Many-valued logic system (need help from math philosophers)

[Ben_Sinner]

Dear math philosopers,

Recently I discovered that not everyone believes in bivalent logic (things can only be true or false), the law of contradiction or excluding the middle.

Some people believe in a trivalent logic system or ‘many-valued’ logic where there are more than just true or false.

In a trivalent/many-valued logic system, if P is true, ~P could be false, or, indeterminate. If ~P is true then ~~P could be false or indeterminate

I want to know what the Catholic position on this would be.

One poster on an earlier thread referenced a philosopher named Michael Dummett who came up with intuitionistic logic, which consisted of this statement

Under this system, one can go from P to ~~P but not ~~P to P

The poster was confused on why this would be so, and so am I. How would one even go about refuting that position? Why would the position be false, etc?

Is mathematical logic an accurate portrayal of reality?

Alot of questions, but I’m very confused with all of this and I can’t shake it off because it feels like this is something I would hear from a cult leader or something, not a mathematical philosopher.

 

[Rhubarb]

I know we were bouncing back and forth in the other thread but I’ll do it here too. The thing to remember about logic is that the results inside a logical framework are only relevant within that framework. Each logical system has its own rulebook that tells you how to treat different sentences and connectives. Logical properties like “truth” and “validity” can only be measured within the confines of the framework used. So using Sentence Logic we can do a lot of things and analyze sentences. But, SL breaks when you try to use sentences that need the existential or universal quantifier. Consider the classic syllogism.

1. All men are mortal
2. Socrates is a man

---

1. Socrates is mortal

This looks right, and it is right. But if I translate the above into Sentence Logic we get this:

1. A
2. S

---

1. M

And we can’t do anything with that. So when we encounter these sorts of sentences we use first-order (predicate) logic. Then we can analyze the argument and make some headway. There are statements and arguments that require different modal logics, and so on. Logic is just a tool-kit we use to analyze language. Nothing more It’s just that some logics seem to map onto our world very well, and some do not.

I did some digging regarding the intuitionist logic and ~~P not being equivalent with P in that system. The reason is because if P is true, then ~~P is also necessarily true. But not the other way around. It’s the same thing as IF P THAN Q doesn’t entail IF Q THAN P. Negating a negation introduces the chance of returning indeterminate, which is why ~~P doesn’t entail P in intuitionist logic.

I’ll keep digging and see if I can come up with anything else. But the main thing I want to stress is that logic is a way to analyze language. When logical systems don’t seem to map onto realty, you need not feel weirded out by it. I mean, by simple first-order logic the sentence “If Santa Claus exists, than Rhubarb is the President of the Universe” is absolutely true. But it’s still looks wrong to say. The truth of logic is not the Truth of the Church. They just happen to coincide.

 

[JohnGerard]

Some people believe in a trivalent logic system or ‘many-valued’ logic where there are more than just true or false.

I can’t help you. But you’ve got me curious about something…

…I wonder if the people who believe in the idea of a trivalent logic system where there is more than just true or false, believe that that very idea has to be true…or false?

 

[inocente]

If you ask me whether it’s raining, I can answer yes, no, or don’t know. We all need that third answer for most of the questions we could be asked. Three-value logic deals with the three possibilities, for instance to provide rules for a computer program to follow when something isn’t known.

 

[SimmieKay]

Dear math philosopers,

Recently I discovered that not everyone believes in bivalent logic (things can only be true or false), the law of contradiction or excluding the middle.

Some people believe in a trivalent logic system or ‘many-valued’ logic where there are more than just true or false.

In a trivalent/many-valued logic system, if P is true, ~P could be false, or, indeterminate. If ~P is true then ~~P could be false or indeterminate

I want to know what the Catholic position on this would be.

One poster on an earlier thread referenced a philosopher named Michael Dummett who came up with intuitionistic logic, which consisted of this statement

Under this system, one can go from P to ~~P but not ~~P to P

The poster was confused on why this would be so, and so am I. How would one even go about refuting that position? Why would the position be false, etc?

Is mathematical logic an accurate portrayal of reality?

Alot of questions, but I’m very confused with all of this and I can’t shake it off because it feels like this is something I would hear from a cult leader or something, not a mathematical philosopher.

Intuitionistic logic began as a controversy about what kinds of proofs should be acceptable in mathematics. There are what are known as “non-constructive proofs”, where one can prove that some mathematical object (such as a number) meeting a certain criterion exists, without being able to give any examples of such an object. For example, there are numbers known as Chaitin’s constants - we can prove these numbers exist, but we can also prove that we can’t know what they are. Classical mathematicians do not have a philosophical problem with these kinds of proofs, intuitionistic/constructivist mathematicians do. So, the motivation for intuitionistic logic, was to come up with a logic which banned the types of proofs these mathematicians saw as problematic.

There are certain mathematical questions we can prove we cannot answer. For example, we can ask the question of what is the value of the nth digit of a Chaitin’s constant, but we can also prove that it is impossible for mere humans to answer that question for arbitrarily large n. Classical mathematicians would say these questions have answers, even if we can never know what they are. Intuitionistic/constructivist mathematicians would say that if we can never know the answer to a question, the question has no answer. So, consider propositions like “the millionth decimal digit of (some particular) Chaitin’s constant is 7”. Classical mathematics says this proposition is either true or false, even though we can never know which one it is (baring divine intervention, such as an oracle for the halting problem). Intuitionistic mathematics says that it is neither true nor false. Seen in that light, the denial of the law of the excluded middle seems to me to be a reasonable position, as opposed to some radical rejection of rational thought.

Simon