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Propositional logic

Rules of Replacement

Transposition (Trans):

( p ⊃ q) :: (∼q ⊃ ∼p)

All i see here is some kind of a modus tollens case I don't see how did they get from

( p ⊃ q)

To

(∼q ⊃ ∼p)

or why it is considered an equivalent.

(I mean by a logical process or explanation not by truth table,even if it's true by the truth table, a truth table isn't a proof for something, there is no rational there only to notice something like equivalent and i don't think i can use a truth table as a proof, that is a very weak proof.)

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  • "...a logical process or explanation not by truth table,even if it's true by the truth table, a truth table isn't a proof for something, there is no rational there only to notice something like equivalent..." is plainly wrong. In propositional logic [logical equivalence] (en.wikipedia.org/wiki/Logical_equivalence)is defined in terms of truth table. Aug 29 at 12:29
  • But if you want a proof based on a derivation, of course we can have it, provided that you specify a Proof system: axioms+rules. Aug 29 at 12:31
  • I’m voting to close this question because this is not a homework forum for logic class Aug 29 at 13:10
  • 2
    Where do you see a homework question? It's a hypothetical question on the rule itself. @ Swami
    – Dasem
    Aug 29 at 14:18
  • @Swami Vishwanana I'm voting to vote down your comment but i don't see an option for that.
    – Dasem
    Aug 29 at 14:41
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For clarification, given the way you've written the question, I'm supposing that your question is about the rule of transposition in classical logic and that you are using ⊃ to represent material implication. The term 'transposition' has a different though related meaning in Aristotelian logic.

If you are struggling to understand why the rule is correct, you might like to start just by considering some examples. "If a thing is a bird then it has wings" has the same truth conditions as, "If a thing does not have wings, then it is not a bird". "If it rains on Friday the match is cancelled" has the same truth conditions as, "If the match was not cancelled, it didn't rain on Friday". "If the buyer did not make payment by the specified date, the contract was voided" has the same truth conditions as, "If the contract was not voided, the buyer made payment by the specified date".

Note that I'm speaking here of truth conditions, because conditionals in the real world can carry all kinds of additional meanings and implicatures. Material implication is a special kind of conditional that boils the conditional down to its bare truth conditions only. P ⊃ Q is equivalent to ¬(P ∧ ¬Q). It holds that the truth of P is a sufficient condition for the truth of Q, and the truth of Q is a necessary condition of the truth of P. Hence, if Q is false, by implication P is false.

It is not irrelevant or weak to demonstrate the equivalence by truth table, and it is easily done. We can also demonstrate the equivalence using natural deduction. The exact form will depend on what rules you use, but if we allow ourselves a little freedom, then the proof is very simple:

1. P ⊃ Q              Premise
2. ¬(P ∧ ¬Q)          From 1, by equivalence of implication
3. ¬P ∨ Q             From 2, by de Morgan 
4. ¬Q                 Assumption
5. ¬P                 From 3, 4, by disjunctive syllogism 
6. ¬Q ⊃ ¬P            From 4, 5, discharging the assumption in 4. 

The converse can be proved in the same way with the use of double negation elimination.

It is worth noting that not all logics have this rule. For example, in intuitionistic logic, P → Q entails ¬Q → ¬P, but the converse does not hold. Also, transposition, and the corresponding rule of contraposition, does not hold in probability logic, because if the conditional is uncertain, a high value for P(B|A) does not entail a high value for P(¬A|¬B). Also, contraposition does not hold in general for counterfactual conditionals.

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  • I believe 5 is by Disjunctive syllogism , and i didn't learn about step 6 yet, thank you.
    – Dasem
    Aug 29 at 17:12
  • But i will kee your answer in my attention, maybe after i learn about the last step (6) everything will come together, i usually don't like to advance in the study while i don't understand something but maybe in this case it's the right thing to do.
    – Dasem
    Aug 29 at 17:20
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    @Dasem Yes, well spotted. I've fixed it.
    – Bumble
    Aug 29 at 17:27
  • How do you get math symbols into your answers? I've read the help on markdown but it doesn't mention special symbols. Aug 29 at 18:42
  • @DavidGudeman I just copy the unicode characters. It is not ideal, but unfortunately the Philosophy site does not support LaTeX.
    – Bumble
    Aug 29 at 19:43
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A => B or "If A is true, B must be true", means that if B is found to be false, A must be false too because if it was true B would be too (and A can only be true or false according to the excluded middle principle).

You can picture it using the equivalence of implication with inclusion: figure 2 circles A and B, with A entirely inside of B. This represents "A => B": if a point is inside A it must be inside B. But you can also see that there is no point outside of B that is inside of A, therefore "~B => ~A".

By the way, truth tables are a perfectly valid demonstration.

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  • I'm not familiar with the symbol => , if it means " if and only if " then it's not the case of the question in the post, my question is about implication and equivalent.
    – Dasem
    Aug 29 at 14:34
  • It means the same as the symbol you use, implication, as stated in the answer: "if A is true, B must be true" (but not the reverse). If and only if is <=>.
    – armand
    Aug 29 at 15:31
  • Ok. Here is an example " if the kid is smiling then he is happy" , if the kid isn't happy.. that doesn't necessarily mean he isn't smiling..(he can fake the smile) this is just an example to test the Transposition equivalent rule, but my main question is about the proof i want to know if there is a logical explanation that show how you get the equivalent from ( if p then q).
    – Dasem
    Aug 29 at 16:51
  • @Dasem, you have discovered that the if . . .then used in math is not identical to how humans who speak English use the same words. In Mathematical logic if . . . .THEN means something specific outside of English grammar. You will often hear necessary or sufficient being used in Math --but outside of Math it is a different story. So there is no way for them to answer sufficiently All conditionals are if . . . Then ... construction. They are not always as math says either necessary or sufficient. English grammar has more options than just those two.
    – Logikal
    Aug 29 at 20:49
  • @dasem now I see the problem: you didn't understand what implication means. "Child smiles => child happy" means the child can't fake it. If he smiles, he is happy. Period. In the above example your natural language sentence is closer to "if the child smiles he is most probably happy". Basic formal logic can't apply to such a case. But note that if we mind this, the transposition still works: "if the child is not smiling, he is most probably not happy" is still a valid consequence of "smiling children are most probably happy".
    – armand
    Aug 29 at 22:38
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To prove logical equivalence syntactically you need to show that $\phi \vdash \psi$ and $\psi \vdash \phi$, where $\psi$ and $\phi$ are wffs.

To do that, start by taking $\phi$ as a premise and then deriving $\psi$. Followed by taking $\psi$ as a premise and then deriving $\phi$.

Another way to do it is by showing $\vdash \phi \to \psi$ and $\vdash \psi \to \phi$, which allows us to infer $\vdash \phi \leftrightarrow \psi$.

Modus Tollens, conditional proof, and how to handle double negation is all that's required for the proof.

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