 2: That's right.
In this revised conversation, the use of definite descriptions ("the building with the highest architectural height" and "the building with the highest occupied floor") helps us avoid confusion and have a more productive conversation.
2. Precise and explicit assumptions and premises:
Suppose we are trying to prove that all prime numbers are odd. Without using definite descriptions, our proof might go something like this:
Proof:
Let P be a prime number.
P is not even, because if it were, it could be divided by 2.
Therefore, P is odd.
In this proof, the second step is not explicitly stated, so it is not clear what the proof is based on. By using a definite description, we can be more precise and explicit in our assumptions and premises:
Proof:
Let P be a prime number.
Let E be the set of even numbers.
P is not an element of E, because if it were, it could be divided by 2.
Therefore, P is odd.
In this revised proof, the use of definite descriptions ("the set of even numbers" and "an element of E") helps us make our assumptions and premises explicit and clear.
3. Valid logical structure:
Suppose we are trying to prove that all triangles have three sides. Without using definite descriptions, our proof might go something like this:
Proof:
Let T be a triangle.
T has three sides.
Therefore, all triangles have three sides.
In this proof, the second step does not logically follow from the first step, so the argument is not valid. By using a definite description, we can make the logical structure of the argument more clear:
Proof:
Let T be a triangle.
Let S be the set of polygons with three sides.
T is an element of S.
Therefore, all triangles have three sides.
In this revised proof, the use of definite descriptions ("the set of polygons with three sides" and "an element of S") helps us make the logical structure of the argument more clear and valid.
Monday, December 19, 2022
Symbolic Logic and Law
I was talking to chatGPT about logic. It was asked to show how symbolic logic is used in law. the AI provides great introductions to subjects. However, one still has to do one’s own research to understand them beyond a shallow level.
Symbolic logic is often used in law to help reason and argue more effectively and rigorously. In law, logical arguments are used to support or challenge the validity of legal principles, rules, and decisions. Symbolic logic provides a formal and precise language for representing and analyzing these arguments, which can help lawyers and judges evaluate their quality and validity.
One way in which symbolic logic is used in law is to represent and analyze the logical structure of legal arguments. In symbolic logic, we can represent the premises and conclusion of a legal argument using logical symbols and operators, such as the connectives "and," "or," and "if-then." This allows us to clearly and precisely express the logical relationships between the premises and the conclusion of the argument, and evaluate whether the argument is valid or invalid.
Another way in which symbolic logic is used in law is to identify and evaluate the assumptions and implications of a legal argument. In symbolic logic, we can represent the assumptions and implications of an argument using quantifiers and variables, such as "all" and "some." This allows us to clearly and precisely express the scope and implications of a legal argument, and evaluate whether the assumptions and implications are reasonable and consistent with the law.
Overall, the use of symbolic logic in law can help lawyers and judges reason and argue more effectively and rigorously, and improve their critical thinking skills. By using the tools and techniques of symbolic logic, they can better evaluate the quality and validity of legal arguments, and make more informed and reasonable legal decisions.
Here is an example of how symbolic logic might be used in a legal case:
Suppose a defendant is charged with the crime of robbery. The prosecution's case against the defendant is based on the following three pieces of evidence:
1. A witness saw the defendant leaving the scene of the robbery with a bag of money. 2. The defendant was arrested shortly after the robbery with a bag of money in his possession. 3. The victim of the robbery identified the defendant as the perpetrator.
In this case, the prosecution might use symbolic logic to represent and analyze the logical structure of their argument as follows:
1. Let W be the witness who saw the defendant leaving the scene of the robbery with a bag of money. 2. Let D be the defendant. 3. Let B be the bag of money. 4. Let A be the arrest of the defendant. 5. Let V be the victim of the robbery.
Premise 1: W saw D leaving the scene of the robbery with B. Premise 2: A occurred, and during A, D had B in his possession. Premise 3: V identified D as the perpetrator.
Conclusion: Therefore, D is guilty of robbery.
In this argument, the premises are statements 1-3, and the conclusion is statement 6. The relationship between the premises and the conclusion is that the premises provide evidence or support for the conclusion.
This argument can be evaluated for its logical validity by using the rules of symbolic logic. For example, we can use the rule of modus ponens to show that the premises logically imply the conclusion:
1. If W saw D leaving the scene of the robbery with B, and A occurred and during A, D had B in his possession, and V identified D as the perpetrator, then D is guilty of robbery. (This is the rule of modus ponens.) 2. W saw D leaving the scene of the robbery with B. (This is premise 1.) 3. A occurred and during A, D had B in his possession. (This is premise 2.) 4. V identified D as the perpetrator. (This is premise 3.)
Therefore, D is guilty of robbery. (This is the conclusion.)
In this way, symbolic logic can be used to represent and analyze the logical structure of a legal argument, and evaluate its validity. This can help lawyers and judges reason and argue more effectively and rigorously, and make more informed and reasonable legal decisions.
Symbolic logic is often used in law to help reason and argue more effectively and rigorously. In law, logical arguments are used to support or challenge the validity of legal principles, rules, and decisions. Symbolic logic provides a formal and precise language for representing and analyzing these arguments, which can help lawyers and judges evaluate their quality and validity.
One way in which symbolic logic is used in law is to represent and analyze the logical structure of legal arguments. In symbolic logic, we can represent the premises and conclusion of a legal argument using logical symbols and operators, such as the connectives "and," "or," and "if-then." This allows us to clearly and precisely express the logical relationships between the premises and the conclusion of the argument, and evaluate whether the argument is valid or invalid.
Another way in which symbolic logic is used in law is to identify and evaluate the assumptions and implications of a legal argument. In symbolic logic, we can represent the assumptions and implications of an argument using quantifiers and variables, such as "all" and "some." This allows us to clearly and precisely express the scope and implications of a legal argument, and evaluate whether the assumptions and implications are reasonable and consistent with the law.
Overall, the use of symbolic logic in law can help lawyers and judges reason and argue more effectively and rigorously, and improve their critical thinking skills. By using the tools and techniques of symbolic logic, they can better evaluate the quality and validity of legal arguments, and make more informed and reasonable legal decisions.
Here is an example of how symbolic logic might be used in a legal case:
Suppose a defendant is charged with the crime of robbery. The prosecution's case against the defendant is based on the following three pieces of evidence:
1. A witness saw the defendant leaving the scene of the robbery with a bag of money. 2. The defendant was arrested shortly after the robbery with a bag of money in his possession. 3. The victim of the robbery identified the defendant as the perpetrator.
In this case, the prosecution might use symbolic logic to represent and analyze the logical structure of their argument as follows:
1. Let W be the witness who saw the defendant leaving the scene of the robbery with a bag of money. 2. Let D be the defendant. 3. Let B be the bag of money. 4. Let A be the arrest of the defendant. 5. Let V be the victim of the robbery.
Premise 1: W saw D leaving the scene of the robbery with B. Premise 2: A occurred, and during A, D had B in his possession. Premise 3: V identified D as the perpetrator.
Conclusion: Therefore, D is guilty of robbery.
In this argument, the premises are statements 1-3, and the conclusion is statement 6. The relationship between the premises and the conclusion is that the premises provide evidence or support for the conclusion.
This argument can be evaluated for its logical validity by using the rules of symbolic logic. For example, we can use the rule of modus ponens to show that the premises logically imply the conclusion:
1. If W saw D leaving the scene of the robbery with B, and A occurred and during A, D had B in his possession, and V identified D as the perpetrator, then D is guilty of robbery. (This is the rule of modus ponens.) 2. W saw D leaving the scene of the robbery with B. (This is premise 1.) 3. A occurred and during A, D had B in his possession. (This is premise 2.) 4. V identified D as the perpetrator. (This is premise 3.)
Therefore, D is guilty of robbery. (This is the conclusion.)
In this way, symbolic logic can be used to represent and analyze the logical structure of a legal argument, and evaluate its validity. This can help lawyers and judges reason and argue more effectively and rigorously, and make more informed and reasonable legal decisions.
Friday, December 16, 2022
How are premises and conclusions different from assumptions and implications?
I was talking to chatgpt about logic and how it works. This question arose since in natural language premises and assumptions are similar, as well as conclusions and implications. These concepts are used in specific ways in logic. I thought the AI gave useful explanations and examples as to how they are different. More could be said about implication though.
In a logical argument, the premises are the reasons or evidence that are offered to support the conclusion. The conclusion is the main point or claim that the argument is trying to establish. The relationship between the premises and the conclusion is that the premises are supposed to provide evidence or support for the conclusion.
In contrast, assumptions and implications are not directly part of the argument itself, but are related to the argument in other ways. Assumptions are the background beliefs or conditions that are necessary for the argument to be valid or reasonable. Implications are the logical consequences or implications of the argument.
For example, consider the following argument:
Premise: All dogs are mammals.
Premise: Fido is a dog.
Conclusion: Therefore, Fido is a mammal.
In this argument, the premises are "All dogs are mammals" and "Fido is a dog," and the conclusion is "Fido is a mammal." The relationship between the premises and the conclusion is that the premises provide evidence or support for the conclusion.
Now suppose that one of the assumptions of this argument is that "mammals are a type of animal." This assumption is not explicitly stated in the argument, but it is necessary for the argument to be valid. Without the assumption that mammals are a type of animal, the conclusion "Fido is a mammal" would not be meaningful or relevant.
Finally, suppose that one of the implications of this argument is that "Fido is not a reptile." This implication is not directly stated in the argument, but it follows logically from the premises and conclusion. If we accept the premises and conclusion of the argument, then it logically follows that Fido cannot be a reptile, because reptiles are not mammals.
In summary, the difference between premise and conclusion vs. assumption and implication is that premises and conclusions are directly part of the argument, while assumptions and implications are related to the argument in other ways. Premises provide evidence or support for the conclusion, while assumptions are necessary for the argument to be valid, and implications are logical consequences of the argument.
In a logical argument, the premises are the reasons or evidence that are offered to support the conclusion. The conclusion is the main point or claim that the argument is trying to establish. The relationship between the premises and the conclusion is that the premises are supposed to provide evidence or support for the conclusion.
In contrast, assumptions and implications are not directly part of the argument itself, but are related to the argument in other ways. Assumptions are the background beliefs or conditions that are necessary for the argument to be valid or reasonable. Implications are the logical consequences or implications of the argument.
For example, consider the following argument:
Premise: All dogs are mammals.
Premise: Fido is a dog.
Conclusion: Therefore, Fido is a mammal.
In this argument, the premises are "All dogs are mammals" and "Fido is a dog," and the conclusion is "Fido is a mammal." The relationship between the premises and the conclusion is that the premises provide evidence or support for the conclusion.
Now suppose that one of the assumptions of this argument is that "mammals are a type of animal." This assumption is not explicitly stated in the argument, but it is necessary for the argument to be valid. Without the assumption that mammals are a type of animal, the conclusion "Fido is a mammal" would not be meaningful or relevant.
Finally, suppose that one of the implications of this argument is that "Fido is not a reptile." This implication is not directly stated in the argument, but it follows logically from the premises and conclusion. If we accept the premises and conclusion of the argument, then it logically follows that Fido cannot be a reptile, because reptiles are not mammals.
In summary, the difference between premise and conclusion vs. assumption and implication is that premises and conclusions are directly part of the argument, while assumptions and implications are related to the argument in other ways. Premises provide evidence or support for the conclusion, while assumptions are necessary for the argument to be valid, and implications are logical consequences of the argument.
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