I once asked a professor what formal logic is good for. He gave me two answers.
It helps you to be certain that an argument (yours or someone else’s) is deductively valid.
Learning formal logic clarifies your mind.
Those seem like pretty good reasons for learning to use formal logic. In that light, I was wondering why Objectivists think it is so horrible.
Give me an example of any of those putative benefits and I will show you.
Give me an example of any of those putative benefits and I will show you.
The example he gave for #1 was the discovery of a modal fallacy in an argument against the existence of God. (I know, I know, but surely we want to know which arguments against God are actually good arguments against God, right?) This website gives a discussion of the specific fallacy. Briefly, in natural language you can’t distinguish very easily between logically distinct phrases like I and II below, which can lead to problems like the one discussed in the link.
I) If P, then necessarily Q.
II) Necessarily, if P, then Q.
But formal logic lets us write them like this:
I’) P→ □Q
II’) □(P→ Q)
So, per benefit #1, the more rigorous formal logic helps us see tricky little fallacies like the above in our reasoning. Benefit #2 seems to be a more subjective benefit, so I think I would just have to take his word for it. (Although actually, just doing this bit of research helped me think a bit more clearly…)
I fully expect someone to point out that Objectivists do not accept necessity. True, but what we did with “necessarily” above could be done with other modifiers, e.g. “metaphysically given” or “from moral obligation”. So,
I) M(P -> Q)
II) P -> MQ
could be
I) It is metaphysically given that if P, then Q.
II) If P, then it is metaphysically given that Q.
or
I) It is obligatory that if P, then Q.
II) If P, then it is obligatory that Q.
Which, if nothing else, seem to be useful distinctions to be able to make with precision.
For starters, there is no example there of helping you be certain that an argument is deductively valid. The formal sequence is one allowed by the rules of deduction, so this is actually a nice case arguing against formal logic, if one wanted to make the case. (It’s an argument against inept logic). Arguments are made in natural language, and in order to formally validate an argument, you have to be able to perform a correct translation from natural language. That’s what this “argument” didn’t do. Let’s see how that goes. I’ll just take the first two lines of the argument
I won’t pick on their English much (bad use of “have”). By symbol substitution you can represent 1 as 1′: P. You cannot do that with 2 to 2′. Let’s reword 2 as 2a:
so by symbol substitution 2a you can get 2′: P→ □Q. I’m just focusing on the formal part. I will henceforth correct the wording. And then you can get by ordinary modus ponens 3. Necessarily the situation will have that outcome, formally □Q.
Introducing a new assumption 4. If necessarily the situation will have that outcome, then no human in the situation F has free will, we can symbolize that as 4′ □Q→S.
Now, here is the technical problem. The symbolization of 4 with Q is invalid, since Q is already “taken”. Q is actually logically complex, and the Q of 3 is not the same as the Q of 4, because both Q’s use two variable, corresponding to natural language “the situation” and “that outcome”. It is not guaranteed that these refer to exactly the same things, therefore you cannot translate the two (distinct) propositions into the same letter. The antecedent in 4 must be R, or some other letter.
In addition, the validity of the axioms of modal logic are (unlike those in first order logic) of dubious truth value. The validity of rules pertaining to “&” derive from the fact that it reflects the semantics of “and”; but that isn’t the case for modals.
So before getting to any Objectivist-type content, we are not seeing any use of formal logic to establish logical validity of an argument (you are seeing a misuse). You are not even seeing proper application of formal logic.
Now we could turn to the Objectivist content, of course. As you know, false and arbitrary propositions are not possible premises in a deduction; thus the premises must be proven first. Hence this attempt doesn’t even get out of the barn.
To turn to your addendum, your candidates
I) M(P -> Q)
II) P -> MQ
aren’t formally well-formed. You can’t just ram letters and parentheses together. Eventually you might come up with a system where those are well-formed expressions, but so what, because we still need to relate them to an argument – which is a linguistic expression. Until you have a proven translation from natural language to formalization, you don’t have an object that can be formally evaluated and found valid or invalid.
So that’s a nutshell of the reason why formal logic is not useful for evaluating the validity of an argument. It is neither necessary nor sufficient (in fact, it is not presently possible).
Damn good post, David.
It helps you to be certain that an argument (yours or someone else’s) is deductively valid.
Learning formal logic clarifies your mind.
Whatever the merit of those claims, there are other ways of describing the role of formal logic, especially in mathematics. in the study of formal languages (which include mathematical languages and computer languages), and, famously, in such heuristic applications as switching circuits, for example.
The example he gave for #1 was the discovery of a modal fallacy in an argument […]
I would think that modal logic would not be the best place to start with examples, since the notion of validity (or what are appropriate axioms) has not nearly as much consensus in modal logic as with other forms of formal logic (which themselves don’t have perfect consensus).
your candidates
I) M(P -> Q)
II) P -> MQ
aren’t formally well-formed.
Sure they are. There’s no problem at all in giving recursive formation rules for modal sentential operators, and the formulas above (where ‘M’ is obviously standing for the modal ‘box’ operator) are perfectly formed in an ordinary syntax for a language of modal logic.
Sure they are.
Snore. There’s a difference between “It could be well-formed, if you defined a system in which it’s well-formed” and “It is well-formed”. Until ctrl-y sets forth that system, there’s no point in talking about something that might be well formed. The question is about formal systems, so let’s stick to that, and not hypothetical formal systems.
For starters, there is no example there of helping you be certain that an argument is deductively valid. The formal sequence is one allowed by the rules of deduction, so this is actually a nice case arguing against formal logic, if one wanted to make the case. (It’s an argument against inept logic). Arguments are made in natural language, and in order to formally validate an argument, you have to be able to perform a correct translation from natural language. That’s what this “argument” didn’t do.
The website intended to show that the proper interpretation of the argument is invalid, and it did. That is easier to establish in formal logic than in natural language. Just because the website did not lay out the proper interpretation of the argument in stepwise form does not mean that it did not establish its deductive invalidity.
Let’s see how that goes. I’ll just take the first two lines of the argument
- God knows which outcome situation F will have at time t
1
. (premise or hypothesis)2. If God knows which outcome situation F will have at time t
1
, then the situation will necessarily have that outcome. (premise)I won’t pick on their English much (bad use of “have”). By symbol substitution you can represent 1 as 1′: P. You cannot do that with 2 to 2′. Let’s reword 2 as 2a:
- If God knows which outcome situation F will have at time t
1
, then necessarily the situation will have that outcome.so by symbol substitution 2a you can get 2′: P→ □Q. I’m just focusing on the formal part. I will henceforth correct the wording. And then you can get by ordinary modus ponens 3. Necessarily the situation will have that outcome, formally □Q.
Introducing a new assumption 4. If necessarily the situation will have that outcome, then no human in the situation F has free will, we can symbolize that as 4′ □Q→S.
Now, here is the technical problem. The symbolization of 4 with Q is invalid, since Q is already “taken”. Q is actually logically complex, and the Q of 3 is not the same as the Q of 4, because both Q’s use two variable, corresponding to natural language “the situation” and “that outcome”. It is not guaranteed that these refer to exactly the same things, therefore you cannot translate the two (distinct) propositions into the same letter. The antecedent in 4 must be R, or some other letter.
The natural language obviously intended them to refer to the same situation and outcome.
In addition, the validity of the axioms of modal logic are (unlike those in first order logic) of dubious truth value. The validity of rules pertaining to “&” derive from the fact that it reflects the semantics of “and”; but that isn’t the case for modals.
Okay, but surely you can see that there are two sorts of propositions here that are easier to distinguish in formal logic than in natural language.
So before getting to any Objectivist-type content, we are not seeing any use of formal logic to establish logical validity of an argument (you are seeing a misuse). You are not even seeing proper application of formal logic.
Now we could turn to the Objectivist content, of course. As you know, false and arbitrary propositions are not possible premises in a deduction; thus the premises must be proven first. Hence this attempt doesn’t even get out of the barn.
Which premise is false or arbitrary? The one about God? But surely if God exists, then God just is a being with the power described. Even if the idea of God is all shot through with contradictions, we can imagine someone saying he will ignore everything about God except this one ability, and argue against God’s existence from that.
If you must press this objection, then there are other cases of the modal fallacy without any premises you would call false or arbitrary. See this page and particularly this instance.
To turn to your addendum, your candidates
I) M(P -> Q)
II) P -> MQ
aren’t formally well-formed. You can’t just ram letters and parentheses together. Eventually you might come up with a system where those are well-formed expressions, but so what, because we still need to relate them to an argument – which is a linguistic expression. Until you have a proven translation from natural language to formalization, you don’t have an object that can be formally evaluated and found valid or invalid.
So that’s a nutshell of the reason why formal logic is not useful for evaluating the validity of an argument. It is neither necessary nor sufficient (in fact, it is not presently possible).
That might be a problem for the “metaphysically given” predicate. However, there are already a number of systems that employ “obligatory.”
The focus so far has been on establishing that the benefits listed in my OP do not exist. While I understand that the claimant (in this case the “formal logician”) bears the burden of proof, I assume Rand made some sort of argument against formal logic at some point, and I was wondering what the gist of that argument was.
There’s a difference between “It could be well-formed, if you defined a system in which it’s well-formed” and “It is well-formed”. Until ctrl-y sets forth that system, there’s no point in talking about something that might be well formed. The question is about formal systems, so let’s stick to that, and not hypothetical formal systems.
The poster gave enough description for a reasonably informed and alert reader to understand that his context is of an ordinary language for modal logic with ‘M’ standing for the box symbol. His formula is clearly well formed in such context. Any logician in the world would recognize it as a well formed formula of a language for modal logic in which the character ‘M’ stands for the box symbol. Your objection as to this particular point is frivolous.
Snore.
Charming.
The website intended to show that the proper interpretation of the argument is invalid, and it did.
Then I don’t understand your question or claim. Are you claiming that it is possible to use formal logic to demonstrate the validity of an argument? That was what you implied before, in which case that web page would be irrelevant.
That is easier to establish in formal logic than in natural language. Just because the website did not lay out the proper interpretation of the argument in stepwise form does not mean that it did not establish its deductive invalidity.
It means exactly that. If a symbolic sequence is syntactically well-formed by the rules of some system of logic, all it shows is that the symbolic sequence is syntactically well-formed by the rules of that system of logic. It shows nothing about an argument. It might show something about an argument, if you can demonstrate that the argument does translate to that symbolic sequence. And, as I pointed out, it does not.
What is your evidence, or indeed metric, that leads you to conclude that something is “easier” to demonstrate in a symbolic system that in natural language?
The natural language obviously intended them to refer to the same situation and outcome.
Do you have any evidence to support that assertion? Really, what you need is a formal system that maps natural language utterances onto symbolic sequences, so that you could prove the relationship that you’re claiming (in the same way that you “prove” the symbolic theorem). But I’d settle for some kind of evidence. Remember, you wanted to focus on formal logic, so I’m taking you at your word. You ask why Objectivists reject meaningless symbolic gobbledygook, and I am answering your question – because it is meaningless. It refers to nothing. You have no model-theoretic interpretation.
Okay, but surely you can see that there are two sorts of propositions here that are easier to distinguish in formal logic than in natural language.
No, I do not see that. Why don’t you come up with a better and clearer example of what you are talking about? (Let me help you here: symbolic formalism does allow you to distinguish logical constituent relations – via the use of parentheses – which may (but only optionally) map to a single surface form in natural language. If your point is that there are ambiguities in some natural language utterances which tend to be avoided in logical formalism, then I agree, but that is something that can be easily dealt with using the resources of natural language).
Which premise is false or arbitrary?
All of them. None of the premises have been validated, and they are not axiomatic.
That might be a problem for the “metaphysically given” predicate. However, there are already a number of systems that employ “obligatory.”
Are you really committed to making this about modal logic? The fact that there are a number of people who are mistaken does not change the fact of the mistake. As a starting point, come to grips with the fact that all “is”'s are necessary.
what you need is a formal system that maps natural language utterances onto symbolic sequences
Since natural languages are not formal languages or formal objects, it’s hard to conceive how a formal system could map from a natural language. On the other hand, there is a formal version of the OPPOSITE direction in two senses, one formal the other still informal: (1) Formal. We map symbols of the formal language to formal objects and relations and then, based on that mapping, we employ a recursively defined function that maps formulas of the formal language to truth values (and I know that you object to the formal notion of truth values as represented by 0 and 1 or similar formal means). (2) Informal. We map symbols of the formal language to objects and relations (objects and relations that may be described in natural language) and then, based on that mapping, we employ a method that is just an informal version of using a recursively defined function that maps formulas of the formal language to announce that they are true or false.
You ask why Objectivists reject meaningless symbolic gobbledygook, and I am answering your question – because it is meaningless. It refers to nothing. You have no model-theoretic interpretation.
It’s not gobbledygook. And as to model-theoretic interpretation, OF COURSE, we may provide model-theoretic interpretation. That is what the formal semantics IS (and has the informal counterpart I mentioned, also). As to modal languages specifically, a formal semantics is more complicated than for classical predicate calculus, but there are methods of models to be applied to modal languages (with informal counterpart too).
Meanwhile, given just an ordinary textbook explanation, a reasonably bright junior high school student should have little problem understanding that the following is a valid argument [here ‘M’ is being used as a 1-place predicate symbol, not, as with the other poste, to stand for a modal operator]:
Ax(Hx -> Mx)
Hs
Ms
as, in particular
Every human is mortal.
Socrates is human.
Therefore, Socrates is mortal.
is valid, and as can be determined by its FORM which is given symbolically above.
If your point is that there are ambiguities in some natural language utterances which tend to be avoided in logical formalism, then I agree, but that is something that can be easily dealt with using the resources of natural language).
There are plenty of examples of formulas of mathematics, even in finitistic investigations in number theory and finitistic combinatorics that would be quite unmanagable and virtually impossible to understand if we attempted to convey them in natural language alone. Moreover, much of the point of a formal logical caclulus is not just to acheive unambiguity but also to allow that it is mechanical (thus perfectly objective) to check whether a purported proof (such as a mathematical one) is indeed a proof (and I know your own view that such deductions are meaningless anyway unless [fill in here whatever paraphrase of the rest of your view].
On the other hand, in Mathematics there are symbols used to represent concepts that are meaningless in reality. For example, the quite common concept of negative numbers. The concept of negative numbers is fundamentally used to represent subtraction - the taking away of some quantity of some entity. But mathematically, the symbolic use of negatives is interpreted as being a negative item. For example 3x*(-1) = -3x. That’s a perfectly valid formula, but the interpreted meaning of the formula is that there are minus 3 of something. Then there’s i - the square root of -1, implying that not only is -1 real but it can have a square root that, when multiplied by itself, equals a negative - and even worse, -i and ∞.
Mathematically these symbols are perfectly sensible - but in reality there is no such thing as a negative. There is no “2 apples plus a negative apple” or “three apples (x) times negative 1 = three negative apples”. You can start with 2 apples and remove one, and the one is simply gone - its not eliminate by a negative apple. And you’ll never, ever see -i apples or ∞ apples because neither can exist in reality.
I like the idea of symbolic or formal logic - but its problem is that of mathematics - just because you can represent something with it doesn’t make it real or true, just as in any other language. Saying “God is a fuzzy blue sky dragon” neither makes God nor the fuzzy blue dragon in the sky real.
I like the idea of symbolic or formal logic - but its problem is that of mathematics - just because you can represent something with it doesn’t make it real or true, just as in any other language. Saying “God is a fuzzy blue sky dragon” neither makes God nor the fuzzy blue dragon in the sky real.
This is the point that defenders of symbolic logic as a means of understanding truth typically fail to grasp. There is nothing wrong with studying methods of reasoning with such precision that you can actually reduce aspects of logic to simple and general symbols, but those symbols and formal operations have no relationship to logic unless they are actually rigorously connected to reality. A formal symbol is valid as a model of logic only if it does in fact correspond to a real aspect of logic. If you completely cut the cord and recognize “I’m just creating a formal system without claiming that it relates to logic”, then there is no objection to pursuing a system where 
←§|∆₡
↓גּ is a well-formed formula.