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Sorry for asking such a dumb question.

I am CS student and I am trying to understand the basic tenets of Logic. I am new to the subject and I am lost understanding Implication.

In formal logic, $(P\rightarrow Q)$, The sentence letter $P$ is premise and $Q$ is conclusion, right?

Now, we know that $(P\rightarrow Q)$ is equivalent to $-(P\wedge -Q)$ i.e. It is not the case that $P$ and not $Q$. Here, $Q$ acts as premise, right?

So, why we treat sentence letter $Q$ as conclusion in form and premise in other?

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    $\begingroup$ What do you mean by "premise" in a statement of the form $\lnot (P \land \lnot Q)$? The terms "premise" and "conclusion" only have any real meaning when talking about a concrete implication, not equivalent reformulations thereof. $\endgroup$ – dkaeae Feb 1 at 15:21
  • $\begingroup$ @dkaeae P and Q represent sentences which are premises in ¬(P∧¬Q), right. If not, then what are P and Q? $\endgroup$ – Ubi hatt Feb 1 at 15:26
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    $\begingroup$ Would you please add the definition for "premise" and "conclusion" to the question? I have never seen the terms used for something else than the parts of an implication (at least not in a CS context). $\endgroup$ – dkaeae Feb 1 at 15:29
  • $\begingroup$ @dkaeae I guess, I have to read the book again. I have lost completely :( $\endgroup$ – Ubi hatt Feb 1 at 15:38
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    $\begingroup$ You can't take a random logical formula and speak about "premises" and "conclusions". Usually, those concepts are only defined when the formula is a implication. $\endgroup$ – chi Feb 1 at 16:32
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I completely agree with others that the terms "premise" and "conclusion" don't apply to things that aren't implications. By analogy, in $P\land Q$, $P$ and $Q$ are called "conjuncts", but it would make no sense to ask what the conjuncts of $P\to Q$ are.

However, there is terminology that could make sense of the connection that is more commonly used in type theory than logic but is just as applicable to logic. We can talk about "positive" and "negative" positions. In $P\to Q$, the premise, $P$, is in negative position and the conclusion is in positive position. These senses "multiply" in that in $(R\to S)\to Q$, $S$ is in negative position, but $R$ is in positive position (because it's in a negative position within a formula that is itself in negative position). Since $\neg P\equiv (P\to \bot)$, ($\bot$ is the always false formula) negation effectively swaps negative and positive. In $P\land Q$ and $P\lor Q$, both positions are positive. (The general intuition is that propositions in positive positions are possibly asserted, while propositions in negative position are possibly being assumed. They might not actually be asserted/assumed, e.g. in $P\lor Q$.)

This positive/negative notion can be a quick sanity check for equivalences and can help guide proof search. (Negative propositions will likely need to be assumed at some point and positive ones proven. Also, particularly for purely logical theorems, you are often trying to bring negative and positive occurrences together, e.g. as in $P\to P$.) For your equivalence $$P\to Q \equiv \neg (P\land\neg Q)$$ we can see on the right-hand side that $P$ occurs in a positive position within a formula in negative position and so is negative, while $Q$ occurs in negative position within a formula in negative position and so is positive. This matches the left hand side.

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  • $\begingroup$ Thank you very much :) I re-read the chapter and I want to say, I messed up with the terminology and goofed up in the big way. Thank you so much for explaining basic concepts in such a great detail. Unfortunately, studying further I presumed, that P and Q always stands for premises. I strayed away from the simple fact that P and Q represent propositions (which can be either premises or conclusions). Thanks!!! $\endgroup$ – Ubi hatt Feb 3 at 10:57
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No, it's not correct to call $Q$ as the premise of $\neg(P \land \neg Q)$. You can only talk about the premise of a formula if it is of the form $\text{something} \to \text{something}$, i.e., if it is an implication. If it doesn't have that form, then it doesn't have a premise.

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