Let ##A##, ##B## and ##C## be propositions. Then the following formulas are equivalent.
\begin{align*}
&A\to (B\to C)\\
&A\land B\to C\\
&B\to (A\to C)\end{align*}Therefore there are many ways to construct equivalent statements using contrapositive. Below ##F\equiv G## means that ##F## and ##G##...
Here is a couple of formulas saying that a natural number p is prime.
\begin{align*}&p>1\land \forall d\,(d\mid p\to d=1\lor d=p)\\
&p>1\land \forall d\,(1<d\land d<p\to \neg(d\mid p))\end{align*}The original variant (∀d∈N)[(d>=1∧p≠1)∧(d|p⇒((d=1)∨(d=p))] is incorrect because it says in...
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...
And I am telling you again:
And knowing your tendency to conceal the definitions you use, I expect you to ask this question several more times without revealing the definition or the reason for your question.
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.
So you assume a definition according to which the answer to the original question ("Are the inference...
No.
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...
Instead of writing another example it would be more useful to provide a definition of a valid argument.
Your argument involves not just propositional variables, but propositional constants (such as "London is in England"), which have fixed truth values.
Since the term "argument" is not usually used in books on mathematical logic, as I wrote above, I am not familiar with the definition of a valid argument when it involves digits, + and =. If the statements occurring in an argument are proposition or predicate formulas that consist of some...
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...
"An argument" is not a term that is used in most textbooks of mathematical logic. What Copi calls a valid or invalid argument is a true (correspondingly, false) claim about logical (or semantic) consequence. This claim is usually denoted by $\Gamma\models A$ where $\Gamma$ is a set of formulas...
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.