Let's Learn Some Logic

  • the law of identity (A=A)
  • the law of non-contradiction (A does not equal ~A)
  • and the law of the excluded middle (either A or not A but not both A and ~A)

So in what senses/respects are these laws universal, absolute and relative?
What happens if we use them on everything, something, something else and/or nothing?

Off the top of my head I can't think of any examples of something contradicting law #1 or law #2. Not sure about the law of the exluded middle. According to Wiki:

Many modern logic systems reject the law of excluded middle, replacing it with the concept of negation as failure. That is, there is a third possibility: the truth of a proposition is unknown. A classic example illustrating the difference is the proposition: "It is not safe to cross the railroad tracks when one knows a train is coming". One should not deduce it is safe to cross the tracks if one doesn't know a train is coming. The principle of negation-as-failure is used as a foundation for autoepistemic logic, and is widely used in logic programming. In these systems, the programmer is free to assert the law of excluded middle as a true fact; it is not built-in a priori into these systems.

http://en.wikipedia.org/wiki/Law_of_excluded_middle
 
Well, these laws of thought/logic hold for something, but there is a limit to something versus everything.
"The same attribute cannot at the same time belong and not belong to the same subject and in the same respect." - Metaphysics G, 3,1005b18-20)
Note same attribute, time, subject and respect. The limit to this was already noted by the old Greeks - If something is as it is then how can it change and still be it? In other words logic only hold for the same thing as "frozen for" attribute, time, subject and respect. Take A is A, A=A, A⇔A and A≡A. All these examples "face" the same problem - the first A is not the second A for all attributes, times, subjects and respects; i.e. they exist differently.
Off the top of my head I can't think of any examples of something contradicting law #1 or law #2. ...
I can because A=A and A does not equal ~A are contradicted by everything.
  • Reality is not a word.
  • Reality is only a word.
  • Reality is also a word.
 
I can because A=A and A does not equal ~A are contradicted by everything.
  • Reality is not a word.
  • Reality is only a word.
  • Reality is also a word.

Are any of the examples above examples of A=A or else of A does not equal ~A? It seems to me that you have three examples of A=B.

A=A would be Reality = Reality and Reality does not = ~Reality would be the law of non contradiction.

Therefore, essentially an example of the law of non-contradiction would be: something is either real or it is not real but cannot be both real and not real at the same time.

The law of identity states that: Reality is reality.
 
Are any of the examples above examples of A=A or else of A does not equal ~A? It seems to me that you have three examples of A=B.

A=A would be Reality = Reality and Reality does not = ~Reality would be the law of non contradiction.

Therefore, essentially an example of the law of non-contradiction would be: something is either real or it is not real but cannot be both real and not real at the same time.

The law of identity states that: Reality is reality.

"Now "why a thing is itself" is a meaningless inquiry (for—to give meaning to the question 'why'—the fact or the existence of the thing must already be evident—e.g., that the moon is eclipsed—but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical, unless one were to answer, 'because each thing is inseparable from itself, and its being one just meant this.' This, however, is common to all things and is a short and easy way with the question.)" - Metaphysics, Book VII, Part 17
"The same attribute cannot at the same time belong and not belong to the same subject and in the same respect." - Metaphysics G, 3,1005b18-20)

Okay, so is reality a thing/subject in the singular sense or are reality the sum of all cases in time and over time for same attribute, time, subject/thing and respect?

In other words if you treat reality is as a singular thing only in the present time then I, as a part of this singular reality, am reality and you are not you, but me. Hell, I haven't changed since I was born, since reality is.

Further:
You - Reality is reality.
Me - No!
Now please explain the real, metaphysical/ontological, epistemological and logical status of this no and how it is in relationship to reality.
 
Okay, so is reality a thing/subject in the singular sense or are reality the sum of all cases in time and over time for same attribute, time, subject/thing and respect?

In other words if you treat reality is as a singular thing only in the present time then I, as a part of this singular reality, am reality and you are not you, but me. Hell, I haven't changed since I was born, since reality is.

Further:
You - Reality is reality.
Me - No!
Now please explain the real, metaphysical/ontological, epistemological and logical status of this no and how it is in relationship to reality.

Do I follow correctly that you are saying that because things change over time therefore A=A is not always the case because "reality now" might be different from "reality 5 minutes from now"?

Again aren't we dealing with two different things or essentially dealing with A=B? "Reality now" and "reality later" is NOT an example of A=A. One is one thing and the other is something different. By definition reality now is not the same as reality later if you are positing them as being different. A=A simply means all things are identical with themselves. Reality now would be identical with reality now and reality later would be identical with reality later.

Regardless of what reality is, whatever it is, it is most certainly itself. Meaning that which you refer to as reality (whatever it may be) is identical with itself. Granted this is not saying much of anything earthshaking it would seem but from my understanding all logic has A=A within its ultimate foundation.

Otherwise I'm not sure of what you mean. :confused:

EDIT: Also "reality" is a very vague term and we should probably be careful of the possibility of a fallacy of equivocation here or some form of amphiboly.
 
1.3 DEDUCTION AND INDUCTION

Generally speaking a deductive argument is one in which the conclusion follows necessarily from the premises and an inductive argument is one in which the conclusion follows probably from the premises. If the premises of an argument support the conclusion in such a way that it cannot be the case the conclusion is false, then we say it is a deductive argument.

Example of a deductive argument:

Socrates is a man
All men are mortal
Therefore Socrates is mortal​

Example of an inductive argument:

Every time it thunders my dog runs under the table and starts shaking.
Therefore if it thunders tonight my dog will probably run under the table and start shaking.​

In the later example of an inductive argument it does not necessarily follow that if it thunders tonight my dog will run under the table and start shaking but rather it is probably a good assumption.

Part I. The following are some argument forms which are typically considered inductive:

A prediction is typically inductive. It takes an occurance in the past or present and uses that occurance to make a prediction about a future event.

An argument from analogy depends upon similarities between two things to derrive a conclusion. An example of an argument from analogy would be to argue that since my Honda Civic gets good gas mileage that your Honda Civic must also get good gas mileage. Such a conclusion obviously does not follow necessarily from the premise but it is rather probably a reasonable assumption.

In an inductive generalization one makes a conclusion about a group of things based upon what is known about a sample of that group. For instance if 9 out of 10 dentists surveyed recommend a particular toothpaste, then it may be reasonable to make the assumption that a similar ratio might persist if 100 dentists were surveyed. Obviously it is possible that a particular part of a group may not be a good representative sample and therefore such an argument is inductive and not deductive.

An argument from authority makes the assumption that if someone is an authority on a topic that it is likely that what that person states about that topic has a good probability of being true. So for instance if one argues that because most climatologists believe in man made accelerated global climate change that therefore man made accelerated global climate change is probably the case, then one is making an argument from authority. Obviously the majority of climatologists could be wrong and it does not follow by necessity that they are correct but it is still considered to be a relatively strong inductive argument.

Another inductive form of argument is an argument based on signs. An example of such an argument would be to make the assumption that because a person is showing high blood pressure that therefore the person is more likely to have a heart attack than someone with lower blood pressure. Again, it does not follow necessarily that such a person is more likely to have a heart attack but it could be a reasonable assumption.

A causal inference involves deriving a conclusion based upon a known cause and effect relationship. So for instance I may conclude that because I put a can of cola in the freezer overnight it probably exploded. It is possible that for some reason the can has not exploded but it is probably a reasonable induction that it has.​

Part II. Some examples of deductive arguments are:

Arguments based on mathematics are generally considered deductive. For instance I may deductively conclude that because two of the angles of a triangle add up to 100 degrees therefore the third angle must be 80 degrees. An exception to this are arguments based upon statistics.

Arguments from definition derive their conclusion based upon a definition in the premise(s). For instance I may conclude that because X is a bachelor therefore X is an unmarried man.​

A syllogism is defined as an argument which contains exaclty 2 premises and one conclusion. Deductive forms of syllogism include:

Categorical syllogisms are almost always treated as deductive arguments and begin each statement with either "all" "some" or "no".

Example:

Socrates is a man
All men are mortal
therefore Socrates is mortal.​

Hypothetical syllogisms are syllogisms containing conditional statements in at least one of its premises.

Example:

If Socrate is a man, then he is mortal.
If someone is mortal, then it means they will die some day.
Therefore If Socrates is a man, then he will die some day.

or

If Socrates is a man, then he is mortal
Socrates is a man
Therefore Socrates is mortal

Disjunctive syllogisms are syllogisms containing a disjunctive statement in at least one of it's premises. A disjunctive statement is basically an "either...or..." statement.

Example:

Either Socrates is immortal or he will die some day.
Socrates is not immmortal
Therefore Socrates will die some day.​
 
EXERCISES FOR SECTION 1.3 (Hurley, pp. 35-38)

Directions from the book: Determine whether the following arguments are best interpreted as being inductive or deductive. Also state the criteria you use in reaching your decision (i.e. the presence of indicator words, the nautre of the inferential link between premises and conclusion, or the character or form of argumentation.)

NOTE: Some of these were particularly tough for me to define, especially the ones regarding physics and planetary motion. Are such arguments deductive arguments from mathematics or inductive causal inferences? I'm especially interested here in hearing others' interpretations of them.

#2. The plague on the leaning tower of Pisa says that Galileo performed experiments there with falling objects. It must be the case that Galileo did indeed perform those experiments there.

My answer:
Spoiler :
Inductive argument, argument based on signs


#3. The rainfall in Seattle has been more than 15 inches every year for the last thirty years. Therefore, the rainfall next year will probably be more than 15 inches.

My answer:
Spoiler :
Inductive argument, prediction


#6. The longer a pendulum is, the longer it takes to swing. Therefore, when the pendulum of a clock is lengthened, the clock will slow down.

My answer:
Spoiler :
Inductive argument, causal inference


#9. Although both front and rear doors were found open after the burglary, there were pry marks around the lock on the rear door and deposits of mud near the threshold. It must be the case that the thief entered thorugh the rear door and left through the front.

Your turn to answer...

#11. Cholesteral is endogenous with humans. Therefore, it is manufactured inside the human body.

My answer:
Spoiler :
Deductive argument, argument from definition.


#14. If cigarette smoking costs $65 billion annually in health care and lost productivity, then cigarette taxes should be raised to reflect this cost. But cigarette smoking does cost $65 billion annually in health care and lost productivity. Therefore, cigarette taxes should be raised to reflect this cost.

Your turn to answer...

#17. Because the apparent daily movement which is common to both the planets and the fixed stars is seen to travel from the east to the west, but the far slower single movemets of the single planets travel in the opposite direction from west to east, it is therefore certain that these movements cannot depend on the common movement of the world but should be assigned to the planets themselves.
(Johannes Kepler, Epitomy of Copernican Astronomy)

My answer:
Spoiler :
Deductive, argument from mathmatics, clue words: "therefore certain".


#18. Reserves of coal in the United States have an energy equivalent of 33 times that of oil and natural gas. On a world-wide basis the multiple is about 10. By shifting to a coal-based economy, we could satisfy our energy requirements for at least a century, probably longer.
(William L. MAsterson and Emil J. Slowinski, Principles of Chemistry)

My answer:
Spoiler :
Inductive, prediction, key word: "probably".


#21. That [the moons of Juupiter] revolve in unequal circles is manifestly deduced from the fact that at the longest elongation from Jupiter it is never possible to see two of these moons in conjunction, whereas in the vicinity of Jupiter they are found united two, three, and sometimes all four together.
(Galileo Galilei, The Starry Messenger)

Your turn to answer...

#24. The interpretation of the laws is the proper and peculiar province of the courts. A constitution is, in fact, and must be regarded by the judges, as a fundamental law. It therefore belongs to them to ascertain its meaning, as well as the meaning of any particular act proceeding from the legislative body.
(Alexander Hamilton, Federalist Papers, No. 78)

My answer:
Spoiler :
Inductive, argument from authority. I think here we are relying on perception of Hamilton as an authority on government and legislation to validate the argument above. Not 100% on that though.


#27. Ordinary things that we encounter every day are electrically neutral. Therefore, since negatively charged electrons are a part of everything, positively charged particles must also exist in all matter.
(James E. Brady and Gerard E. Humiston, General Chemistry)

Your turn to answer... (this is another tough one in my opinion.)

#30. Because the moon moves relative to the earth so that it returns to the same position overhead after about 25 hours, there are two high and two low tides at any point ever 25 hour.
(Douglas C. Giancoli, The Ideas of Physics, 3rd ed.)

My answer:
Spoiler :
Inductive, causal inference, the conclusion does not follow necessarily from the premise.


 
Therefore, essentially an example of the law of non-contradiction would be: something is either real or it is not real but cannot be both real and not real at the same time.

Okay, I will try from another angle, because it might have some part in how we each individually understand these 3 laws of thought/logic.
What does - it is not real - mean? What is not real? How do you know that something is not real?
 
Okay, I will try from another angle, because it might have some part in how we each individually understand these 3 laws of thought/logic.
What does - it is not real - mean? What is not real? How do you know that something is not real?

Honestly I don't have too much of a clue as to a singular, most correct, meaning of "real" or "not real". The term "real" appears to be used in some different senses. I may say "this diamond is 'not real' but a 'fake'." Essentially meaning that it is not what it is supposed to be. Or, "this vision is a mirage and not 'real'." Meaning something along the lines of what does or doesn't exist. I would venture to say that in many cases it is difficult to know if something is "real" or "not real". This is surely a matter of epistemology. Prima facie, I don't see a connection between knowing what is real or unreal and the law of identity so I'll await your further explanation.

But, for sake of argument and to give you an answer you have requested, let's go ahead and say that generally the way I know that something is not real is by inference from other data. So for instance I may know something is a mirage by walking toward it only to have it disappear in front of me. Or I may know a diamond is not "real" by using the methods of a jewler to assertain its "realness".
 
Okay, here it comes. It is a variant of Descartes' evil demon and Bertram Russell's five minute hypothesis. Ask yourself this - can I know whether I am as I am or I am a computer simulation turned on five minutes ago and which will be turned off in five minutes.
In the abstract sense we got this -
Reality is either as it seems or it is something else. R is either R or not R.
I am either as I seem or I am something else. I am either I or not I.
Something(A) is either A or not A.

Now how do you know that you are A or not A?
 
#6. Shut the cage door you fool! The lions are escaping into the streets!

By Hurley's definition of "argument" this can't be one, since the would-be conclusion ("shut the cage door!") is not a proposition. It's an imperative, as lovett pointed out. But to my mind, this just points to a flaw in the definition. And plenty of philosophers have thought that imperatives can appear in arguments, including the conclusion.
 
Okay, here it comes. It is a variant of Descartes' evil demon and Bertram Russell's five minute hypothesis. Ask yourself this - can I know whether I am as I am or I am a computer simulation turned on five minutes ago and which will be turned off in five minutes.
In the abstract sense we got this -
Reality is either as it seems or it is something else. R is either R or not R.
I am either as I seem or I am something else. I am either I or not I.
Something(A) is either A or not A.

Now how do you know that you are A or not A?

Does the fact of me NOT knowing without remote possibility of doubt serve as a case that refutes either the law of identity or the law of non-contradiction though? So for instance Descartes imagined that 2 + 3 might equal something other than 5 in his thought experiment of the evil genius. Perhaps an evil genius is tricking him into thinking that 2 + 3 equals 5 when in fact it does not. Does this disprove the theorems of math? Do you therefore think we have proven that 2 + 3 may not equal 5 because we can perhaps hypothesize an evil genius having the power to deceive us about something which seems obvious to us? This seems like a bit of a stretch. At the very best maybe one can raise possibility of doubt but probably not good reason to justify doubt. I don't think it can be said that Descartes overturned the basic axioms of math in his meditations.

Descartes believed that his radical doubt was a once in a lifetime endeavor to discover what one can know with absolute certainty. He didn't suggest one should spend all his or her time doubting everything. That is pretty much impossible. I couldn't walk a step if I doubted the floor was really there to support me from falling into an endless abyss.

By Hurley's definition of "argument" this can't be one, since the would-be conclusion ("shut the cage door!") is not a proposition. It's an imperative, as lovett pointed out. But to my mind, this just points to a flaw in the definition. And plenty of philosophers have thought that imperatives can appear in arguments, including the conclusion.

Hi Ayatollah So,

Thanks for sticking with the thread. It was getting a little lonely here other than the valliant ongoing efforts of Global Skeptic to undermine the foundations of logic on me. :)

This one is considerably over my head.

I certainly don't have any knowledge of "imperative logic". That is something new to me. If Hurley is correct and a logical argument is meant to prove something, then do imperatives purport to prove anything? Unfortunately the book seems to leave us mostly with classical logic as it is an "introductory" book. I tried looking up "imperative logic" on the Stanford Encyclopedia of Philosophy but didn't come up with anything under the search terms.

http://plato.stanford.edu/search/searcher.py?query=imperative+logic

Just a cursory look at the Wiki page... Imperatives are certainly things we can weigh over whether or not we want to obey them. But it seems odd to consider such things as being either "valid" or "invalid". One example on the Wiki page:

P1. Take all the books off the table!
P2. Foundations of Arithmetic is on the table.
C1. Therefore, take Foundations of Arithmetic off the table!

Wiki seems to imply that this can be considered a "valid" argument. Does that mean something like:

P1. Everyone who is 5'11", go jump off a cliff!
P2. I am 5'11".
C1. Therefore I go jump off a cliff.

Is a "valid" argument?

Jumping ahead (to coincidently the next chapter I will be highlighting), Hurley appears to state that validity is a term relative to deductive arguments. An argument is said to be valid if "The premises support the conclusion in such a way that if they are assumed true, it is impossible that the conclusion is false." (Hurley p. 40)

I would hate to think that me jumping off a cliff necessarily follows from the premises above. What would be the truth value of something like "remove all the books from the table" or "everyone who is 5'11", go jump off a cliff?" How would you construct a truth table for it?
 
Sorry, instead of posting a new response to Ayatollah So I suppose I should have edited my response above. I've done that so if a mod wishes to delete this post please feel free to. My apologies again.
 
Does that mean something like:

P1. Everyone who is 5'11", go jump off a cliff!
P2. I am 5'11".
C1. Therefore I go jump off a cliff.

Is a "valid" argument?

[...]
How would you construct a truth table for it?

It's valid if you phrase it correctly:

C1. Therefore, self, go jump off a cliff!

Which is an imperative, not (as your version comes off sounding) a prediction.

You wouldn't call the tables used to evaluate these arguments "truth" tables. Maybe something like "correctness" tables. You'd need more symbols inside the tables; not just T and F but also ! and :nono: ;)
(Maybe ! and !)
 
It's valid if you phrase it correctly:

C1. Therefore, self, go jump off a cliff!

Which is an imperative, not (as your version comes off sounding) a prediction.

You wouldn't call the tables used to evaluate these arguments "truth" tables. Maybe something like "correctness" tables. You'd need more symbols inside the tables; not just T and F but also ! and :nono: ;)
(Maybe ! and !)

Interesting. I suppose there is a kind of logic to imperatives. If someone says, "clear the books off the table" then clearing the pencils off the table would not follow from the imperative. However, imperatives don't seem to really deal with truth or falsity for the most part, though some statements in an imperative may involve truth or falsity such as what is a book and what isn't.

Certainly this is something to perhaps look into for the future but in the meantime does the fact that there is such a thing as imperative logic nullify Hurley's definition of arguments as purporting to prove something? It sounds like imperative logic isn't about trying to prove anything, so it isn't about arguments. It's about imperatives.
 
Hi Gary Childress.

First off, sorry for derailing in a sense, but even more for doing a piss pour job out of it.
So I will try again. If you liken the 3 laws of thought/logic to a hammer, then what I am trying to ask you is this: Not what the hammer can do, but what it can't do?
Further as to logic and epistemology - I take for granted that you know that A=A means A=A, but if that is the case then logic is connected to epistemology. It is not A=A, but you know A=A.
More later.
 
Hi Gary Childress.

First off, sorry for derailing in a sense, but even more for doing a piss pour job out of it.
So I will try again. If you liken the 3 laws of thought/logic to a hammer, then what I am trying to ask you is this: Not what the hammer can do, but what it can't do?
Further as to logic and epistemology - I take for granted that you know that A=A means A=A, but if that is the case then logic is connected to epistemology. It is not A=A, but you know A=A.
More later.

Hi Global Skeptic,

Don't worry, you are not "derailing" anything. So long as the topic is logic then the thread is very much "on topic". Plus I welcome the fact that, like I, you seem to have a passion for delving into the nuts and bolts of logic to come to a better understanding of things. I'm just glad at least a few others out there share my interest in learning or sharpening our logic skills. I'll withhold any comments until you finish your explanation above. :)
 
This came up in another thread, and I'm curious if your textbook has anything to say about it:

What are the consequences of rejecting the law of non-contradiction, ie. a proposition cannot be both true and not true?
 
I haven't gotten to anything yet that states it specifically. However, off the top of my head, I would think it would really mess up truth tables a lot, which would in turn probably mess up a lot of rules of logic.

According to traditional logic if P is true, then ~P must therefore be false and visa versa (~ means not the case). Later down the road I hope to cover propositional logic which uses symbols to analyze arguments. One thing you can do in propositional logic is deduce the value of P in a long convoluted argument if you already know the value of ~P or visa versa. Say for instance that we know P is true, then we should know that ~P is false. Therefore everywhere in an argument where ~P shows up we can give it the value of false and everywhere P shows up we can give it the value of true. Such problem solving would be almost impossible if P can be both true and false and ~P can be both true and false as well.

Take a very simple valid and sound argument for example called modus ponens:

If Socrates is mortal then he will die someday
Socrates is Mortal
Therefore he will die someday

This can effectively be represented as:

If P, then Q
P is the case
Therefore Q is the case

Normally, because I have asserted P is the case, I can therefore conclude that Q is the case if it is established that "if P, then Q" is True. However, if ~P could also be the case, then Q does not follow from P. For instance, if Socrates is mortal, it will not follow that he will die some day if he can also be not the case mortal at the same time.

Hopefully I will get the opportunity to go over the next chapter which introduces the terms, truth, validity, soundness, strength, and cogency. Probably not tonight though, just got a new job and will be working tomorrow. In brief, validity and soundness apply to deductive arguments and strength and cogency apply to inductive arguments. Both deductive and inductive arguments rely on the truth values of their premises and conclusion to derrive their respective merrits.

If P and ~ P can both be the case, then there is no such thing as a valid and sound deductive argument, nor is there such a thing as a strong and cogent inductive argument. Under such circumstances I don't see how anything would be provable.
 
Under such circumstances I don't see how anything would be provable.
I've heard the opposite account: that from a contradiction, everything is provable. Though I don't know how such a proof would work.
 
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