Albert Einstein said "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality". Let us examine why it is so. Consider these two statements (or instances of propositional symbols):
P: I wake up late
Q: I go to office late
The deductive apparatus in formal logic consists of two things: A set of premises (called axioms) and the rules of inference. The rules of inference could be anything like Modus Ponens (MP) or Modus Tollens. MP basically says, the statement if P then Q taken together with P will imply Q. Which kinda looks true in our case: If I wake up late, then I go to office late and I wake up late would mean I go to office late. Among the basic ideas in formal logic is the idea that the deductive apparatus or syntactics should be kept aside from the semantics of the language. One may then naturally ask what is the relation between the syntactics and the semantics of the language. It is here the idea of axioms come in. In general, in mathematics, the main goal is to come up with the set of premises so that the syntactics is matched with the semantics. In other words, we keep the rule of inference fixed and come up with our axioms that deduces statements that are true or false. Although we can change the rules of inference as much as the axioms, mathematicians don't attempt that. If P then Q and P need not always make Q true. For instance, take
P: We men going to mars
Q: Saturn is full of chocolates
Now, if P then Q and P is true does not make Q true - for sure we know that right, although P is a real possibility (and not true now). Mathematicians ignore the general notions of truth and reduce them to validate their statements by building axioms rather than equally exploring the rules of inference. Thus the notion of truth or falsity in mathematics may not conform to the real world. Here is where temporal logic and relevance logic come to rescue!
P: I wake up late
Q: I go to office late
The deductive apparatus in formal logic consists of two things: A set of premises (called axioms) and the rules of inference. The rules of inference could be anything like Modus Ponens (MP) or Modus Tollens. MP basically says, the statement if P then Q taken together with P will imply Q. Which kinda looks true in our case: If I wake up late, then I go to office late and I wake up late would mean I go to office late. Among the basic ideas in formal logic is the idea that the deductive apparatus or syntactics should be kept aside from the semantics of the language. One may then naturally ask what is the relation between the syntactics and the semantics of the language. It is here the idea of axioms come in. In general, in mathematics, the main goal is to come up with the set of premises so that the syntactics is matched with the semantics. In other words, we keep the rule of inference fixed and come up with our axioms that deduces statements that are true or false. Although we can change the rules of inference as much as the axioms, mathematicians don't attempt that. If P then Q and P need not always make Q true. For instance, take
P: We men going to mars
Q: Saturn is full of chocolates
Now, if P then Q and P is true does not make Q true - for sure we know that right, although P is a real possibility (and not true now). Mathematicians ignore the general notions of truth and reduce them to validate their statements by building axioms rather than equally exploring the rules of inference. Thus the notion of truth or falsity in mathematics may not conform to the real world. Here is where temporal logic and relevance logic come to rescue!
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