Rule of Inference: A->B - When is B?

[philiprdutton]

I have a basic question about the rule of inference.

Given:
A->B

That is to say, "if A then B". I am curious, "if A WHEN B?" Basically, when is B? Is B after A or before A? Actually I am asking about this in temporal terminology but, I am sincerely wondering if mathematical logic allows one to take both the non-temporal interpretation and the temporal interpretation OR just one of the two. Which interpretation is the default? Can I have both? Or just one? Which one? Am I even asking a valid question? If not then why?

 

[EnumaElish]

I don't see time having a special designation over other dimensions. In math, one can not only "forward time travel" but one can also travel backward in time. The question "when" has no distinction from the question "where."

 

[Rogerio]

I have a basic question about the rule of inference.
Given:
A->B
That is to say, "if A then B". I am curious, "if A WHEN B?" Basically, when is B? Is B after A or before A?

Take a look:
In the sentence "IF I ask a question THEN I'll get an answer" , B is after.
In the sentence "IF he got an oppinion THEN he had made a question" , B is before.
In the sentence "IF it was a lap year THEN there were 366 days" , B before or after makes no sense.

So, inference rules have nothing to do with time relations.

 

[fopc]

I have a basic question about the rule of inference.

Good luck.

 

[honestrosewater]

Basically, when is B?

When is B what? True? Proven? If you mean this as a rule of inference, it is saying something about the conditions under which A and B are true or provable.

If there is some confusion here, A and B denote formulas in some special, formal language, say, L, is that right? If you mean A -> B to also denote a formula in L, i.e., if -> is a symbol of L, then A -> B is not a rule of inference. A rule of inference says something about L. It is not a formula of L. It is a formula of one of L's metalanguages, e.g., English. It is a relation on L-formulas (or on sets of them or some other objects of L). The turnstiles, |- (syntactic implication) and |= (semantic implication), are common ways of denoting inferences. So you might write the rule of inference Modus Ponens as

1) (A -> B), A |- B

which contains and says something about three L-formulas ((A -> B), A, B) but is itself a formula in the metalanguage that you are using (English). Both -> and |- do include the idea of inference or implication, but they are not interchangeable.

One way to implement a concept of time is as an ordering. Do you see a way that "before" and "after" are similar to "less than" and "greater than"? (Or it might be clearer to say "A occurred before B", "A is less than B", etc.) You could add symbols and interpretations to L or your metalanguage (or both) to represent this order, but implication will not do the trick. Syntactic and semantic implications (|- and |=) are relations, but they are not order relations. You would have to add something new, which might say something like "A precedes B" as implication says "A implies B". You could add this to L as a function symbol or logical operator too, though I'm not sure how you would use it logically since I can't see how it is truth-functional. How do you think temporal order is relevant to proofs or truth-values?

Alternatively, you could define temporal orders on implication relations, thinking of an implication relation as telling you what's true or what you can prove at each instant and of time as just arranging those instances into some sequence.

 

[philiprdutton]

infer

When is B what? True? Proven? If you mean this as a rule of inference, it is saying something about the conditions under which A and B are true or provable.

If there is some confusion here, A and B denote formulas in some special, formal language, say, L, is that right? If you mean A -> B to also denote a formula in L, i.e., if -> is a symbol of L, then A -> B is not a rule of inference. A rule of inference says something about L. It is not a formula of L. It is a formula of one of L's metalanguages, e.g., English. It is a relation on L-formulas (or on sets of them or some other objects of L). The turnstiles, |- (syntactic implication) and |= (semantic implication), are common ways of denoting inferences. So you might write the rule of inference Modus Ponens as

1) (A -> B), A |- B

which contains and says something about three L-formulas ((A -> B), A, B) but is itself a formula in the metalanguage that you are using (English). Both -> and |- do include the idea of inference or implication, but they are not interchangeable.

One way to implement a concept of time is as an ordering. Do you see a way that "before" and "after" are similar to "less than" and "greater than"? (Or it might be clearer to say "A occurred before B", "A is less than B", etc.) You could add symbols and interpretations to L or your metalanguage (or both) to represent this order, but implication will not do the trick. Syntactic and semantic implications (|- and |=) are relations, but they are not order relations. You would have to add something new, which might say something like "A precedes B" as implication says "A implies B". You could add this to L as a function symbol or logical operator too, though I'm not sure how you would use it logically since I can't see how it is truth-functional. How do you think temporal order is relevant to proofs or truth-values?

Alternatively, you could define temporal orders on implication relations, thinking of an implication relation as telling you what's true or what you can prove at each instant and of time as just arranging those instances into some sequence.

So basically the rules of inference might be re-described as the rule of co-existence? If A exists then B exists but that we don't know anything about when they came into existence?

 

[philiprdutton]

boundary

I don't see time having a special designation over other dimensions. In math, one can not only "forward time travel" but one can also travel backward in time. The question "when" has no distinction from the question "where."

I also feel this way about time not having any place unless you put it there. However, let me ask you a philosophical question related to your comments: how does a mathematical system (a formal system like Peano's axioms) .. how does the system know when to "stop"?? If I ask what is 23+10 (two great athletic numbers) then how does the Peano formal system know to stop when it does (allowing the user to then take the answer for his/her use)?

If anyone out there knows the name of the philosophical question I am asking I would appreciate it. I am quite interested in this idea in relation to formal systems. I assume it gets standard coverage in the textbooks.

 

[Hurkyl]

What do you mean by "exist"? You've clearly indicated (if not explicitly) that A and B were part of your formal language, so it's obvious that A and B both exist in the usual sense...

Ordinary logic doesn't have any temporal component -- questions like "when" something comes into existence don't make sense.


One can do other things. For example, Boolean logics can be interpreted so that truth values are relations on some set -- if it was a set of "points", in some sense, then a truth value can be viewed as saying at which points something is true. But Boolean logic operates "pointwise"; you can't interpret a rule of inference as saying that something comes into existence "later".

One can do other things; people have studied various kinds of modal operators, or one can attach additional structure to a language or a theory, such as a measure of the "complexity of theorems".


I'm not trying to say you can't try and capture some temporal notion in logic -- I just want to make sure it's clear that there is no temporal notion in ordinary logic, so you have to either use a revised notion of logic, or attach additional data that represents temporal ideas.

 

[Hurkyl]

how does a mathematical system (a formal system like Peano's axioms) .. how does the system know when to "stop"?? If I ask what is 23+10 (two great athletic numbers) then how does the Peano formal system know to stop when it does (allowing the user to then take the answer for his/her use)?\

This doesn't sound like logic; it sounds more like you're thinking about computability.

+ is a binary operator. 23+10 is equal to 33, precisely because of the definition of +. There is no process involved, there is nothing to "stop"; it just is. The truth of the assertion that 23+10=33 can be derived from the axioms of Peano arithmetic and the definitions of all the symbols involved in that expression.


Now, if you're talking about computability, then one might wonder about an algorithm for computing 23+10 that's built out of some basic components -- the typical example is to ask if the computation can be carried out by a Turing machine. Integer addition is, in fact, a computable operation; we can write down an explicit algorithm, prove that the algorithm always terminates, and if the input is (x, y) and the output is z, then you can prove that x+y=z.


Computability theory, incidentally, is very closely intertwined with formal logic.

 

[philiprdutton]

if A exists

What do you mean by "exist"? You've clearly indicated (if not explicitly) that A and B were part of your formal language, so it's obvious that A and B both exist in the usual sense...

If this is the standard way to speak in logic:
A -> B means "if A is true then B is true..." << is this legal interpretation?

Then what makes something true? I think I could rephrase the above:

A -> B means "if A exists then B exists"

Pending the math community's approval, I dare say that my version also is correct in a philosophical sense. I also feel it is has a tangible quality.

 

[honestrosewater]

If this is the standard way to speak in logic:
A -> B means "if A is true then B is true..." << is this legal interpretation?

You tell us. Is that meant to be a semantic implication? If so, then yes. If it is instead meant to be a syntactic implication or material implication, then no, not generally.

Also, if you stop and think for a second about what a rule of inference is, having a rule that let's you derive an arbitrary formula (B) from another arbitrary formula (A) has some obvious major consequences. Would you want that as a rule? Do you think any popular systems have that as a rule? The same goes for truth considerations.

Then what makes something true?

You do. You just say what objects have the value "true", if "true" is one of your truth-values. You might want to do this in some systematic way and study these systems (i.e., study logic or model theory (or perhaps some innate reasoning faculty in humans or whatever, if you want to go that way)), but you have to start somewhere.

I think I could rephrase the above:

A -> B means "if A exists then B exists"

You could also say "if A sneezes, then B sneezes". But what makes you think that A and B are objects that can exist or sneeze? And what makes you think that either is of any relevance to logic, which is primarily concerned with truth and provability?

Pending the math community's approval, I dare say that my version also is correct in a philosophical sense.

In what philosophical sense? What does yours mean?

 

[dodo]

A -> B means "if A exists then B exists"

I think the A in A->B is not the same A in "if A exists...". The logical proposition, "A exists", can be true or false. So "if A exists..." becomes A' -> B', where A', B' are true/false propositions talking about the existence of some A and B.

 

[honestrosewater]

I think the A in A->B is not the same A in "if A exists...". The logical proposition, "A exists", can be true or false. So "if A exists..." becomes A' -> B', where A', B' are true/false propositions talking about the existence of some A and B.

The questions in this thread involve 2 languages and 3 types of implication, not counting the nontechnical types of implication already in English. I find it easier to adopt some additional symbols* instead of using "A -> B" for these several different types of formulas. But suit yourselves.

If "A -> B" is meant to be a semantic implication, then "A" and "B" are English expressions that represent formulas in L, or L-formulas. L-formulas are already logical propositions that can be assigned truth-values. In the paraphrase of this "A -> B", "if A is true, then B is true", "A is true" is not a formula in L but a formula in English. If philiprdutton is just replacing "is true" with "exists", the two "A"s are the same; they are English expressions that represent some formula in L, and the predicate "exists" is English.

Of course, the L-formula that this "A" represents could very well say something like "A exists", with this "exists" being a predicate of L and this "A" being something else. No one has said enough about L to rule this out, and maybe this is what was meant.

Still, even if you mean "A -> B" as material implication, i.e., as an L-formula, "A" and "B" must still represent L-formulas, which, even if you mean them to be atomic, are still propositions that can be assigned truth-values, so I'm not sure what your objection was.

If you want symbols to do double-duty and can't rely on context for disambiguation, then you need to say each time what they represent. What are each of the symbols in "A -> B" and "A' -> B'" meant to represent? (If "->" is meant to be semantic implication in both, you actually have three levels of languages, as the second is a relation on English formulas (which is what "A'" and "B'" must then be).) * For example, using A, B, C, etc. for L-formulas, -> for the material implication symbol of L, |- for the syntactic implication symbol of English, and |= for the semantic implication symbol of English.

 

[philiprdutton]

make formulas

...

Also, if you stop and think for a second about what a rule of inference is, having a rule that let's you derive an arbitrary formula (B) from another arbitrary formula (A) has some obvious major consequences. Would you want that as a rule? Do you think any popular systems have that as a rule? The same goes for truth considerations.
...

Creating formulas from formulas- Are the "formulas" the only "objects" of logic systems? In a sense, one uses logic to create new formula objects... like a substitution mechanism? For the simpler logics, can I just think of them as substitution systems? (I use the word "substitution system" informally).

Thank you,

 

[EnumaElish]

In relation to the OP, I couldn't help but notice honestrosewater's signature "I am a not a man (but am possibly a woman or a robot (or both))."

One can posit "not man --> (woman or robot)," which are simultaneous events. One can also posit "not born a man --> currently (woman or robot)," which is a cause-and-effect relationship (abstracting from bio-psycho-social implications, e.g. sex vs. gender). Finally, one can posit "currently not a man --> (born a woman or built as a robot)," which is an inference.

These examples prove that "-->" does not have a necessary relationship with the flow of time.

 

[honestrosewater]

Creating formulas from formulas- Are the "formulas" the only "objects" of logic systems? In a sense, one uses logic to create new formula objects... like a substitution mechanism? For the simpler logics, can I just think of them as substitution systems? (I use the word "substitution system" informally).

It depends on what you mean.

Do you realize that you might be asking questions similar to "Should it be legal for plants to have abortions? And if so, should the mother be required to inform the father?"? Those are at least odd questions, if they are acceptable at all, since they make assumptions that fail to be true normally or generally: that flowers are legal entities under human laws, are capable of being informed, reproduce sexually, etc.

Look carefully at what you're asking. If you are still talking about formal logics, it looks plainly contradictory. A formal logic is not an informal system -- hence the name -- so no, you cannot "rightly" think of a simpler (formal) logic as what you admit is an informal system. You might need to add a new type of system to your repertoire. I understand your not wanting to start completely from scratch (if such a thing is possible), but if you aren't willing to acknowledge that the study of logic might contain structures that are not perfectly analogous to any structures that you are already familiar with, I don't think I can help you.

I might be able to give you something of a bird's-eye view of what's going on, though, which I think is what you're going for. Are you comfortable with the mathematical concepts of sets, relations, or functions?