I would like to know the usage of 'If and only if' and the...

Answer by Mikey
Submitted on 9/28/2003
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Ritesh, "If and only if" is generally used in common language to emphasize to someone that ONLY "if" a specific condition is met, then a given response should be made. A seargent might instruct a private to inform him if, and ONLY if, he sees Captain Jones coming. That lets the private know that ol' sarge doesn't want to be interrupted every 15 seconds because Colonel Smith is coming or Lieutenant Arbuckle is coming or anyone or anything else is coming. Also, that terminology has an identical twin that many of us are all too familiar with from our wild youth. It would occur as we separately tried to excuse or defend our way out of a jam amidst the verbal assails of an angry parent: "You will answer me when and ONLY when I ASK you for an answer, young man, is that CLEAR?" (downtrodden silence) "ANSWER me!" (yes, ma'am/sir") ;-) John, In logic, "if and only if" is redundant. The word "if" implies that a tested state may not exist. Therefore, if a tested state is true, then the response state must also be true. And since the response state can not be true if the tested state is false, then "and only if" is redundant. That's the beauty of logic: unlike human imperatives as I described to Ritesh above, logic neither understands nor requires emphatic reinforcement to "do its job". ;-) Your "X if and only if Y" can be fully restated without losing anything as "X if Y". BTW, in digital logic (AKA: Boolean logic), you have defined a simple gate. This particular gate is used in digital circuitry for 2 main reasons, to "clean up" a signal by using a Schottky device, or most often to strengthen a "weak" input signal to a "stronger" one (in electronics, strengths and weaknesses come in various flavors) so the signal can be used by certain other components. In these capacities, these gates are called "drivers". Since digital inputs are usually described as "x" (or Xn) or often "a/b/c/d..." and outputs are defined most often by "y" or "q" (or Yn, Qn), your "X if Y" would be reversed and restated as "If X then Y";its truth table being: X Y 1 1 0 0 So if the input to our little gate were high, then the output is also high. If the input goes low,then the output goes low. No "and only if" needed! ;-)

Answer by John Nowakowski
Submitted on 9/30/2003
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Mr. Mikey: I humbly apologize for my rudeness in objecting to your error. However, I believe it is important to properly resolve this issue. 1) It in not the case that "X if and only if Y" can be restated "X if Y" without losing anything. 2) "X if Y" is different from saying "X only if Y" 3) Citation: "The word 'if' used alone introduces the antecedent of a conditional." Moore and Parker, Critical Thinking., McGraw-Hill, 2004, pg 314. 4) Citation: "The phrase 'only if' introduces the consequent of a conditional." id. 5) Therefore, "X, if Y" can be restated only as "If Y, then X" 6) Whereas, "X, only if Y" can be restated only as "If X, then Y" 7) The claim "If X, then Y" differs significantly from "If Y, then X" 8) The word "and" has a logical conjunctive function. "and" links two propositions. 9) "X if and only if Y" is a biconditional. It expresses two different conditional statements. 10) "if X, then Y AND if X, then Y" is a different statement than "If Y, then X" standing alone. 11) "X, if" is the same as "If Y, then X." [see #5] The Truth Table for "If Y, then X" X Y If Y, then X ------------------------------ T T T T F T F T F F F T (See Moore, pg 322) 12) The Truth Table for the biconditional "X if and only if Y" is as follows X Y X if and only if Y ------------------------------------ T T T T F F F T F F F T (Citing Cambridge Dictionary of Philosophy, Robert Audi ed., Cambridge University Press, 1996, pg 76) 13) You can see the difference in truth values when X is true and Y is false. "X, if Y" is true under these values "X if and only if Y" is false under the same 14) It is irrelevant to me what laypeople might mean by it in common language. This is logic. 15) "X if and only if Y" is sometimes written as "X iff Y", spelled with an extra f. also as "X just in case Y" "X <---> Y" or "X" (triple bar) "Y" (id.)

Answer by Mikey
Submitted on 10/1/2003
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Ha! Thought you could get rid of me that easily, eh? You ARE correct (except in one place), but I figured I'd at least defend myself and try to mitigate my "error". I can see you are well versed in formal logic. I am well versed in digital logic. As I mentioned before, digital logic is AKA Boolean Logic, which is a subset of formal logic. I don't know if formal logic has a place for single input/single output systems or not, but digital circuitry does. Using X as input and Y as output, this 1-in/1-out system is called a "driver" if (Y if X) or an "inverter" if (NOT Y if X). In fact, inverters used to be called "NOT gates" in some VERY early circles. Now in the case of a driver, (Y if X). This is equivalent to (Y=X). An inverter, of course is (NOT Y if X) or (NOT Y=X). Suppose lighting a match were TRUE. Then: for a driver, ("lit match" = "lit match") an inverter, ("unlit match" = "lit match") The truth tables would be: driver inverter XY XY 11 10 00 01 or: ---driver---- ---inverter---- X (Y if X) X (NOT Y if X) 1 1 1 0 0 0 0 1 So you can see that in this case, and only in this case<g>, that (Y if X) is identical to (Y if and only if X). Interestingly, there is a special case of drivers that have what are known as tri-state outputs. This gate has another input that literally turns the gate on. If the gate is off, the output "floats", that is, its logic level follows whatever is driving the external circuit at the time, independently of the driver input. Now. If you were to tie this tri-state trigger to the driver input, such that the gate activated ONLY when the input were true, then you would get: ---driver---- ---inverter---- X (Y if X) X (NOT Y if X) 1 1 1 0 0 * 0 * Where "" denotes an indeterminable state based on X. Now we can start making this even more interesting (or boring...) Suppose instead of the actual truth table above, we took all values of XY (input & output values) and checked to see if they satisfy the system definition of (Y if X). Then we could lay it out thusly: -normal driver- -tri-state dvr- X Y (Y if X) X Y (Y if X) 1 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 * Hmmm... Recognize the "normal driver"? It's your previous #12 example. It's also the truth table for what would be called an "exclusive NOR" gate. If they made one, that is. They don't, so we have to use an XOR gate and hang an inverter off of it. ;-) Now don't forget that all I've done is list all possible values for X & Y to see which ones can satisfy the definition of this single-ended system. Note also that an XOR gate (or NOR or OR etc,) is a 2 input/1 output system. For a 2-input gate, the X & Y both are inputs, I'll call the output Q. XOR gate: _ X Y Q Q 1 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 Okay...........Now I mentioned earlier that you were correct except for one place. That place is your example in #11, where you stated: > The Truth Table for "If Y, then X" > X Y If Y, then X > ------------------------------ > T T T > T F T > F T F > F F T Well,John, I'll be damned if I could come up with a simple system that followed that rule. Now I know that somebody swapped your fingers on you while you were typing it, so no big thing. However, just for kicks, here's an inverter and a NAND gate that'll fill the bill.... ;-) \|\ ___ X --\| >O----\| \ \|/ \| )O---- Q Y ----------\|___/ X Y Q 1 1 1 1 0 1 0 1 0 0 0 1

Answer by John Nowakowski
Submitted on 10/1/2003
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Mikey First, let me say I am not necessarily disagreeing with your pronouncements on Boolean logic. However, from the question Ritesh asked, I do not find it justified to assume that he meant George Boole's 1847 algebra when he asked how 'if and only if' is used in a sentence. Likewise, I did not assume he meant to use a logic other than the two-valued system, etc. ---- Let's take a common language example, derived from yours. A) Sgt. Xavier is giving an order to Pvt. Yeoman. (X and Y, respectively) Assume Sgt. Xavier says: "Pvt. Yeoman, you will speak to me IF I, Sgt. Xavier ask you a question." Assume this is the only order Sgt. Xavier gives to Pvt. Yeoman. This would take the form of If X, then Y. If Xavier asks, Yeoman answers. So, if Xavier asks a question, Yeoman must answer. Otherwise, he did not follow the order. X=T, Y=T, X-->Y=T If Xavier does not ask a question, Yeoman does not have to answer. He would not be breaking the order. X=F, Y=F, X-->Y=T Yeoman is in real trouble if Xavier asks and Yeoman does not answer. He is clear violation of the order. X=T, Y=F, X-->Y=F However, Yeoman is NOT constrained to ONLY speak if Xavier asks. Therefore, if Yeoman is speaking, we do not necessarily know that Xavier asked a question. There are no limits on when Yeoman CAN speak. Yeoman can talk to anyone about anything. The order only says that he cannot remain silent when Xavier asks a question. So Yeoman can be a motormouth and still follow the order. X=F, Y=T, X-->Y=T B) Assume that Xavier had given a different order. "Yeoman, I am tired of your motormouth. You have one order and one order only. Here it is: Yeoman is to speak ONLY IF Xavier asks a question of you." This would take the form If Y, then X. Since Yeoman wants to follow the order, he better make sure that if Yeoman is speaking, then it is only in response to Xavier. If we knew of this order and Yeoman's intent to follow it, if we heard Yeoman speaking, then we'd know Xavier asked a question. X=T, Y=T, Y-->X=T Likewise, if Xavier did not ask a question, Yeoman better NOT be speaking. X=F, Y=F, Y-->X=T If Xavier did not ask Yeoman anything, but Yeoman is speaking, then he has clearly violated his single order and is in trouble. X=F, Y=T, Y-->X=F However, Xavier's order has left a different loophole. This time, Yeoman does not HAVE to answer Xavier when he asks a question. Xavier could fire away questions all day, without a shred of response from Yeoman. Yeoman would still be following the order, because he does not have to speak. Yeoman only has to make sure that, when he does speak, it is only if Xavier has spoken. X=T, Y=F, Y-->X=T C) Assume that Xavier does not like either of these alternatives. If Xavier asks a question of Yeoman, he wants to get an immediate response from Yeoman, without giving Yeoman the luxury of deciding whether he will respond or not. Also, Xavier does not want Yeoman flapping his gums at anyone he feels like, Xavier wants Yeoman to respond only to him. So, what orders can Xavier give Yeoman to close the loopholes and effect the results he wants? He can give both! "Yeoman, you have two orders. First, you answer IF I ask a question. Second, you will answer ONLY IF I ask a question." OR "Pvt. Yeoman, you will answer IF AND ONLY IF I ask you a question." Now, if Xavier speaks, Yeoman must answer. X=T, Y=T, X<-->Y=T If Xavier has not spoken, Yeoman must be silent. X=F, Y=F, X<-->Y=T Yeoman can violate his order(s) two ways. First, if Xavier speaks and he does not answer, he has violated his orders. X=T, Y=F, X<-->Y=F Second, if Yeoman begins speaking, but Xavier has not spoken. X=F, Y=T, X<-->Y=F I hope this helps. Best, John

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Question by Ritesh
Submitted on 9/15/2003
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I would like to know the usage of 'If and only if' and the meaning of the same in a sentence. thank you Ritesh

Answer by marv
Submitted on 11/9/2003
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Wanted to Thank you both, John and Mikey, and Ritesh, for the question that started that great discourse. Micky, your answer, including the 1st paragraph of your reply to John, covered 3 different levels, something for everyone! The logic lesson made me work, and also gave me the opp. to exercise dormant, read lazy, neurons and synapses. It's also a lot of fun to see rather then hear, a good argument. Thanks to both, Marv

Answer by John Nowakowski
Submitted on 9/22/2003
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"X if and only if Y" expresses the 'material equivalence' of X and Y. Logically it is the conjunction of both "If X, then Y" AND "If Y, then X"

Answer by Mikey
Submitted on 10/3/2003
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Rats... Stand by, John. I haven't left in a huff, it's just that the "real world" keeps getting in the way of my "quality internet time"... ;-) BTW, I see you've gotten your name up in lights at the top of the page. Congratulations!

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