Formal Logic 101 – Part 5: “Not” This Again
Last time around, we briefly touched on “And”, “Or”, and “Not”. Today I would like to delve a bit deeper into “Not” because “Not” can be used in several ways, and the symbolization changes just a bit depending on how “Not” is used in a sentence. Consider the phrase:
"It’s not true that the world will end if there’s a nuclear war."
If we remove the “not”, we’re left with “the World will end if there’s a Nuclear war.” From previous lessons we should know that that is symbolized as:
N→W (if Nuclear war, then World ends).
So it may be tempting to say that the way to symbolize the full phrase would be
~N→W
However, if we translate the symbolization back to plain English we’re left with “if there’s not nuclear war, the world will end” which is a radically different statement.
Ok so what about:
~N→~W (If there’s not nuclear war, then the world won’t end)
That seems closer, but that’s still addressing what happens if there’s NOT a nuclear war, rather than what happens if there is one.
To make the negation apply to the whole statement, we include parentheses to make:
~(N→W)
This now says, “the statement ‘the world will end if there’s nuclear war’ is not true.” Which is functionally the same as our original phrase.
In short, the Not only applies to the symbol that immediately follows it, unless a set of parentheses extends its reach.
Not interacts in a similar fashion with "And", with the position of the Not varying depending on the meaning of the original phrase.
Consider the phrase “It’s not true that Obi-wan Kenobi was a Jedi and had a purple lightsaber.”
By now we should be able to determine the propositions here without too much trouble. Let J be our shorthand for “Obi-wan was a Jedi” and P be our shorthand for “Obi-wan had a Purple lightsaber”.
Looking at the full phrase, ask yourself: what specifically is that phrase saying is not true? It’s not saying “Obi-wan Kenobi was not a Jedi and did not have a purple lightsaber” (which would be ~J^~P). It’s also not specifying which of the two propositions is false (either J^~P or ~J^P). What is being denied is the “And”. What the “Not” in this phrase is doing is saying “Anywhere from zero to one of these propositions is true, but not both.” So the correct phrasing is ~(J^P).
(Note: The setup ~(A^B) is useful if you need to use an “exclusive” Or. You can set that up by stating (AvB) ^ ~(A^B). Translated to plain English, it reads “Either A or B, but not A and B”, or "Either A or B, but not both.")
As one might expect, Not interacts with "Or" in a much similar way to And.
Consider the difference between the phrases:
They will either Pay me, or I will not do the Job. Pv~J
They will neither Pay me, nor will I do the job. ~(PvJ)
The former establishes that this is an “Or” situation. It’s established that at least one side of the Or statement is true.
The latter denies that this is even an “Or” situation. It is stating that both sides of the Or statement are false. (As we will learn later, ~(PvJ) is actually logically equivalent to ~P^~J.)
*****
A. If the economy improves and consumers increase borrowing, interest rates will rise.
B. The economy will not improve and interest rates will not rise if either consumer spending falls or unemployment rises.
C. Either interest rates or unemployment will rise, but not both.
D. Interest rates will not rise if the economy improves, provided consumers do not increase borrowing.
E. The deficit will be reduced and the economy will improve if taxes are raised and interest rates do not rise.
F. Neither taxes nor interest rates will rise if the deficit is reduced, but if the deficit is not reduced, then both taxes and interest rates will rise.
My symbolization (again, recall you can use any letters you like as long as you’re consistent)
R= Interest rates will rise; E= The economy improves; B= Consumers increase borrowing; S= Consumer spending falls; U=Unemployment rises; D=Deficit will be reduced; T= Taxes are raised
"It’s not true that the world will end if there’s a nuclear war."
If we remove the “not”, we’re left with “the World will end if there’s a Nuclear war.” From previous lessons we should know that that is symbolized as:
N→W (if Nuclear war, then World ends).
So it may be tempting to say that the way to symbolize the full phrase would be
~N→W
However, if we translate the symbolization back to plain English we’re left with “if there’s not nuclear war, the world will end” which is a radically different statement.
Ok so what about:
~N→~W (If there’s not nuclear war, then the world won’t end)
That seems closer, but that’s still addressing what happens if there’s NOT a nuclear war, rather than what happens if there is one.
To make the negation apply to the whole statement, we include parentheses to make:
~(N→W)
This now says, “the statement ‘the world will end if there’s nuclear war’ is not true.” Which is functionally the same as our original phrase.
In short, the Not only applies to the symbol that immediately follows it, unless a set of parentheses extends its reach.
Not interacts in a similar fashion with "And", with the position of the Not varying depending on the meaning of the original phrase.
Consider the phrase “It’s not true that Obi-wan Kenobi was a Jedi and had a purple lightsaber.”
By now we should be able to determine the propositions here without too much trouble. Let J be our shorthand for “Obi-wan was a Jedi” and P be our shorthand for “Obi-wan had a Purple lightsaber”.
Looking at the full phrase, ask yourself: what specifically is that phrase saying is not true? It’s not saying “Obi-wan Kenobi was not a Jedi and did not have a purple lightsaber” (which would be ~J^~P). It’s also not specifying which of the two propositions is false (either J^~P or ~J^P). What is being denied is the “And”. What the “Not” in this phrase is doing is saying “Anywhere from zero to one of these propositions is true, but not both.” So the correct phrasing is ~(J^P).
(Note: The setup ~(A^B) is useful if you need to use an “exclusive” Or. You can set that up by stating (AvB) ^ ~(A^B). Translated to plain English, it reads “Either A or B, but not A and B”, or "Either A or B, but not both.")
As one might expect, Not interacts with "Or" in a much similar way to And.
Consider the difference between the phrases:
They will either Pay me, or I will not do the Job. Pv~J
They will neither Pay me, nor will I do the job. ~(PvJ)
The former establishes that this is an “Or” situation. It’s established that at least one side of the Or statement is true.
The latter denies that this is even an “Or” situation. It is stating that both sides of the Or statement are false. (As we will learn later, ~(PvJ) is actually logically equivalent to ~P^~J.)
*****
Practice
Symbolize the following:A. If the economy improves and consumers increase borrowing, interest rates will rise.
B. The economy will not improve and interest rates will not rise if either consumer spending falls or unemployment rises.
C. Either interest rates or unemployment will rise, but not both.
D. Interest rates will not rise if the economy improves, provided consumers do not increase borrowing.
E. The deficit will be reduced and the economy will improve if taxes are raised and interest rates do not rise.
F. Neither taxes nor interest rates will rise if the deficit is reduced, but if the deficit is not reduced, then both taxes and interest rates will rise.
Answers
R= Interest rates will rise; E= The economy improves; B= Consumers increase borrowing; S= Consumer spending falls; U=Unemployment rises; D=Deficit will be reduced; T= Taxes are raised
A. (E^B)→R; Remember, both E and B need to happen for R to happen, so E and B go in parentheses.
B. (SvR)→(~E^~R)
C. (RvU)^~(R^U); “R or U, but not both”
D. ~B→(E→~R); Remember, “provided” is one of our “If” equivalents.
E. (T^~R)→(D^E)
F. [D→~(TvR)]^[~D→(T^R)]; [D→(~T^~R)]^[~D→(T^R)] would also be correct.
B. (SvR)→(~E^~R)
C. (RvU)^~(R^U); “R or U, but not both”
D. ~B→(E→~R); Remember, “provided” is one of our “If” equivalents.
E. (T^~R)→(D^E)
F. [D→~(TvR)]^[~D→(T^R)]; [D→(~T^~R)]^[~D→(T^R)] would also be correct.
Comments
Post a Comment