Describe similarities and differences between the usage of logic and accepted mo
ID: 2985512 • Letter: D
Question
- Describe similarities and differences between the usage of logic and accepted modes of "proof" in everyday language or non-mathematical contexts as compared to the formal proof methods and propositional logic you have learned about in this module.
- Explain the importance of formal logic and proof in modern mathematics.*
- What does it mean for an implication to be "trivially" or "vacuously" true? How do you understand this concept?*
- What is the difference between classifying an argument as "valid" or "invalid", versus classifying an argument and "sound" or "unsound"?*
- Is it possible to have a valid argument which is not sound?*
- Is it possible to have a sound argument which is not valid?*
- What about the possibility of an unsound, invalid argument?*
- What would it mean for an argument to be both valid and sound?*
- Be sure to provide some examples!
Explanation / Answer
It is easier to list the differences than similarities between the two kinds of logic.
Mathematical logic is the branch of mathematics that studies mathematical activity. It has all the usual properties of a mathematical branch. It uses the standard mathematical methods, such as the axiomatic method, informal set theory, and symbolic notation. It idealizes away from the actual situation by ignoring many aspects of actual mathematical activity, for example, it focuses mostly on how mathematical statements are proved and what they mean, but says little about conjectures, analogies, elegance, or about mathematics as a human activity. It phrases its results in terms of mathematical theorems (as opposed to, say, critical essays or historical studies).
Philosophical logic on the other hand attempts to attack its object of interest, which we could broadly characterize as reasoning, as a whole and from many different angles, as is customary in philosophy. Thus, apart from using the deductive method, we might consider linguistic aspects of logic, or logic as it relates to religion, we might learn something about logic by looking at its historical development, we may put it in the sociological context, etc. Consequently, no single treatment of logic will be accepted as a comprehensive one by (good) philosophers. In fact, it will be hard to get philosophers to agree on what precisely philosophical logic is.
It is naive to think of mathematical logic as being superior to philosophical logic. Certainly, mathematical logic is superior to philosophical logic in certain aspects, but one should never forget that the scope of mathematical logic is very narrow and it is therefore not surprising that mathematical logic boasts with deeper and technically more complicated insights than philosophical logic.
All arguments then can be classified as valid or invalid. If valid, they are sound or unsound. If invalid, they are strong or weak and then, depending on the premises, cogent or not cogent. Note that a strong argument by definition cannot be valid, and a valid argument by definition cannot be strong.
A sound argument is an argument that satisfies three conditions. -True premises -Unambiguous premises -Valid logic
If one of these conditions is unsatisfied then the argument is unsound, though in the case of ambiguous premises, not necessarily so.
The first, second and fourth arguments (depending of course, on who "I" is - it is being assumed it is the author!) are all sound. Here are some more:
5 Grass is green. The sky is blue. Snow is white. Therefore coal is black.
6 Grass is green. The sky is blue. Snow is white. Therefore pigs can fly.
7 2+2=4; hence 2+2=4
8 2+2=4; hence 2+2=5
Note that the truth value of the conclusion plays no role in determining whether an argument is sound; the only consideration is the premises.
A sound argument always has consistent premises. This must be the case, since there is a possible situation (namely reality) in which they are all true.
Of greater interest to the logician are valid arguments. A valid argument is an argument for which there is no possible situation in which the premises are all true and the conclusion is false.
Of the above arguments 2, 3 and 7 are valid. The reader should consider whether argument 1 is valid (read Meditations on First Philosophy by Descartes, chapters 1, 2).
It does not matter whether the premises or conclusion are actually true for an argument to be valid. All that matters is that the premises could not all be true and the conclusion false. Indeed, this means that an argument with inconsistent premises is always valid. There is no situation under which such an argument has all premises true and so there is no situation under which such an argument has all premises true and conclusion false. Hence it is valid. Similarly, an argument with a necessary conclusion can in no situation have all true premises and a false conclusion, since there is no situation in which the conclusion is false.
An argument with the single premise 'The conclusion is true.' is valid (regardless of the conclusion). An argument with the conclusion 'The premises are all true.' is also valid.
According to the definition of truth given previously, if the conclusion is false, its negation is true. Hence a valid argument can also be defined as an argument for which there is no possible situation under which the premises and the negation of the conclusion are all true. Hence, a valid argument is an argument such that the set of its premises and the negation of the conclusion is inconsistent. Such a set (the union of the set of premises and the set of the negation of the conclusion) is known as the counter-example set. It is called the counter-example set for the following reason: if a possible situation is found in which the members of this set are all true (and so the set is found to be consistent), this situation provides a counter-example to the arguments being valid, i.e. the existence of such a situation proves that the argument is not valid.
Counter-examples do not exist only for arguments, but also for statements:
Prime numbers are always odd 2 provides a counter-example (a number) to this statement. All animals have four legs Human beings provide a counter-example (a type of animal). Years are 365 days long Leap years provide a counter-example (a type of year). Years designated by a number divisible by four are leap years The year 1900 provides a counter-example (a particular year). It always rains in England A singularly sunny day in September (today, when written - a particular interval of time) provides a counter-example.
Counter-examples to declarative sentences refute their truth and are classes of things (thing being understood very broadly here) or particular things. Counter-examples to arguments refute their validity and are possible situations designated by sets of sentences (the counter-example set). Some clarification of the situation is often needed.
For example, take argument 4. It will be modified slightly as follows:
John has never had an accident; therefore, John is a safe driver.
It will be assumed here that accident means car accident and driver means motorist and safe means not liable to cause an accident. The counter-example set is:
John has never had an accident. John is not a safe driver.
Clarification by example: it may be that John has never driven in his life (and so never had an accident) because he is blind (and so cannot be considered a safe driver).
As mentioned, an inconsistent counter-example set implies that a conclusion is valid because it means that there is no situation under which the premises are all true and the conclusion is false (the negation of the conclusion is true).
Take argument 2; the counter-example set is:
All men are mortal. Socrates was a man. Socrates was not mortal.
These sentences cannot all be true at once. If Socrates was a man and he was not mortal, it could not be that all men are mortal. If all men are indeed mortal and Socrates was not mortal, he could not have been a man. If all men are mortal and Socrates was a man, he must have been mortal. Hence the counter-example set is inconsistent and the argument is valid.
Use a similar approach to show that arguments 3 and 7 are valid (and use it to consider argument 1). This method is known as reductio ad absurdum (which translates literally from Latin as "reduction to absurdity"). The negation of the conclusion is absurd given the truth of the premises and so the conclusion must be true.
The soundness of an argument is not determined by whether it is right or wrong, but by the relation of its premises to its conclusion: if all the premises are true, so is the conclusion. An argument that is found to be wrong may still be sound and valid.
While the ultimate goal of logic is to discover truth and recognize arguments that are right, a prerequisite is in developing arguments that are sound. To have the appearance of being right, while being unsound, is ultimately to fail.
It is easier to list the differences than similarities between the two kinds of logic.
Mathematical logic is the branch of mathematics that studies mathematical activity. It has all the usual properties of a mathematical branch. It uses the standard mathematical methods, such as the axiomatic method, informal set theory, and symbolic notation. It idealizes away from the actual situation by ignoring many aspects of actual mathematical activity, for example, it focuses mostly on how mathematical statements are proved and what they mean, but says little about conjectures, analogies, elegance, or about mathematics as a human activity. It phrases its results in terms of mathematical theorems (as opposed to, say, critical essays or historical studies).
Philosophical logic on the other hand attempts to attack its object of interest, which we could broadly characterize as reasoning, as a whole and from many different angles, as is customary in philosophy. Thus, apart from using the deductive method, we might consider linguistic aspects of logic, or logic as it relates to religion, we might learn something about logic by looking at its historical development, we may put it in the sociological context, etc. Consequently, no single treatment of logic will be accepted as a comprehensive one by (good) philosophers. In fact, it will be hard to get philosophers to agree on what precisely philosophical logic is.
It is naive to think of mathematical logic as being superior to philosophical logic. Certainly, mathematical logic is superior to philosophical logic in certain aspects, but one should never forget that the scope of mathematical logic is very narrow and it is therefore not surprising that mathematical logic boasts with deeper and technically more complicated insights than philosophical logic.
All arguments then can be classified as valid or invalid. If valid, they are sound or unsound. If invalid, they are strong or weak and then, depending on the premises, cogent or not cogent. Note that a strong argument by definition cannot be valid, and a valid argument by definition cannot be strong.
A sound argument is an argument that:
(1) Is valid, in that if the premises are true, then the conclusion must be true.
(2) The premises are true, or at least reasonable to believe.
So the difference between a sound and valid argument is that an argument that is valid may have false or absurd premises, but still have validity in its structure.
But a sound argument must not only have a valid structure (in that truth of the premises guarantee the truth of the conclusion) but also true (or reasonable) premises.
In this way, premises or conclusions are not, strictly speaking, sound or unsound, valid or invalid. They are either true or false, reasonable or unreasonable
Sound arguments
A sound argument is an argument that satisfies three conditions. -True premises -Unambiguous premises -Valid logic
If one of these conditions is unsatisfied then the argument is unsound, though in the case of ambiguous premises, not necessarily so.
The first, second and fourth arguments (depending of course, on who "I" is - it is being assumed it is the author!) are all sound. Here are some more:
5 Grass is green. The sky is blue. Snow is white. Therefore coal is black.
6 Grass is green. The sky is blue. Snow is white. Therefore pigs can fly.
7 2+2=4; hence 2+2=4
8 2+2=4; hence 2+2=5
Note that the truth value of the conclusion plays no role in determining whether an argument is sound; the only consideration is the premises.
A sound argument always has consistent premises. This must be the case, since there is a possible situation (namely reality) in which they are all true.
Valid arguments
Of greater interest to the logician are valid arguments. A valid argument is an argument for which there is no possible situation in which the premises are all true and the conclusion is false.
Of the above arguments 2, 3 and 7 are valid. The reader should consider whether argument 1 is valid (read Meditations on First Philosophy by Descartes, chapters 1, 2).
It does not matter whether the premises or conclusion are actually true for an argument to be valid. All that matters is that the premises could not all be true and the conclusion false. Indeed, this means that an argument with inconsistent premises is always valid. There is no situation under which such an argument has all premises true and so there is no situation under which such an argument has all premises true and conclusion false. Hence it is valid. Similarly, an argument with a necessary conclusion can in no situation have all true premises and a false conclusion, since there is no situation in which the conclusion is false.
An argument with the single premise 'The conclusion is true.' is valid (regardless of the conclusion). An argument with the conclusion 'The premises are all true.' is also valid.
According to the definition of truth given previously, if the conclusion is false, its negation is true. Hence a valid argument can also be defined as an argument for which there is no possible situation under which the premises and the negation of the conclusion are all true. Hence, a valid argument is an argument such that the set of its premises and the negation of the conclusion is inconsistent. Such a set (the union of the set of premises and the set of the negation of the conclusion) is known as the counter-example set. It is called the counter-example set for the following reason: if a possible situation is found in which the members of this set are all true (and so the set is found to be consistent), this situation provides a counter-example to the arguments being valid, i.e. the existence of such a situation proves that the argument is not valid.
Counter-examples do not exist only for arguments, but also for statements:
Prime numbers are always odd 2 provides a counter-example (a number) to this statement. All animals have four legs Human beings provide a counter-example (a type of animal). Years are 365 days long Leap years provide a counter-example (a type of year). Years designated by a number divisible by four are leap years The year 1900 provides a counter-example (a particular year). It always rains in England A singularly sunny day in September (today, when written - a particular interval of time) provides a counter-example.
Counter-examples to declarative sentences refute their truth and are classes of things (thing being understood very broadly here) or particular things. Counter-examples to arguments refute their validity and are possible situations designated by sets of sentences (the counter-example set). Some clarification of the situation is often needed.
For example, take argument 4. It will be modified slightly as follows:
John has never had an accident; therefore, John is a safe driver.
It will be assumed here that accident means car accident and driver means motorist and safe means not liable to cause an accident. The counter-example set is:
John has never had an accident. John is not a safe driver.
Clarification by example: it may be that John has never driven in his life (and so never had an accident) because he is blind (and so cannot be considered a safe driver).
As mentioned, an inconsistent counter-example set implies that a conclusion is valid because it means that there is no situation under which the premises are all true and the conclusion is false (the negation of the conclusion is true).
Take argument 2; the counter-example set is:
All men are mortal. Socrates was a man. Socrates was not mortal.
These sentences cannot all be true at once. If Socrates was a man and he was not mortal, it could not be that all men are mortal. If all men are indeed mortal and Socrates was not mortal, he could not have been a man. If all men are mortal and Socrates was a man, he must have been mortal. Hence the counter-example set is inconsistent and the argument is valid.
Use a similar approach to show that arguments 3 and 7 are valid (and use it to consider argument 1). This method is known as reductio ad absurdum (which translates literally from Latin as "reduction to absurdity"). The negation of the conclusion is absurd given the truth of the premises and so the conclusion must be true.
The soundness of an argument is not determined by whether it is right or wrong, but by the relation of its premises to its conclusion: if all the premises are true, so is the conclusion. An argument that is found to be wrong may still be sound and valid.
While the ultimate goal of logic is to discover truth and recognize arguments that are right, a prerequisite is in developing arguments that are sound. To have the appearance of being right, while being unsound, is ultimately to fail.
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