The basis of logic?

[Qoeleth]

Let us consider a logical argument. How do we know that it is a valid or invalid formulation?

Take the following:
“If it is day, it is light.
It is light, therefore it is day.”

We know that this contains the error of ‘affirming the consequent’. Now, how do we actually know that ‘affirming the consequent’ leads to a false conclusion? It would seem we only establish it empirically (having observed that, for example, there may be light, yet it may not be day.) Now, if logical laws is rendered valid only by observation of ‘the real world’, what is the point of logical demonstration?- given that it requires observation of reality to verify the soundness of its arguments. Logical formulations seem to come then only by induction. There is no way of establishing they are true in all cases. Moreover, one might find that in 90% of cases, ‘affirming the consequent’ actually leads to a true conclusion.

In other words, the validity of an argument is accepted only if we already observe its conclusion to be true. Therefore, logic in itself seems to prove nothing.

 

[GEddie]

Logic isn’t really about physical observation. It is about categorization.

Affirmation of the consequent fails as a rule, although it may in fact give a true result even most of the time, because it leads to categorizations that may not hold.

Consider the logical rule, modus ponens, which is the true counterpart to AofC:

A -------> B (if A then B)

A​

:. B. (Therefore B)

A----> B states that the category of A falls within B.

An example would be: All men are human beings.

I am a man,

therefore I am a human being.

This works because the category of man falls within the larger category of human being. This can be shown by a Venn diagram.

It doesn’t in general work backward from the larger category to the smaller, however, because there are elements in the larger category that are nor in the smaller. For example, women are human beings, but are not men.

ICXC NIKA

 

[Qoeleth]

Logic isn’t really about physical observation. It is about categorization.

Affirmation of the consequent fails as a rule, although it may in fact give a true result even most of the time, because it leads to categorizations that may not hold.

Consider the logical rule, modus ponens, which is the true counterpart to AofC:

A -------> B (if A then B)

A​

:. B. (Therefore B)

A----> B states that the category of A falls within B.

An example would be: All men are human beings.

I am a man,

therefore I am a human being.

This works because the category of man falls within the larger category of human being. This can be shown by a Venn diagram.

It doesn’t in general work backward from the larger category to the smaller, however, because there are elements in the larger category that are nor in the smaller. For example, women are human beings, but are not men.

ICXC NIKA

But this example, I believe, demonstrates what I mean.

Consider that premise, “all men are human beings.” It would seem this can be known either by induction from observed cases, or known by definition.

If this is known by induction (by observation of many men, then inducing it as a general rule), the premise can only be upheld if you already observe yourself, as a human being, to be a man. If you did not observe that, as a human being, you were also a man, the premise (if it is discovered by induction) could no longer be upheld. So, it would seem that the premise depends upon the inference.

If, on the other hand, it is not kinown by induction, but by definition, i.e. “man” is defined as “a male human being”- then you would need to know already that you are a human being (as well as a male), in order to be able to say “I am a man”, since a man is nothing other than “a male human being.”

I agree that the argument is logically valid- but it does not really seem to prove anything that does not need to be known already. (As shown, whether the premise if known by induction or definition, nothing is actually proven which is not already necessarily known).

 

[Khalid]

(Deductive) logic - that which produces certain conclusions given correct premises, unlike merely inductive logic - is inherently a priori in its methodology (as famously stated, a priori is that which is prior to experience: “you can know that it’s true laying on a couch”: whereas inductive reasoning is inherently a posteriori). Some argue this reduces down to intuitionism, for lack of a better word, but all systems, no matter how rigorous, do reduce down, at the lowest level of abstraction, to a set of axioms that can not be proved within the system itself - this is most famously proved in mathematics (with the Peano axioms), of which logic is a subset (or, according to some, a superset).

Aristotle affirmed that poor argumentation could lead to correct conclusions, just with no possibility of being able to reproduce the results, essentially. As is said, “Even a broken clock is right twice each day”, and in computer programming, “garbage in, garbage out”.

Combine those, and if you put garbage in (false premises or bad reasoning), you’re likely to get garbage out (false conclusion), but even the broken clock is right occasionally - but rarely. By reasoning to confirm a presupposition, as well, it seems that you are committing the much-more-informal “Texan sharpshooter” fallacy: “shoot and see what sticks”, or, “shoot at the side of a barn, and then paint a bullseye where you hit”.

 

[Perplexity]

Let us consider a logical argument. How do we know that it is a valid or invalid formulation?

Take the following:
“If it is day, it is light.
It is light, therefore it is day.”

We know that this contains the error of ‘affirming the consequent’. Now, how do we actually know that ‘affirming the consequent’ leads to a false conclusion? It would seem we only establish it empirically (having observed that, for example, there may be light, yet it may not be day.) Now, if logical laws is rendered valid only by observation of ‘the real world’, what is the point of logical demonstration?- given that it requires observation of reality to verify the soundness of its arguments. Logical formulations seem to come then only by induction. There is no way of establishing they are true in all cases. Moreover, one might find that in 90% of cases, ‘affirming the consequent’ actually leads to a true conclusion.

In other words, the validity of an argument is accepted only if we already observe its conclusion to be true. Therefore, logic in itself seems to prove nothing.

Using truth-tables, we can mathematically prove beyond a shadow of a doubt that certain inferences will always result in a true conclusion, so long as the premises are true. After having established a few of these inferences (e.g., Modus Ponens, Modus Tollens, etc.) we can suppose their logical validity, and use them to try and deduce the conclusions of other inferences. This is known as natural deduction. There’s no need at all to get empirical.

It’s easy to prove that ‘affirming the consequent’ is logically invalid using truth-tables:

p | q | p → q |​

T | T | T |
F | F | T |
T | F | F |
F | T | T |

These are all of the possible combinations of truth-values your ‘affirming the consequent’ example can take. They’ve exhaustively been listed. Now, it’s logically valid if and only there is no row in which both premises are true but the conclusion is false.

What’s the first premise? ‘p → q’. What’s the second? ‘q’, and the conclusion is ‘p’. Is there any row in which both ‘p → q’ and ‘q’ are true, but ‘p’ is false? Yes: the last one. Therefore, affirming the consequent is logically invalid.

 

[Qoeleth]

Using truth-tables, we can mathematically prove beyond a shadow of a doubt that certain inferences will always result in a true conclusion, so long as the premises are true. After having established a few of these inferences (e.g., Modus Ponens, Modus Tollens, etc.) we can suppose their logical validity, and use them to try and deduce the conclusions of other inferences. This is known as natural deduction. There’s no need at all to get empirical.

It’s easy to prove that ‘affirming the consequent’ is logically invalid using truth-tables:

p | q | p → q |​

T | T | T |
F | F | T |
T | F | F |
F | T | T |

These are all of the possible combinations of truth-values your ‘affirming the consequent’ example can take. They’ve exhaustively been listed. Now, it’s logically valid if and only there is no row in which both premises are true but the conclusion is false.

What’s the first premise? ‘p → q’. What’s the second? ‘q’, and the conclusion is ‘p’. Is there any row in which both ‘p → q’ and ‘q’ are true, but ‘p’ is false? Yes: the last one. Therefore, affirming the consequent is logically invalid.

I agree that valid argument with true premises lead to true inferences. But, it would seem (looking at textbooks and example), the inferences are necessarily things known already (i.e. that if the premisses are known, the inference must also already be known).

Let me give an example-
Either it is day, or it is night.
It is not day.
Therefore it is night.

Now, in knowing the second premise, it is necessary that the inference is already known. Indeed, it would seem necessary to know the inference in order to know the second premise!

Another example:
All lemons are yellow.
This is a lemon.
Therefore it is yellow.

Now, if the premise is accepted as a definition (i.e. that something must be yellow in order to be a lemon), it will not be possible to affirm the second premise until the inference (that the object in question is yellow) is already known.

Can you think of an example of a logical argument where the conclusion is not already as evident as the premisses?

 

[MPat]

I agree that valid argument with true premises lead to true inferences. But, it would seem (looking at textbooks and example), the inferences are necessarily things known already (i.e. that if the premisses are known, the inference must also already be known).

Let me give an example-
Either it is day, or it is night.
It is not day.
Therefore it is night.

Now, in knowing the second premise, it is necessary that the inference is already known. Indeed, it would seem necessary to know the inference in order to know the second premise!

Another example:
All lemons are yellow.
This is a lemon.
Therefore it is yellow.

Now, if the premise is accepted as a definition (i.e. that something must be yellow in order to be a lemon), it will not be possible to affirm the second premise until the inference (that the object in question is yellow) is already known.

Can you think of an example of a logical argument where the conclusion is not already as evident as the premisses?

What exactly do you mean by “evident” and “things known already”?

If you meant that someone who knows the premises must (at the same time) also see and accept the conclusion, then it is only true for simple reasoning. Otherwise the Mathematics would be very easy to teach: just show the students the axioms and definitions, and you’re done… Unfortunately, it doesn’t seem to be that way… Thus many proofs of theorems can be used as examples you wanted.

But if you meant that everything one can logically prove using given premises can be found using no other premises, then it is obviously true.

By the way, if you would like to know what logic is good for, you might wish to install some Prolog interpreter and play with it a little (having a textbook might be useful in that case).

 

[Qoeleth]

What exactly do you mean by “evident” and “things known already”?

If you meant that someone who knows the premises must (at the same time) also see and accept the conclusion, then it is only true for simple reasoning. Otherwise the Mathematics would be very easy to teach: just show the students the axioms and definitions, and you’re done… Unfortunately, it doesn’t seem to be that way… Thus many proofs of theorems can be used as examples you wanted.

But if you meant that everything one can logically prove using given premises can be found using no other premises, then it is obviously true.

By the way, if you would like to know what logic is good for, you might wish to install some Prolog interpreter and play with it a little (having a textbook might be useful in that case).

Granted, mathematics (if it is the same as logic) can arrive at inferences which are not necessarily as evident as the premisses.

But does this apply to non-mathematical logic? Is there an example of deductive reasoning arriving at a inference which is not already as evident as its premisses, or whose premise does not depend upon the inference? I have looked at many examples, and they all seem to infer things which are just as evident as the premises…

Often, in deductive reasoning- the premise seems to have been reached via induction- which then makes the whole thing circular. e.g.;
All pizza tastes great.
This particular food is a pizza
Therefore it tastes great.

But the premise from which the conclusion is derived seems only possible by induction from particulars. Therefore, nothing is really proved, if the premise is arrived as by induction.The induced premise seems to depend upon the particular.

I am open to being corrected, if I can see a non-mathematical logical argument which arrives at an inference which is not just as evident as the premise.

 

[Perplexity]

I agree that valid argument with true premises lead to true inferences. But, it would seem (looking at textbooks and example), the inferences are necessarily things known already (i.e. that if the premisses are known, the inference must also already be known).

Let me give an example-
Either it is day, or it is night.
It is not day.
Therefore it is night.

Now, in knowing the second premise, it is necessary that the inference is already known. Indeed, it would seem necessary to know the inference in order to know the second premise!

Another example:
All lemons are yellow.
This is a lemon.
Therefore it is yellow.

Now, if the premise is accepted as a definition (i.e. that something must be yellow in order to be a lemon), it will not be possible to affirm the second premise until the inference (that the object in question is yellow) is already known.

Can you think of an example of a logical argument where the conclusion is not already as evident as the premisses?

Sure:

1. If aliens exist, then Obama will rule for another 4 years.
2. Aliens exist.
3. Therefore, Obama will rule for another 4 years.

This is logically valid, but its soundness isn’t evident.

 

[Qoeleth]

Sure:

1. If aliens exist, then Obama will rule for another 4 years.
2. Aliens exist.
3. Therefore, Obama will rule for another 4 years.

This is logically valid, but its soundness isn’t evident.

It’s logically valid- but again it illustrates my point, that the inference is already equally (or more) evident than the premise. It would seem that the premise “If aliens exist, then Obama will rule for another 4 years” could only be known by observation. So, until the inference (“that Obama rules for another 4 years”) is known, the premise could not possibly be known.

 

[MPat]

Granted, mathematics (if it is the same as logic) can arrive at inferences which are not necessarily as evident as the premisses.

But does this apply to non-mathematical logic? Is there an example of deductive reasoning arriving at a inference which is not already as evident as its premisses, or whose premise does not depend upon the inference? I have looked at many examples, and they all seem to infer things which are just as evident as the premises…

Often, in deductive reasoning- the premise seems to have been reached via induction- which then makes the whole thing circular. e.g.;
All pizza tastes great.
This particular food is a pizza
Therefore it tastes great.

But the premise from which the conclusion is derived seems only possible by induction from particulars. Therefore, nothing is really proved, if the premise is arrived as by induction.The induced premise seems to depend upon the particular.

I am open to being corrected, if I can see a non-mathematical logical argument which arrives at an inference which is not just as evident as the premise.

Well, humans are good enough at performing simple reasoning tasks and thus I am not sure there are many examples to be discovered… Yet I did think of something:

Meat of mammals is (tends to be) eatable.
Kangaroos are mammals.
Therefore, meat of kangaroos is (is likely to be) eatable.

In practice it is going to be an analogy:

Meat of pigs, cows and the like is eatable.
Kangaroos are similar to pigs, cows and the like in some ways.
Therefore, it is likely that the meat of kangaroos is also eatable.

Would that count?

 

[Qoeleth]

Well, humans are good enough at performing simple reasoning tasks and thus I am not sure there are many examples to be discovered… Yet I did think of something:

Meat of mammals is (tends to be) eatable.
Kangaroos are mammals.
Therefore, meat of kangaroos is (is likely to be) eatable.

In practice it is going to be an analogy:

Meat of pigs, cows and the like is eatable.
Kangaroos are similar to pigs, cows and the like in some ways.
Therefore, it is likely that the meat of kangaroos is also eatable.

Would that count?

Yes, that does produce a non-evident inference, by assuming, on the basis of similarities between kangaroos and other animals, that they will also be similar in respect to edibility.

But imagine this:
Fish are edible.
A stonefish is a fish.
Therefore a stonefish is edible.

Now, if the inference is not correct- then the first premise can no longer be held to be correct. So the truth of the premise seems to depend upon the truth of the premise.

Strictly speaking, is it a logical procedure to assume, on the basis of some similarities between different things, that they will also be similar in respect to other aspects? e.g.

The kangaroo, like the cow, is a grass eating mammal. The cow is edible. Therefore the kangaroo is (probably) edible.

Is this valid?
A is like B in respect to D and C.
Therefore, A is (probably) like B in respect to E.

 

[WillP]

Let us consider a logical argument. How do we know that it is a valid or invalid formulation?

In other words, the validity of an argument is accepted only if we already observe its conclusion to be true. Therefore, logic in itself seems to prove nothing.

Yet you support your thesis, namely, that logic proves nothing, by a process based on logic. The reason we predicate our conclusions on logic is that it is a function of our rational natures to do so. Hence, by our very nature, we are compelled to logic. Logic or rationally based solutions also turn out the right answers.

If you rely upon “faith” in deciding to jump off a cliff because you are told that there is a net below, you lead with your chin, and, it is only if you’re lucky that you may turn out to be right. If I put my faith in my older brother who tells me that there is a net, I can be right or wrong depending on his whim. If I place faith in imaginary sky gods, then what happens is a function of physics because there are no sky gods, and, the laws of physics are indifferent to the purposes of man.

 

[hicetnunc]

Let us consider a logical argument. How do we know that it is a valid or invalid formulation?

This is a very interesting post…thank you for sharing it!

St Thomas draws a distinction between sense, reason and intellect. Sense knows only particular things; reason knows through discursion; and intellect knows its object immediately, through intuition (ST 1a.q59.a1). The way we come to know the truth, he says, is through a process of discursive reasoning, as through the use of syllogisms and arguments; this is how all our knowledge is attained (ST 2a2ae.q9a1). However, the intellect, which knows by intuition, is able to recognise the truth when it discovers it: the itnellect grasps something to be true as the result of discursion. But there is no a priori knowledge of the truth which precedes this discursive reasoning but simply the act of recognition when reason operates correctly.

I think St Thomas anticipated your concern about the seemingly circular nature of reason. It’s sometimes called Meno’s paradox, from Plato’s dialogue Meno: how will you recognise virtue when you go looking for it, unless you already know what it is in order to recognise it? But if you already know what it is, why do you need to look for it?

St Thomas’ answer is not that we already know what virtue is (or what particular things are), but because all things participate in the True, this includes our intellect. So when the intellect encounters the true in nature, or in logic, it recognises it because there is a match so to speak between the intellect and the truth encountered (he goes into this in some fascinating detail lin 2a2ae.q180,a6,ro2),

Great food for thought!

Many thanks.