there may be two senses of the term "valid" in a logical context; one formal, the other informal and that evaluating an argument with either definition may cause different conclusions as to whether a given argument is valid. — NotAristotle

Of course.

But the ordinary formal definition is itself not equivocal. It is definite. It gives an 'if and only if' with a definiens in which all the terms are themselves defined back to primitive rubric.

Meanwhile, there are other formal definitions that differ from the ordinary formal definition. And they may also be not equivocal, though probably more complicated than the ordinary definition.

Meanwhile, there are different informal senses. If there is one in particular that you propose as being definite enough to avoid the kind of subjectivity equivocation in everyday discourse, then you're welcome to state it.

Or even if just one (but not all) of the premises is false and the conclusion is false (I am having trouble thinking of an example that meets this description). — NotAristotle

Having a false premise and a false conclusion does not in and of itself make an argument invalid. You have forgotten or did not understand the definition.

Okay, I actually do get that the example I just gave has "an interpretation wherein all the premises are true and the conclusion is false" such that it is "not valid." " Would you care to formalize the validity definition as it concerns arguments and do so using logical operators? I was trying to apply De Morgan's laws to your definition but I don't think it worked. On a side note, Banno I can hear your laughter and it is most unwelcome at this time.

:zip:

I can't see how it could matter if we designated a name for that special class of modus ponens described in the OP, where it is structurally consistent with modus ponens but is logically inconsistent. This thread strikes me as more of a primer in formal logic nomenclature than in logic qua logic. — Hanover

Whatever "structurally consistent" means there, a clear and simple way to say it is: The argument is and instance of modus ponens.

And no instance of modus ponens is inconsistent. What are consistent or inconsistent are sets of sentences. What is inconsistent in the argument is the set of premises.

This thread strikes me as more of a primer in formal logic nomenclature than in logic qua logic. — Hanover

This thread strikes me as having posters in it that are commenting on formal logic while knowing virtually nothing about it.

The original thread question would naturally be taken by many people to pertain to ordinary formal logic. But it can also be taken to pertain to other informal contexts, including everyday speaking and reasoning, and also can be discussed in context of alternative formal logics.

But when answers were given in terms of ordinary formal logic, certain posters commented as to the formal logic, while knowing virtually nothingabout it. Thus, correct explanations of it are exemplary meaningful, informative and generous posting.

I've said it maybe fifty times in this forum: Ordinary formal logic with its material conditional does not pertain to all contexts. But that is not a basis that one should not say how ordinary formal logic handles a question and not a basis that one should not explain ordinary formal logic to people who are talking about it without knowing about it.

Would you care to formalize the validity definition as it concerns arguments and do so using logical operators? — NotAristotle

Your question is answered by looking at the method of truth tables.

That is in chapter 1 of any book in 'Logic 1'.

I mean take the definition of validity, and write it as an expression using symbols and logical operators; is that something that can be done?

I don't mean examples of valid arguments, I am referring to the definition itself.

I was re-composing my post while you were posting your reply.

Meanwhile, do you have any thoughts about offering your own unequivocal definition of 'valid argument'?

It has been a long time since I learned some logic and I wasn't great at it, but I do know what truth tables are and I think how to use them.; I don't see how that implies a definition of "validity" using classical logic.

Here are two ideas for defining validity: (1) an argument is valid when all the relevant rules are followed. Or, (2) an argument is valid when the meaning of the premises leads to the meaning of the conclusion.

They don't imply a definition of validity.

If you read chapter one, you'll understand that we have:

(1) a definition of 'valid argument'

(2) a definition of 'interpretation' such that an interpretation assigns truth values to sentence letters

(3) a stipulation for how the truth value of a compound sentence is reckoned per an interpretation, so that for any interpretation and any compound sentence, we can reckon the truth value of that sentence.

(4) To determine whether an argument is valid or not, apply (1), (2), (3).

And, if I recall, I even showed in this thread how to set that up with truth tables.

/

In a nutshell: I defined 'valid argument'. And earlier in this thread, I defined 'true in an interpretation' for compound sentences. So apply the definition of 'valid argument' by considering the truth values of the compound sentences per each of the interpretations.

What relevant rules? What makes a rule relevant? Whose rules? What if people use different sets of rules from one another? What if the rules are unclear or ambiguous?

What is the meaning of a sentence? How do we unequivocally, let alone objectively, determine the meaning of sentences? What theory of meaning? What if people take different meanings of sentences from one another? What if someone takes 'valid' to mean causal connection but they don't take causality and meaning to be the same? What about people who consider 'valid' to require that all the premises are true?

Of senses of 'valid' different from yours, are they wrong? Or can there be different reasonable senses of 'valid'? The ordinary formal sense cannot be among those different reasonable senses?

Relevant rules like conditionals "And" "Or" operators-- when those are used correctly the rules are followed and the argument may be considered valid. Any rule that is such that if it weren't followed, the conclusion would be different, is a relevant rule. The rules would ideally be universal and based on logical intuition; if people use different sets of rules, then the rules must be clearly communicated so that that "logic" can be understood or followed.

The meaning of the premise and conclusion depends on the expressions used (I guess this definition isn't unequivocal as it would only apply to ordinary natural language, not to formal logic). I don't know any theories of meaning so I can't answer that. If the meanings differ, then I'm not really sure what the result would be, seems like communication is out the door let alone logic if we can't agree on the same meaning of words and sentences.

And a relevant rule is correctly followed just in case.. if it were the case that all the premises were true and the relevant rule is followed, then the conclusion must also be true.

Relevant rules like conditionals "And" "Or" operators-- when those are used correctly the rules are followed and the argument may be considered valid. Any rule that is such that if it weren't followed, the conclusion would be different, is a relevant rule. The rules would ideally be universal and based on logical intuition; if people use different sets of rules, then the rules must be clearly communicated so that that "logic" can be understood or followed.

The meaning of the premise and conclusion depends on the expressions used (I guess this definition isn't unequivocal as it would only apply to ordinary natural language, not to formal logic). I don't know any theories of meaning so I can't answer that. If the meanings differ, then I'm not really sure what the result would be, seems like communication is out the door let alone logic if we can't agree on the same meaning of words and sentences. — NotAristotle

What is an example of rule that if it weren't followed then the conclusion would be different? Different from what?

The rules of formal systems in mathematical logic and computing are not just clearly communicated, but they can be checked algorithmically for correctness of application.

What are some examples of your rules that are communicated differently?

What if two people both like the same rules, but have different intuitions as to whether they're being correctly applied?

How are your rules for your propositional logic different from those in ordinary formal logic?

Of course, the meanings depend on the expression. But I'm asking, given an expression, what determines its meaning, or its meanings?

People disagree about meanings often. We can't do logic with expressions because people disagree about their meanings?

relevant rule is correctly followed just in case.. if it were the case that all the premises were true and the relevant rule is followed, then the conclusion must also be true. — NotAristotle

You say a relevant rule is on such that if all the premises are true then the conclusion is true.

That is the ordinary definition of 'valid' in formal logic.

And it doesn't say anything about meaning other than truth and falsehood.

So what are you adding to the ordinary formal definition or how are you disagreeing with it?

You're circular. You say that an argument is valid only if the rules used are relevant, but also rules are relevant only if they are truth-preserving. But truth-preserving is the same as valid.

P->Q. P. Therefore, not-Q. would both flout the meaning of the conditional, and in such a way that it changes the conclusion. It's different than what the conclusion should be (namely Q).

I don't understand the second question.

Third question answered as correctly used rules is defined.

I don't know the difference between propositional logic and ordinary formal logic so I do not know how to answer this one.

The meaning of an expression depends on what the speaker intended by it - natural language I would think would go along way in dissolving confusion over what is meant.

Right, where there is disagreement over a meaning, that meaning is not well-formed and not suitable for logical operations. I would expect something like that to be true for your definition of validity as well.

I guess I agree with the ordinary definition of valid in formal logic. That is not the definition you cited earlier in the thread - the definition that I am suggesting an alternative to.

I do not see truth preservation as synonymous with validity; I defined validity as rule following; a rule is followed correctly if it preserves truth; I didn't define validity as truth-preserving. Truth preservation is a consequence of validity, namely, following the relevant rules correctly.

So actually, I would say my definition of valid is different from the ordinary formal logic definition in that I am defining validity in terms of rule-following, not in terms of truth-preservation; truth-preservation is more like a consequence of the definition.

(1) What is the meaning of the conditional?

(2) A set of premises can prove more than one conclusion. So what is "the" conclusion that "should be"?

(3) I wrote: "What are some examples of your rules that are communicated differently?"

I meant: What are some examples of your rules that are communicated clearly?

(4) How do you know what the speaker intended? What if there is not a particular speaker? People disagree about what speakers intend often. And people misunderstand and disagree as to what was said often. You say your answer goes a long way to making clear what the meanings are. Well, yes, often it's pretty clear what a speaker intends, but not so often that it would determine which arguments are valid, since too often it is quite unclear what was intended.

(5) Yes, if the expression is equivocal, then either we reject it of choose one of the candidates for its meaning. But rather how would we determine in an objective way whether there is or is not equivocation and, if so, which candidate to choose?

Formal logic does not presume to know how sort out many of the difficulties in everyday speech; only that if we are given sentences that have a formal relation, we can determine validity of arguments. That is the very point. Suppose someone writes words from a language I don't know:

tarabalu bock meras dan pelrere bosoundo tam.
erofereht, pelrere bosoundo garom

As long as I know that 'dan' means 'and' and 'erofereht' means 'therefore', I can say, "On the assumption that the foreign language expressions are declarative statements, then the argument is modus ponens, and, as an instance of modus ponens, it's valid".

That is, if I know what are the connectives and how we reckon the truth of compound statents, I can tell you about the validity of the argument.

But you have not given such forms, but rather, we have to know the meanings first before determining validity.

So, your offer requires sorting out all the problems about 'meaning' much more than the formal method does.

(6) What definition of 'formal logic' have I given that contradicts any other definition I've given?

(7) An argument form is truth preserving if and only if, for every instance of the form, there is no assignment of truth values such that the premises are all true and the conclusion is false. That is, mutatis mutandis, the same definiens as for 'valid argument'.

(8) Yes, you defined validity as rule following, but then you defined proper rule following as being truth preserving. That's your circularity. You say, a rule is proper only if it is truth preserving, and an argument is valid only if it uses only proper rules. But truth-preservation is validity.

P->Q. P. Therefore, not-Q. would both flout the meaning of the conditional, and in such a way that it changes the conclusion. It's different than what the conclusion should be (namely Q). — NotAristotle

Notice that you didn't say anything about the meanings of P and Q, even if they were translated to a natural language.

Rather, you mentioned only the meaning of the conditional (and I would mention also the meaning of 'not'.)

That's an example of formal logic. We stipulate how the connectives determine the truth or falsehood of compound statements depending on all the assignments of the sentence letters or atomic statements, without having to consult otherwise as to the meanings of those atomic statements.

my definition of valid is different from the ordinary formal logic definition in that I am defining validity in terms of rule-following, not in terms of truth-preservation; truth-preservation is more like a consequence of the definition. — NotAristotle

Nope. You say that your notion of validity is based on proper use of rules, but your notion of proper use of rules goes through the notion of rules being truth-preserving. But truth-preserving is validity, even as you defined yourself "If the premises are true then the conclusion is true".

And a relevant rule is correctly followed just in case.. if it were the case that all the premises were true and the relevant rule is followed, then the conclusion must also be true. — NotAristotle

That is equivalent to saying: A rule is correctly used only if application of it never leads from true premises to a false conclusion.

And "never leads from true premises to a false conclusion" is validity.

You've managed to define validity in terms of correct rules and correct rules in terms of validity.

1. The conditional means that in the event that the antecedent is true, the consequent must be true. It is one of the logical rules that must be followed for the argument to be valid.

2. Provided that a set of all conclusions follows the rules correctly and is exhaustive of all such conclusions, that set encompasses all legitimate conclusions.

3. logical operators.

4. We would have to ask the speaker to clarify.

5. Noted, let's set aside questions concerning meaning; the second definition may have more problems then I can resolve.

6. Okay.

7. So then "the truth of the premises guarantees the truth of the conclusion" is the same as "there is no interpretation (assignment of truth values) such that the premises are all true and the conclusion is false" ?

8. So I think what I am trying to say is that the definition of validity is following the rules correctly. And that following the rules correctly is defined by rule-following that results in truth preservation. Such that, truth preservation is a consequence of rule following, and it is the rule following itself that is responsible for the validity. In other words, the premises themselves don't guarantee the truth of the conclusion, rather the following of the rule(s), given that the premises are all true, is what guarantees the truth of the conclusion. Put another way, truth preservation does not make the argument rule-following, but rule-following is what makes the argument truth preserving. (Truth preservation does make the rule-following "correct.") Not sure if that totally makes sense.

(1) How is your meaning of the conditional different from the ordinary meaning in formal logic?

You use "must"; is that in addition to "is"? Example:

If Bob is smart then Bob knows English history.

In ordinary sentential logic, that is false in an interpretation if and only if Bob is smart and it is not the case that Bob knows English history.

What is your sense of "must"?

Is it that there are no possible circumstances in which Bob is smart and Bob does not know English history? That is, there are no interpretations in which Bob is smart and Bob does not know English history? Or does "must" mean something else for you?

As I shared already, note the difference in ordinary formal logic:

Here 'P' and 'Q' are variables ranging over sentences:

P -> Q
is true in a given interpretation if and only if either P is false in that interpretation or (inclusive 'or') Q is true in that interpretation.

and a different notion:

P entails Q
if and only if there is no interpretation in which P is true and Q is false.

Symbolized :

{P} |= Q

(2) Do you mean, "as long as the rules are followed (applied) correctly" or do you mean "as long as the conclusion follows correctly from the rules"?

And I asked you already, what is "the correct conclusion" when there may be many correct conclusions? And you skipped answering

And "exhausts"? A lot of different conclusions may follow from a given set of premises. How would you know that you exhausted them? I asked that already, and you skipped answering.

(3) Connectives are not rules. Rather, we have rules for connectives. What are your rules for the connectives?

(4) What if the speaker is not around to clarify? What if the speaker is too confused himself? What if there's not a particular speaker but rather the statement is a general public statement? I asked you about that already, and you skipped answering.

(5). Okay, so take out meaning. Now, how is your offer different from ordinary formal logic?

(7) Yes, they are equivalent. The former is a less rigorous way of saying the latter.

(8) You say, "the definition of validity is following the rules correctly. And that following the rules correctly is defined by rule-following that results in truth preservation."

And, as I've shown, that is circular. You have not given an answer to that other than eliding that validity and truth-preservation are the same.

(9) Whether a rule is truth-preserving or not is not based on whether it happens to be a rule, but rather on the fact that it is truth-preserving. One can make any rule one wants to make, and it will or will not be truth-preserving not on the basis that one says, "It's a rule" but rather on the basis that any application of the rule is truth-preserving.

So, still in your situation:

To define 'valid' (truth-preserving) you appeal to correct rules. But what is a correct rule? Well, it's one that is valid (truth-preserving). That's circular.

Let's compare with a non-circular approach in ordinary formal logic:

First, define, 'is an interpretation'.

Second, define 'is true in an interpretation' and 'is false in an interpretation'.

Third, define 'valid argument' without mentioning inference rules, but only mentioning interpretations and true and false.

Fourth, we state rules that are correct in the sense that they provide for only valid arguments.

Meanwhile, I hope you're looking up the method of truth tables.

I might indeed be mixing things up. Formal logic was something I did in my own time over 20 years ago but haven't really used it since then. Happy to brush up again though, since I'm considering making a career switch where argumentation becomes more important again. Any online tips you'd give?

My point is that we know that If P then Q, where P = A and Q = not-A, implies a contradiction where P is true because Q will be true and both A and not-A will be the case. It is counterintuitive to assert that "if it rains then it doesn't rain" and "it rains" therefore "it doesn't rain" is a valid argument. So formal logical inference appears to ignore the obvious rejection any normal person would have with the natural language sentence without going through a logical proof (the resulting logical contradiction is rather obvious).

So we know the premisse is unsound but it seems to be of another order when it's unsound due to a logical contradiction then say because it fails to take into account the fact there are additional causes for a consequent to happen (any time really where correlation isn't causation). I guess "formal" in formal logic really is the main point of that system.

1. I take a conditional to be saying: if the antecedent is true, it can't be the case (there is no circumstances such) that the consequent is false.

2. Rather than a correct conclusion, all we need are conclusions that follow the relevant rules, any and all such conclusions are legitimate.

3. I refer to connectives as rules.

4. Then we are out of luck.

5. I drop the truth preservation condition for validity.

8. If we drop the truth preservation part of the definition, it is not circular. An argument is valid where it follows the relevant rules. Period. I don't think it is necessary for me to stipulate that a rule be followed "correctly," just that it be followed.

- Thank you. Good post. :up:

My point is that we know that If P then Q, where P = A and Q = not-A, implies a contradiction where P is true because Q will be true and both A and not-A will be the case. — Benkei

"If P then Q" means "not P or Q".
"If A then not A" means "not A or not A".

"not A or not A" is not a contradiction.

It is counterintuitive to assert that "if it rains then it doesn't rain" and "it rains" therefore "it doesn't rain" is a valid argument. — Benkei

"If it rains then it doesn't rain" means "it doesn't rain or it doesn't rain".

So the argument is:

P1. it doesn't rain or it doesn't rain
P2. it rains
C1. therefore, it doesn't rain

Notice that P1 and P2 cannot both be true. If P1 is true then P2 is false; if P2 is true then P1 is false.

C1 is irrelevant. It could be anything, as per the principle of explosion.