Are the inference rules of propositional calculus tautologies?

[solakis1]

Are the inference rules of propositional calculus tautologies ,yes or no

 

[Evgeny.Makarov]

Inference rules are relations on formulas, while tautologies are formulas, so the answer is no.

 

[solakis1]

is modus ponens a rule of inference yes or no ?

 

[Evgeny.Makarov]

is modus ponens a rule of inference

Yes. It's a ternary relation on formulas.

 

[solakis1]

but modus ponens is a tautology isn't it??

 

[Evgeny.Makarov]

As I wrote, Modus Ponens is a ternary relation, i.e., a set of ordered triples of formulas. It is not a formula, therefore, not a tautology.

 

[solakis1]

ANY book of logic in the whole Universe will tell you that Modus Ponens ( or law of Detatachment) is a Tuatology
Any truth table generator in the internet will show you that M.Ponens is a tautology
But i will quote you just on only two famous books of logic:

1) Introduction to Logic 5th edition ,by IRVING COPI
On page 301,7lines from the top of the page I COPI writes
"A statement form that has only true substitution instances is a tautologous statement form ,or A tautology"
2)Introduction to logic by , by PATRICK SUPPES
On page 34 on the top of the page has :

A TABLE OF USEFUL TAUTOLOGIES
Underneath this title has 10 useful tautologies and the 1ST of them is the:Law of Detachment ,or M.Ponens

Well, lastly you can form the truth table of M.Ponens yourself [p,(p->q)]->q
and you can find out if this a tautology or not
And M.Ponens is (wwf) well formed formula

 

[topsquark]

ANY book of logic in the whole Universe will tell you that Modus Ponens ( or law of Detatachment) is a Tuatology
Any truth table generator in the internet will show you that M.Ponens is a tautology
But i will quote you just on only two famous books of logic:

1) Introduction to Logic 5th edition ,by IRVING COPI
On page 301,7lines from the top of the page I COPI writes
"A statement form that has only true substitution instances is a tautologous statement form ,or A tautology"
2)Introduction to logic by , by PATRICK SUPPES
On page 34 on the top of the page has :

A TABLE OF USEFUL TAUTOLOGIES
Underneath this title has 10 useful tautologies and the 1ST of them is the:Law of Detachment ,or M.Ponens

Well, lastly you can form the truth table of M.Ponens yourself [p,(p->q)]->q
and you can find out if this a tautology or not
And M.Ponens is (wwf) well formed formula

Once again I must ask. Are you asking questions or simply trying to prove that you know more than we do? Why did you ask this if you "knew" that you already know the answer and just want to fight someone that disagrees with you? If that's how you get your kicks on websites I can recommend a good counselor.

-Dan

 

[Evgeny.Makarov]

1) Introduction to Logic 5th edition ,by IRVING COPI
On page 301,7lines from the top of the page I COPI writes
"A statement form that has only true substitution instances is a tautologous statement form ,or A tautology"

The problem is that according to this definition, a tautology is a special case of a statement form, but modus ponens is not a statement form. I have edition 14 of Copi (2016). Chapter 8, section 7 defines modus ponens as an argument form. According to section 4, an argument form is an array of symbols containing statement variables but no statements, such that when statements are substituted for the statement variables the result is an argument. Finally, chapter 1, section 2 defines an argument as a group of propositions of which one is claimed to follow from the others. Thus, an argument form may be specified by a collection of statement forms, but it is not a single statement form.

A separate remark: I have not used Copi's book a lot, but my impression is that it may be a textbook of logic, but not of mathematical logic. It may be suitable for teaching logic to students of law, philosophy and political science, but not to students of mathematics. For example, every textbook of mathematical logic refers to statement forms and propositional (or sentential) formulas and gives a precise inductive definition. And none of them does this in the middle of the book (page 341 of 665 in Copi). For example, a textbook by Elliott Mendelson gives this definition on p. 28 of 499, and the freely available textbook from the Open Logic Project (recommended) has this definition on p. 61 of 724.

Concerning the textbook by Suppes, it distinguishes between tautologies and corresponding inference rules. It does call the formula $P\&(P\to Q)\to Q$ "the law of detachment", or "modus ponendo ponens", but then it says: "this tautology corresponds to the rule that from $P$ and $P\to Q$ we may infer $Q$". I have not read the book sufficiently to verify whether it consistently applies the name "modus ponens" to the tautology rather than to the corresponding inference rule, but even if it applies it to both, it does not seem to say that inference rules are formulas and in particular tautologies.

Well, lastly you can form the truth table of M.Ponens yourself [p,(p->q)]->q
and you can find out if this a tautology or not
And M.Ponens is (wwf) well formed formula

Could you give a reference to the textbook that contains the definition of a well formed formula that you use? Does this definition allow commas in wffs like in $[p,(p\to q)]\to q$?

Answering to the essence of your question, in regular propositional and predicate logics if an inference rule derives $B$ from $A_1,\ldots,A_n$, then $A_1\land\dots\land A_n\to B$ is indeed a tautology. This is needed in the proof of soundness: if $\Gamma\vdash A$, then $\Gamma\models A$. However, there are logics where this is not the case. For example, modal logic has the rule of necessitation: $\Box A$ is derived from $A$, but $A\to\Box A$ is not an axiom or a tautology. The idea is that this rule must be applied in the absence of open assumptions.

 

[solakis1]

what d

Once again I must ask. Are you asking questions or simply trying to prove that you know more than we do? Why did you ask this if you "knew" that you already know the answer and just want to fight someone that disagrees with you? If that's how you get your kicks on websites I can recommend a good counselor.

-Dan

What do you mean once again.
Because most of my questions are in the challenging subforum

 

[solakis1]

Could you give a reference to the textbook that contains the definition of a well formed formula that you use? Does this definition allow commas in wffs like in $[p,(p\to q)]\to q$?

On page 47 of the book:
Schaum's OUTLINE SERIES, LOGIC,there is the following problem.
PROVE:
~P->(Q->R),~P,Q |- R
As you can see the Authors here instead of usinig the Logical symbol & for the premisses they use the comma
Also in the same book on page 44 they give the 3 rules for the formation of wwf in propositional calculus.
From these rules one can easily conclude that M.Ponens is wwf.
So your claim in post no 6 is not correct
Our subject in concern is not modal or fuzzy or 3 valued logic

 

[solakis1]

Tomorrow i will answer to the rest of your claims

 

[topsquark]

what d

What do you mean once again.
Because most of my questions are in the challenging subforum

Perhaps a language problem, then. The Challenge forum is a place to pose a challenging problem that the author knows how to solve and is challenging the rest of the community to solve. It is not meant to be a debate forum. And, frankly, even if your point is a debate your approach is pretty rude.

-Dan

 

[Evgeny.Makarov]

On page 47 of the book:
Schaum's OUTLINE SERIES, LOGIC,there is the following problem.
PROVE:
~P->(Q->R),~P,Q |- R
As you can see the Authors here instead of usinig the Logical symbol & for the premisses they use the comma

How does it refute anything I said? We have several formulas separated by commas to the left of the turnstile.

Also in the same book on page 44 they give the 3 rules for the formation of wwf in propositional calculus.
From these rules one can easily conclude that M.Ponens is wwf.

Depends on what you mean by modus ponens. Would you like to give a precise definition or provide a reference to a definition? But in most books modus ponens is not considered a formula.

 

[solakis1]

How does it refute anything I said? We have several formulas separated by commas to the left of the turnstile.

Depends on what you mean by modus ponens. Would you like to give a precise definition or provide a reference to a definition? But in most books modus ponens is not considered a formula.

As I wrote, Modus Ponens is a ternary relation, i.e., a set of ordered triples of formulas. It is not a formula, therefore, not a tautology.

In the page 44 Schaum's OUTLINE SERIES, LOGIC the 3 rules to form a formula or well formed formula (wff) are:
1) Any sentence letter P,Q,R...e.t.c is a wwf or a formula
2) If P is wwf then so is ~P
3)If P,Q are wwf ,then so are: (P&Q),(PvQ),(P->Q),(P<->Q)
Now according to the above (i will use small letters to make the result more accessable)

According to rule (1) p,q are wwf
According to rule (3) (p->q) is wwf
According to rule (3) again (p&(p->q)) is wwf....put P=p ,Q=(p->q)
According to rule (3) again [(p&(p->q))->q] is wwf.....put P=(p&(p->q)) ,Q=q
So the formula(wwf) [p&(p->q)->q] is called modus ponens or Law of Det.

Those formulas above, that for all possible substitutions of their letters with the values T,F give ONLY the value T are called tautologies
One very powerfull tool that we have in our disposal to determine whether the above wwf are tautologies or not are the truth tables
Not all wwf are tautologies For example ~(pvq) is not
Is M.Ponens a tautulogy?
You can check it up by using this truthtable generator
Truth Table Generator (logictown.org)
Now,is M.Ponens a rule of inference?
In book Introduction to Logic 5th edition ,by IRVING COPI page-312
M.Ponens is characterised as a rule of inference and a very elementary valid argument
I know that some mathematical logic books like the one of A.margaris ,where propositional calculus is based on an axiomatic development the definition of formula (wwf) is more general,but even using the one that A.Margaris produces M.Ponens is a formula

 

[solakis1]

Inference rules are relations on formulas, while tautologies are formulas, so the answer is no.

Inference rules are relations on formulas, while tautologies are formulas, so the answer is no.

Is the formula P&(~P) A tautology?

 

[Evgeny.Makarov]

So the formula(wwf) [p&(p->q)->q] is called modus ponens or Law of Det.

This is indeed a well-formed formula. However, can you name any books other than the one by Suppes where modus ponens is a formula rather than an inference rule?

I believe in most books modus ponens is an inference rule. Now, what is your precise definition of an inference rule (preferably in general, but also specifically for modus ponens)?

Now,is M.Ponens a rule of inference?
In book Introduction to Logic 5th edition ,by IRVING COPI page-312
M.Ponens is characterised as a rule of inference and a very elementary valid argument

But the book by Copi does not define modus ponens as a formula \(\displaystyle p\,\&\,(p\to q)\to q\). You are conflating two definitions from Suppes and from Copi.

but even using the one that A.Margaris produces M.Ponens is a formula

This is not correct. From what I see, the author never refers to modus ponens as a formula. He does not give a precise definition of a rule of inference but writes: "The rules of logic are called rules of inference. An example of a rule of inference is: For all statements P and Q, the statement P may be inferred from the statement P /\ Q." Nevertheless, the book writes rules of inference with a horizontal line between the premises and the conclusion. Can you point a place in the book where a formula is written using a line? If rules of inference are formulas, then why have two different ways of writing them?

Is the formula P&(~P) A tautology?

Of course not; it's a contradiction. Do you have reasons to believe I think it's a tautology?

 

[solakis1]

tautologies are formulas, post No 2

Maybe there is a misundertanding,by the above you mean that every formula (wwf) is a tautology?

 

[solakis1]

This is not correct. From what I see, the author never refers to modus ponens as a formula. He does not give a precise definition of a rule of inference but writes: "The rules of logic are called rules of inference. An example of a rule of inference is: For all statements P and Q, the statement P may be inferred from the statement P /\ Q." Nevertheless, the book writes rules of inference with a horizontal line between the premises and the conclusion. Can you point a place in the book where a formula is written using a line? If rules of inference are formulas, why then have two different ways of writing them?

Yes, that i was looking for to show that inference rules are tautologies in Copis book
Yes in page 2 the book nentions some rules of inference using as you said a line beteween premises and a conclusion and also gives names for some of them ,like :
Modes ponens
transitivity
proof by cases
contraposition
excluded middle
double negation
Then on page 71 gives a list of tuatologous schemes ,among them are all the rules of inference mentioned above.
transitivity is No 14 on the list
proof by cases is No 26 on the list
contraposition is No 15 on the list excluded middle is No 12 on the list double negation is No 11 on the list
Hence rules of inference according to COPI are tautologies because any substitution instance of a tautologous scheme can become a simple tautology

 

[Evgeny.Makarov]

you mean that every formula (wwf) is a tautology?

No.

Yes, that i was looking for to show that inference rules are tautologies in Copis book

After this phrase you refer to the book by Margaris and not by Copi. This is confusing.

The fact that two different objects are called "proof by cases" does not mean that they are the same object. It simply means that they are related to the same idea. For example, formulas 13 and 14 on p. 71 in the book by Margaris are called "transitivity of implication", but it does not mean that they are the same formula. For another example, "sum of two numbers" may refer to a number of the form $a+b$ for some $a$ and $b$ or to the function (a special case of a binary relation) that takes a pair $(a, b)$ and returns $a+b$.

Note also the none of the tautologies on pp. 71-72 are called "modus ponens", probably because this name is strongly associated only with a rule of inference, not a formula.

Ideally, of course, this abuse of naming should not occur, and indeed it does not in more formal books.

In post #17 I asked you to provide a definition of an inference rule, which you have not done. Arguing about concepts in the absence of their definitions is meaningless.

 

[solakis1]

Yes there was a typo it is Margaris
My definition of a rule of inference is:

Any (wwf) formula is a rule of inference iff is a tautology
The reason that M.Ponens is not the list of tautologies is that in the axiomatic development that Margaris follows the only law of logic used is M.Ponens
You expect him to prove something that allready assumes

 

[solakis1]

As I wrote, Modus Ponens is a ternary relation, i.e., a set of ordered triples of formulas. It is not a formula, therefore, not a tautology.

So the formula(wwf) [p&(p->q)->q] is called modus ponens or Law of Det.

This is indeed a well-formed formula.

So in your post No 7 you claim that modus ponens is not a formula (wwf) and then on your post No 17 you admit that modus ponens is formula (wwf)
Which one is correct?

 

[Evgeny.Makarov]

then on your post No 17 you admit that modus ponens is formula (wwf)

In post #17 I wrote that $p\,\&\,(p\to q)\to q$ is a formula. I never said that modus ponens viewed as an inference rule is a formula. That is, I never said that an inference rule is a formula.

My definition of a rule of inference is:

Any (wwf) formula is a rule of inference iff is a tautology

So you assume a definition according to which the answer to the original question ("Are the inference rules of propositional calculus tautologies") is obvious. But you did not reveal this definition until now. And when I gave an obviously different definition in post #2, you chose to spend the whole thread convincing me that your definition is right and mine is wrong. And you chose to do this even though there is hardly any textbook of mathematical logic that uses your definition. The best argument you presented is that the book by Suppes uses the name "the law of detachment" both for the tautology and for the inference rule. And even this book says on p. 32 that the tautology corresponds to the rule that derives Q from P and P -> Q. It does not say that the tautology is that rule.

Here is my summary. You define an inference rules as a tautology. I define an inference rule as an $(n+1)$-ary relation on the set of formulas for $n\ge1$. These concepts are different since tautologies (which are formulas) are not relations and vice versa. So strictly speaking our definitions are different. You are free to use any definition you like provided you make this clear. Still, I would be surprised if even a single book on mathematical logic uses your definition. Arguing whose definition is the right one (when the definitions themselves and the difference between them are understood) is usually pointless.

 

[solakis1]

In post #17 I wrote that $p\,\&\,(p\to q)\to q$ is a formula.
Ok fine but on post No 6 you wrote that m.ponens it is not a formula

As I wrote, Modus Ponens is a ternary relation, i.e., a set of ordered triples of formulas. It is not a formula, therefore, not a tautology.

And i asked you which of the 2 is correct

 

[Evgeny.Makarov]

I think all your questions have been answered several times in this thread.

 

[solakis1]

I think all your questions have been answered several times in this thread.

yes definitely ,particularly the last one where in one post you say that m.ponens is a formula and in another it is not .
By the way which is the book that suports your definition of an inference rule because up to now i am the one that have produced several books as reference.
And because in your last post you wrote, that and i quote:
"Still, I would be surprised if even a single book on mathematical logic uses your definition"
Here is another book that supports my definition and i hope your reasoning powers (although i doubt that) will accept that:
In the book:
SET THEORY AND LOGIC BY ROBERT R. STOLL on page175 the author writes:
Under the title tautological conditionals
1) (A&(A->B))->B which is the formula of M.Ponens
Then on page 185 at the bottom of the page explains :
Each of the tautological implications GENERATES A RULE OF INFERENCE
for example tautology (1) determines the rule
from A and A->B TO INFER B (infer is the key word that's why are rules of inference)
This is called the rule of detachment or modus ponens
In the book
introduction to elementary mathematical logic ,by Abram Aronovich Stolayar
On page 58 the author writes:
A special role in the algebra of propositions is played by tautologies,which express the laws of logic

 

[Evgeny.Makarov]

yes definitely ,particularly the last one where in one post you say that m.ponens is a formula and in another it is not .

I never said that modus ponens according to my (not Suppes') definition is a formula. Thus, I stated in post #4 that modus ponens is an inference rule, i.e., a ternary relation on formulas. In post #17, which you are referring to, I said that $p\,\&\,(p\to q)\to q$ is a formula. In the message I was replying to you called this formula "modus ponens", but I did not confirm it. I said that the formula was indeed a formula, not that it is commonly known as modus ponens.

Here is another book that supports my definition and i hope your reasoning powers (although i doubt that) will accept that:
In the book:
SET THEORY AND LOGIC BY ROBERT R. STOLL on page175 the author writes:
Under the title tautological conditionals
1) (A&(A->B))->B which is the formula of M.Ponens

I agree that \(\displaystyle A\,\&\,(A\to B)\to B\) is a tautology (and a formula). Who argued with that? But the words "which is the formula of M.Ponens" do not appear in the book. This page says nothing about the relationship of this formula with inference rules. In fact, the phrase "rule of inference" seem to appear for the first time on p. 181 and "modus ponens" appears first on p. 185.

Then on page 185 at the bottom of the page explains :
Each of the tautological implications GENERATES A RULE OF INFERENCE
for example tautology (1) determines the rule
from A and A->B TO INFER B (infer is the key word that's why are rules of inference)
This is called the rule of detachment or modus ponens

Precisely. There is a formula, and there is a rule of inference that has a similar meaning. Still, they are different objects: a formula generates, or determines, a rule of inference. By the way, "infer is the key word that's why are rules of inference" again does not appear in the book. Your way of writing quotations is quite sloppy.

In the book
introduction to elementary mathematical logic ,by Abram Aronovich Stolayar
On page 58 the author writes:
A special role in the algebra of propositions is played by tautologies,which express the laws of logic

I agree with the phrase in bold. But firstly, it does not define the concept "the laws of logic" and does not clarify the relationship between tautologies and the laws. It simply says that tautologies "express" the laws in some sense. Secondly, it does not say that the law of logic are the same as inference rules.

By the way which is the book that suports your definition of an inference rule

Wikipedia says:

A rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q".

So, in the Hilbert-style calculus for propositional logic, which is usually described in textbooks, an inference rule is a function on formulas. Now, a function is a special case of a relation. However, a function is usually assumed to be defined everywhere on the domain, but inference rules accept only functions of a special form. Therefore, it is fair to characterize an inference rule as a relation on formulas.

Harel, D., Kozen, D., Tiuryn, J. Dynamic logic. MIT Press, 2000, p. 69:

A Hilbert system consists of a set of axioms, or sentences in the language that are postulated to be true, and rules of inference of the form

\(\displaystyle \frac{\varphi_1,\quad\varphi_2,\quad\ldots,\quad\varphi_n}{\psi}\)

from which new theorems can be derived. The statements $\varphi_1,\varphi_2,\ldots,\varphi_n$ above the line are called the premises of the rule and the statement below the line is called the conclusion.

To state is more formally, a rule of inference is an ordered sequence $(\varphi_1,\varphi_2,\ldots,\varphi_n,\psi)$ of statements (here statements mean formulas), or at least an ordered pair of a set of formulas and another formula $(\{\varphi_1,\varphi_2,\ldots,\varphi_n\},\psi)$ (however, the latter definition looks unnecessarily complicated). And a set of ordered sequences of formulas is known as a relation on the set of formulas.

Finally, consider Mendelson, E. Introduction to Mathematical Logic. 6th ed. CRC Press, 2015, p. 27. This book is known for its formal approach. It says that a formal theory (also known as a calculus) includes

a finite set $R_1,\ldots, R_n$ of relations among wfs, called rules of inference. For each $R_i$, there is a unique positive integer $j$ such that, for every set of $j$ wfs and each wf $B$, one can effectively decide whether the given $j$ wfs are in the relation $R_i$ to $B$, and, if so, $B$ is said to follow from or to be a direct consequence of the given wfs by virtue of $R_i$.

In the footnote it continues:

An example of a rule of inference will be the rule modus ponens (MP): $C$  follows from $B$ and
$B\Rightarrow C$. According to our precise definition, this rule is the relation consisting of all ordered triples $\langle B, B\Rightarrow C, C\rangle$, where $B$ and $C$ are arbitrary wfs of the formal system.

Admittedly, most logic textbooks use the word "rule" in the dictionary sense, without defining it strictly. For example, The Open Logic Text. Open Logic Project, 2019-02-27, p. 73 says, "A rule of inference is a conditional statement that gives a sufficient condition for a sentence in a derivation to be justified". Here "justified" is used informally, and a rule of inference is thus a statement of the form "If formulas $A_1$, ..., $A_n$ are justified, then so is formula $B$". However, logic textbooks do not usually provide a precise definition of a statement, or proposition. On the other hand, one would like for a formal system (calculus) to be a purely formal object with a precise definition (a ordered sequence of sets, relations and so on.) Therefore it makes sense to view an inference rule as the essential content of the statement above, i.e., an ordered sequence $(A_1,\ldots, A_n, B)$, and describe its use in the definition of a derivation. Another difficulty with viewing inference rule as a statement above is that the term "justified" is undefined. Sometimes, "justified" is replaced by "provable" or "derivable", but again the use of these terms precedes the definition of provable (derivable) formulas. This is because they are in fact a part of the inductive definition of the set of derivable formulas. Inductive definitions is a subject of its own, which is very useful in mathematical logic. Many concepts (formulas, truth values of formulas, free and bound variables, derivations and so on) are conveniently defined using inductive definitions.

 

[Evgeny.Makarov]

A deep account of propositions (formulas), judgments and inference rules can be found in these lecture notes (PDF) by Frank Pfenning from CMU.

 

[solakis1]

I agree that \(\displaystyle A\,\&\,(A\to B)\to B\) is a tautology (and a formula). Who argued with that? But the words "which is the formula of M.Ponens" do not appear in the book. This page says nothing about the relationship of this formula with inference rules. In fact, the phrase "rule of inference" seem to appear for the first time on p. 181 and "modus ponens" appears first on p. 185.

Dont you know the basic fact that every rule of inference can be expressed as a conditional whose antecedent is the conjunofnction of premises and whose consequent is the conclusion
You can find that in any book in basic symbolic logic
The above conditional is the conditional of m.ponens because every rule of inference can be expressed with a line separating the premises of the conclusion as is shown clearly in
Margaris page 2

I agree with the phrase in bold. But firstly, it does not define the concept "the laws of logic" and does not clarify the relationship between tautologies and the laws. It simply says that tautologies "express" the laws in some sense. Secondly, it does not say that the law of logic are the same as inference rules.

]
Rules of logic are called rules of inference Margaris page 2

Admittedly, most logic textbooks use the word "rule" in the dictionary sense, without defining it strictly.

Definition of a rule of inference:
Every elementary rule of inference is an elementary valid argument form
Copi page 312
You see first you learn 3+5=8 and then study real analysis , never the other way round
CONCLUSION:
M.Ponens is a rule of inference or a law of logic and thas a tautology,because a valid argument form is a tautology
I hope you realize if we say 3+5=7 the whole of mathematics collapses

 

[Evgeny.Makarov]

Nice talking to you.

 

[topsquark]

I hope you realize if we say 3+5=7 the whole of mathematics collapses

I think saying Mathematics "collapses" is a bit too strong a statement. After all we don't know everything. Your example is oversimplified but it has often happened that someone has made a statement that has been accepted for centuries only to have it be disproven later on. (Admittedly I don't know of any examples of this in Mathematics...I primarily do Physics.) A false statement obviously needs to be fixed but if we don't know it's false the structure of Mathematics remains.

-Dan

 

[solakis1]

No rules the Universe-PYTHAGORAS 26 centuries ago
God gave us the Nos everything else is human invension- KRONECKER nearly 130 years ago

 

[topsquark]

No rules the Universe-PYTHAGORAS 26 centuries ago
God gave us the Nos everything else is human invension- KRONECKER nearly 130 years ago

Do you have a point?

-Dan

 

[solakis1]

No is ashort writing for Number
Sorry i don't get your question
I simply mentioned famous sayings about Nos