Known Valid Argument Forms - Is the system complete.

TheMadFool

My knowledge of logic is very basic. I think I remember someone referring to Categorical, Propositional and Predicate logic, those I'm familiar with as BABY LOGIC.

Anyway, all the elementary logic books I've read come with a list of valid argumentation forms which are allegedly guaranteed to lead you to the truth given you input true premises. Last I remember there were 18 of them some of which I'll list here to help readers understand the backdrop against which I'll ask a simple question.

Valid argument forms:
1. Modus ponens (p > q, p / q)
2. Modus tollens (p > q, ~q / ~p
3. Disjunctive syllogism (p v q, ~p / q)
4. Hypothetical syllogism (p > q, q > r / p > r)
5. Constructive dilemma (p > r, q > s, p v q / r v s)
6. Conjunction (p, q / p & q)
7. Addition (p / p v q)
8. DeMorgan's laws [ ~(p & q) = ~p v ~ q....~(p v q) = ~p & ~q]
9. Contraposition (p > q = ~q > ~p)
10. Material Equivalence [ p <-> q = (p > q) & (q > p)....p <-> q = (p & q) v (~p & ~q)]
11. Double negation [~~p = p]
12. Association [(p v q) v r = p v (q v r)...(p & q) & r = p & (q & r)]
13. Distribution [p v (q & r) = (p v q) & (p v r)...p & (q v r) = (p & q) v(p & r)]
14. Exportation [ (p & q) > r = p > (q > r)]
15. Tautology (p = p v p)
16. Commutation ( p v q = q v p...p & q = q & p)
17. Material Implication (p > q = ~p v q)
18. Simplification (p & q / p)

The point is that most books don't go beyond the above list which is an implicit claim that no more are needed In other words these forms listed in the books are adequate for.....(my question follows)....

1. Any and all possible logical arguments i.e. the list is globally comprehensive and it's not possible for there to be an argument that isn't in a combination of these valid forms OR that can't be reduced or rephrased with these valid forms.

OR

2. This list only provides for normal/conventional/usual/ordinary argumentation by which I mean that there are arguments possible which both don't need any member of the list and that CANNOT be reduced to a combination of the members in the list.

If choice 1 is true we don't have to worry as we can discover ALL possible truths through any possible argument.

However, if choice 2 is true then there are some truths beyond our reach as we won't be able to comprehend the argument or even realize that such an argument is possible.

Note: I've read in a book that there is a huge number or even infinite number of valid argument forms.

Is our list of valid argument forms complete?

Your comments...

Fine Doubter

Some of the forms of argument that aren't on the list will probably be complex combinations chosen from the above, however there may also be fresh types.

Arguments are dependent, for usefulness, on being "understood".

Nonetheless there are also many kinds of fallacies and textbooks can list those.

Then there will in practice be the employment of combinations of sound and unsound arguments. The more complicated the "better" in some people's eyes.

Rhetoric is a very interesting field of study altogether. Rhetoric can be good. J H Newman and S J Gould were skilled rhetoricians.

Some forms of argument can be used both soundly and unsoundly. For example, under "argument from authority": "Dr J Smith, biology professor, says the British political system will not last more than 30 years in its present shape." It was alleged this was asking to be shot down on the "grounds" that it wasn't his field. BUT I say, he is just as qualified to express this opinion (as every observant member of this society), indeed I see he is a wide and deep thinker about "political animals"!

TheMadFool

↪Fine Doubter

:up:

Fine Doubter

I have got hold of Stanley Jevons and a couple of other decent books. When I get a chance I'll list lots more. There are dozens upon dozens!

I wish I'd been told these things when I was a youngster.

Deleted User

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TheMadFool

↪tim wood

I remember vaguely about how any statement can be rephrased as a NEGATION-OR combination.

I'll add this question: truth tables can be fun, but are in any case tedious and time consuming. Is there a better, faster way to test arguments? — tim wood

What about abbreviated truth tables?

Then there's another method where you assume worlds and check if in any world the argument is invalid. I forgot how it's done.

Terrapin Station

It seems to me that we either see all argumentation as resting on truth tables for the operators/connectives we've defined, in which case stuff like modus ponens wouldn't be a form we'd specify, it's just a possible compound, or the forms would be theoretically endless.

alcontali

Is our list of valid argument forms complete? — TheMadFool

There is also an alternative version of propositional calculus that only has one inference rule, i.e. modus ponens, and two operators, AND and OR, which allows for axiomatizing the core rewrite rules of Aristotelian/Boolean two-valued logic.

The basic and derived argument forms then naturally follow from this axiomatization of propositional logic.

Two-valued (Aristotelian/Boolean) logic -- with only the TRUE and FALSE values supported -- is actually a degenerate form of logic that allows for single-value negation results. Not all paradoxical results in two-valued logic also occur in many-valued logic. Therefore, two-valued logic often ends up being considered naive and unusable. For example, the IEEE 1164 (9-valued) and IEEE 1364 (4-valued) international standards disallow the use of naive 2-valued logic in electronic design automation.

aletheist

Is our list of valid argument forms complete? — TheMadFool

It depends on what you mean by "argument forms." As

↪alcontali

just pointed out, what you seem to be seeking is an axiomatization of classical logic, which typically involves a few primitives and an inference rule. The "existential graphs" of Charles Sanders Peirce are an innovative diagrammatic alternative.

alcontali

The "existential graphs" of Charles Sanders Peirce are an innovative diagrammatic alternative. — aletheist

I was looking for extended definitions for the term "consistency" when either extending the permissible logic language or the permissible truth values.

In simple propositional calculus, inconsistency arises when the theory is capable of generating both $\phi(x)$ and $\neg \phi(x)$ .

In first-order logic, introduction of the universal quantifiers $\forall$ and $\exists$ , allows for a new form of inconsistency, i.e. $\omega$ -inconsistency, where the theory generates both $\forall x (\phi(x))$ and $\exists x (\neg \phi(x))$ .

Higher-order logic allows, for example, quantification over sets: $\forall x \forall A (x \in A \wedge \phi(x))$ . What new mischief is possible here?

So, I was looking for what the term "consistency" means in higher-order, many-valued logic. What new forms of mischief are not allowed? What new forms of inconsistency can be introduced by further increasing the power of the logic language?

The Wikipedia page on consistency only mentions Henkin's theorem in that respect, but I frankly do not understand what exactly this theorem means ... ;-)

TheMadFool

↪Terrapin Station

↪alcontali

↪aletheist

Thanks for your valuable answers. I must confess I didn't understand everything the links were about.

Anyway...

To put it in logical terms I'm looking for a proof that shows that the given set of argumentation forms (listed in the OP) is

1) the exact (no less no more) set of valid forms we need to construct any and all arguments

2) any and all arguments can be rephrased/reduced to a combination of these forms

TheMadFool

I even think (don't know, could be wrong) that all can be reduced to negation, "~," and or, "v." — tim wood

Do you know of a reason why negation and or are preferred over other logical connective like "and" and "implication"?

Is it that "negation" and "or" require the least number of symbols?

TheMadFool

Thank you again for your valuable comments. I was reading another thread of mine concerning axioms and @alcontali mentioned Godel's incompleteness theorems which, to my understanding, proves that there will always be some unprovable truths in any axiomatic system.

Therefore, the no set of valid argument forms is ever going to be enough. There will always be an unprovable truth.

Is there a way to distinguish the provable from the unprovable?

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alcontali

In some systems, not all. Maybe alcontali can tell us which, and why. — tim wood

An example of a system that is weak enough as such that the incompleteness problem does not occur is the Presburger arithmetic. It only uses "0" and "1" as numbers and only addition "+" as operator.

So, it is only capable of saying things of the following form:

1) 0+0=0
2) 0+1=1 and 1+0=1
3) 1+1=0

Universal quantifiers only run over {0,1}. So, in my impression, they may not even be needed. For example, $\forall x$ can be always be replaced by x=0 or x=1 (This is known as quantifier elimination).

The following remark supports that:

The decidability of Presburger arithmetic can be shown using quantifier elimination, supplemented by reasoning about arithmetical congruence (Enderton 2001, p. 188).

Still, there is also the following remark:

Generally, any number concept leading to multiplication cannot be defined in Presburger arithmetic, since that leads to incompleteness and undecidability.

I am not really sure why that is, though. According to the remark above, axiomatizing the following would cause incompleteness. Not sure why, though:

1) 0*0=0
2) 1*0=0 and 0*1=0
3) 1*1=1

I do not completely understand how adding this simple multiplication scheme would prevent quantifier elimination, but the text seems to claim that it does. It has something to do with "arithmetical congruence". (how!?)

So, my take on the matter is that, if quantifier elimination is not possible, because for example that would lead to infinitely long axioms, then the theory really requires first-order logic, and in that case, it will be incomplete.

For example, normal (=Peano) arithmetic runs over an infinite set n $\in$ { 1, 3, 4, ... }. Quantifier elimination would lead to replacing it by sentences that look like: n=1 or n=2 or n=3 or n=4 ... which is an infinitely long sentence. Hence, the use of first-order logic quantifiers cannot be avoided -- to keep the size of the theory's rules finite -- leading to incompleteness.

"For each natural number the following is true" becomes: it is true for 1. It is true for 2. It is true for 3 ... ad infinitum. "There exists a natural number for which the following is true" becomes: It could be true for 1. It could be true for 2. It could be true for 3 ... ad infinitum.

So, my tentative interpretation is that when a theory gets rephrased in propositional logic (=without universal quantifiers), and it will always consist of infinitely-long construction rules, then the theory will be incomplete.

(but issues with "arithmetical congruence" can apparently also cause incompleteness, but I am not finished trying to figure that out yet ...)

TheMadFool

↪tim wood

↪alcontali

:up:

Thank you.

I was just wondering whether Godel's incompleteness theorems proves that any given set of valid argument forms constructed for our convenience is necessarily inadequate since there'll always be some truths that can't be proved at all.

One thing I'd like to know is whether logic - the entire system - is complete or not. I vaguely remember reading somewhere that logic is a complete system.

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aletheist

One thing I'd like to know is whether logic - the entire system - is complete or not. I vaguely remember reading somewhere that logic is a complete system. — TheMadFool

As

↪tim wood

already noted, it depends on which system of logic you have in mind, since Gödel's incompleteness theorem only applies to formal systems "within which a certain amount of arithmetic can be carried out" (Stanford). For example, there are well-established proofs that the now-standard systems of first-order propositional and predicate logic are both (deductively) complete and consistent.

TheMadFool

For example, there are well-established proofs that the now-standard systems of first-order propositional and predicate logic are both (deductively) complete and consistent. — aletheist

You mean it's possible to create an axiomatic system that is complete and consistent as long as it doesn't involve arithmetic?

Do you know why this is the case?

aletheist

You mean it's possible to create an axiomatic system that is complete and consistent as long as it doesn't involve arithmetic? — TheMadFool

I mean exactly what I said, quoting the Stanford article--Gödel's incompleteness theorem only applies to formal systems "within which a certain amount of arithmetic can be carried out" (emphasis mine).

Do you know why this is the case? — TheMadFool

Because there are minimum requirements for a formal system to be able to generate the kind of undecidable sentence that Gödel's incompleteness theorem requires.

↪alcontali

gave an example of a formal system that can do some arithmetic, but not enough for the theorem to apply.

TheMadFool

↪aletheist

:ok: :up:

Thomas Bailey

Thank you for such a valuable information. I was looking for a long time detailed explanation. Really appreciate your reply!

Agent Smith

i) 2 truth values.

ii) 4 binary operators/connectives (conjunction, disjunction, implication, double implication)

iii) 1 unary operator (negation i.e. double negation)

How many permutations are possible?

For binary operators, 2 truth values give us 4 (TT, TF, FT, FF).

Each one of the 4 (above) can be permuted with itself and others: 4 $\times$ 4 = 16 permutations.

There are 16 implication-type argument forms. Not all of them are be valid. In fact only 8 are valid.

Come to equivalence now, there's no limit here but it appears that 10 suffices.

Basically, it's a combinatorics question, more or less. I neither have the wherewithal nor the patience to answer this question.

Thomas Bailey

Thanks a lot for the explanation.

Known Valid Argument Forms - Is the system complete.

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