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Sufficient and Necessary conditions in Symbolic Logic notation

 
 
Reply Wed 8 Sep, 2010 06:52 am
Have logicians been able to come up with a particular notation for cause and effect? Give you an example: say I tell you that "The heat from the sun caused the stone to warm." - What, if any, type of notation would be used? I was thinking that you could use the connective --> (if...then) to symbolize this, but then I realized that the fact that the stone is warmed up by the sun's heat is a sufficent condition for the stone being warmed up (because a stone can be heated up by other means); So I was thinking that there might indeed be a new symbol to show sufficiency (although it could be still be "-->") I was thinking of strict implication, but that deals with necessity.

I was thinking of this, but I am not entirely sure. Seems too easy. But then again I could be overanalyzing.

1. The sun's heat on the stone is a sufficient condition for the stone to heat up.
p: the sun's heat on the stone
q: the stone to heat up.

1. (p -->q)?

Help on this would be appreciated. Especially from individuals who I know that know a thing or two about symbolic logic. *cough* kennethamy *cough*
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kennethamy
 
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Reply Wed 8 Sep, 2010 07:23 am
@Ding an Sich,
Ding an Sich wrote:

Have logicians been able to come up with a particular notation for cause and effect? Give you an example: say I tell you that "The heat from the sun caused the stone to warm." - What, if any, type of notation would be used? I was thinking that you could use the connective --> (if...then) to symbolize this, but then I realized that the fact that the stone is warmed up by the sun's heat is a sufficent condition for the stone being warmed up (because a stone can be heated up by other means); So I was thinking that there might indeed be a new symbol to show sufficiency (although it could be still be "-->") I was thinking of strict implication, but that deals with necessity.

I was thinking of this, but I am not entirely sure. Seems too easy. But then again I could be overanalyzing.

1. The sun's heat on the stone is a sufficient condition for the stone to heat up.
p: the sun's heat on the stone
q: the stone to heat up.

1. (p -->q)?

Help on this would be appreciated. Especially from individuals who I know that know a thing or two about symbolic logic. *cough* kennethamy *cough*


The causal relation can be symbolized by implication as you say (although there are complications arising from the use of material implication). However, although to say that C causes E often means that C is a sufficient condition of E, sometimes it means that C is a necessary condition of E (e,g, the presence of oxygen caused the fire) and sometimes, we mean by "cause" both a sufficient and necessary condition. It depends on the circumstances. For instance when we say that the cause of A's death was strangulation, we do not mean to say that it was not also necessary that A's heart stop beating. We often fix on one event as the cause while assuming (in the background) other conditions without which the event would not cause the effect. There is a nice book by McIntyre called, The Cement of the Universe on causation, and also an older, but good book by H.L.A. Hart (and someone else) on causation in the law.
Ding an Sich
 
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Reply Wed 8 Sep, 2010 07:59 am
@kennethamy,
kennethamy wrote:

Ding an Sich wrote:

Have logicians been able to come up with a particular notation for cause and effect? Give you an example: say I tell you that "The heat from the sun caused the stone to warm." - What, if any, type of notation would be used? I was thinking that you could use the connective --> (if...then) to symbolize this, but then I realized that the fact that the stone is warmed up by the sun's heat is a sufficent condition for the stone being warmed up (because a stone can be heated up by other means); So I was thinking that there might indeed be a new symbol to show sufficiency (although it could be still be "-->") I was thinking of strict implication, but that deals with necessity.

I was thinking of this, but I am not entirely sure. Seems too easy. But then again I could be overanalyzing.

1. The sun's heat on the stone is a sufficient condition for the stone to heat up.
p: the sun's heat on the stone
q: the stone to heat up.

1. (p -->q)?

Help on this would be appreciated. Especially from individuals who I know that know a thing or two about symbolic logic. *cough* kennethamy *cough*


The causal relation can be symbolized by implication as you say (although there are complications arising from the use of material implication). However, although to say that C causes E often means that C is a sufficient condition of E, sometimes it means that C is a necessary condition of E (e,g, the presence of oxygen caused the fire) and sometimes, we mean by "cause" both a sufficient and necessary condition. It depends on the circumstances. For instance when we say that the cause of A's death was strangulation, we do not mean to say that it was not also necessary that A's heart stop beating. We often fix on one event as the cause while assuming (in the background) other conditions without which the event would not cause the effect. There is a nice book by McIntyre called, The Cement of the Universe on causation, and also an older, but good book by H.L.A. Hart (and someone else) on causation in the law.


Thank you. Just out of curiosity, what problems/ complications would arise from me using material implication in this way?

If I am not mistaken saying that something is "a sufficient and necessary condition" can be expressed using "<-->", or material equivalence. Correct?

I will check out the books you made note of in the previous post. Working on Bertrand Russell's "Our Knowledge of the External World" at the moment, but it couldnt hurt to add on some other books to the reading list.
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