Reducible Logic as a Method of Reducing Logic to Mathematics
Hassan Ajami
2024 / 6 / 30
Reducible logic is a plausible branch of logic which aims to reduce the logical rules to mathematical equations and inferences. According to reducible logic, the logical rules could be accurately and successfully reduced to mathematical inferences and equations, such as reducing the logical rule of Modus Barbara to a specific mathematical inference presented in mathematical equations. From this perspective, logic is mathematics in the sense that logic is reducible to mathematics. And hence, logical inferences are mathematical inferences.
Categorical logic contains many valid syllogisms, such as Modus Barbara, Modus Celarent and Modus Darii. Those valid syllogisms in categorical logic could be successfully expressed in mathematical inferences based on mathematical equations.
The logical rule of Modus Barbara is the following:
All M are P.
All S are M.
Therefore, all S are P.
Modus Barbara could be accurately expressed in the following mathematical equations:
M = P.
S = M.
Therefore, it mathematically follows that S = P.
In other words, we could successfully represent the logical rule of Modus Barbara in a mathematical inference presented in mathematical equations, according to which, from "S = M" and "M = P", it mathematically follows that S = P. This, ultimately, supports the view that logical inference is nothing but mathematical inference. In the previous logical inference, "All M are P" is mathematically expressed as M = P, while "All S are M" is mathematically expressed as S = M. And the conclusion "all S are P" is mathematically expressed as S = P.
From the same perspective, we could successfully reduce the logical rule of Modus Celarent to a mathematical inference based on mathematical equations. The logical rule of Modus Celarent is the following:
No M is P.
All S are M.
Therefore, no S is P.
The first two premises of Modus Celarent could be accurately represented in the following mathematical equations:
No M = P.
S = M.
By negating both sides of the equation "S = M", we would validly infer from "S = M" the conclusion that"No S = No M", which is equivalent to "S = M" (given that both "S = M" and "No S = No M" have the same truth value). And, now, we have the first two premises as being the following:"No M = P" and "No S = No M". And, in light of those two premises, we could successfully construct the logical rule of Modus Celarent in the following mathematical way:
No M = P.
No S = No M.
Therefore, it mathematically follows that No S = P.
The logical rule of Modus Celarent could be successfully expressed mathematically, and hence, reduced to a mathematical inference presented in mathematical equations, such that the logical rule of Modus Celarent is nothing but the following mathematical inference: from "No S = No M" and "No M = P" , it mathematically follows that No S = P.
From the same perspective, we can successfully present the logical rule of Modus Darii as a mathematical inference. Modus Darii has the following valid form:
All M are P.
Some S are M.
Therefore, some S are P.
The first premise is mathematically rewritten as "M = P", while the second premise is mathematically rewritten as "Some S = M". From "Some S = M" and "M = P", it mathematically follows that "Some S = P", which mathematically represents the conclusion "some S are P".
As we have seen, through applying simple mathematical operations, we could accurately and successfully present logical rules as mathematical inferences based on simple mathematical equations. Hence, any so-called logical rule which could not be accurately and successfully expressed as a mathematical inference based on mathematical equations is not actually a genuine logical rule. This shows that reducible logic offers a systematic procedure for testing logical rules, aiming at discovering which so-called logical rules are genuinely logical. And this is a virtue in itself, speaking for the plausibility of reducible logic.
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