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Most of the Theorems in mathematics are of the form: $A\implies B$. If i want to prove this statment often i assume A to be true, and then show B is also true. With the help of the truth table \begin{array}{cc|c@{}ccc@{}c} A&B&(&A&\rightarrow&B&)\\\hline 1&1&&1&\mathbf{1}&1&\\ 1&0&&1&\mathbf{0}&0&\\ 0&1&&0&\mathbf{1}&1&\\ 0&0&&0&\mathbf{1}&0& \end{array} one can reason then, that this is sufficent to show that the implication is always true. This is the case because if A is wrong, the implication is true in either case. In the other case if A is true, then i showed B is also true, or in other words i showed in my proof that if A is true, B cannot be false, so the implication is true. Is that right ? So if i want to proof a Theorem, the goal is to show that it is true. Is that right ?

The following question is in regard to understanding implication in mathematical theorems. Does the implication imply a causal relationship, something like if A then B, or is it simply a operator in propositinal logic ?

MathAccount12
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    In "real life", like e.g. in mathematics, we do not use truth table. What we do is to prove a theorem T starting from a set A of axioms. If the proof is valid (i.e. if we have applied correctly the inference rules) what we get is the result: "if A, then T". – Mauro ALLEGRANZA Apr 19 '23 at 08:34
  • The result must be read: "theorem T is a consequence of axioms A", i.e. the theorem holds in every "model" where the axioms hold. Example: Pythagoras Theorem in the context of Euclid's system for geometry. – Mauro ALLEGRANZA Apr 19 '23 at 08:35
  • This means that, in "real life", we are not interested to the two bottom lines of the truth table above, i.e. the cases where the axioms that we are using in the proof do not hold. – Mauro ALLEGRANZA Apr 19 '23 at 08:37
  • Regarding the final question: "Does the implication imply a causal relationship, or is it simply a operator in propositional logic ?" see e.g the post The meaning of implication in logic – Mauro ALLEGRANZA Apr 19 '23 at 09:01
  • @MauroALLEGRANZA Thank you for your input! So you could say mathematics has a form something like that: (Set of axioms)$\implies$ theorem1 $\implies$ theorem 2 and so on. – MathAccount12 Apr 19 '23 at 12:48

1 Answers1

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Does the implication imply a causal relationship?

No. Maybe a real-life example will help.

Consider: If it is raining, then it is cloudy. ($R \to C)$

This does not mean that rain causes cloudiness. It means only that it is now not the case that it is both raining and not cloudy.

$$R \to C~~\equiv ~~ \neg (R \land \neg C)$$

This definition is entirely consistent with the truth table for $R\to C$.

\begin{array}{cc|c@{}ccc@{}c} R&C&R&\rightarrow&C&\\\hline T&T& &T& &\\ T&F& &F& &\\ F&T& &T& &\\ F&F& &T& & \end{array}

As required of logical implication, if both $R$ and $R\to C$ are true (line 1 only), then $C$ must also be true. (The Detachment Rule, aka modus ponens)

Note that we can infer from this "definition" that the implication $R\to C$ is true if $R$ is false (lines 3, 4) or if $C$ is true (lines 1, 3).

Note that, in classic logic, there need not be any other logical relationship between the antecedent and consequent propositions. They may be entirely independent of each other.

Dan Christensen
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