Why is the law of non-contradiction a real law?

Philosophy Forums Philosophy
1

Heres a practical exercise for you ILP folk. But feel free to argue or discuss.

The law of non-contradiction is an utterly beautiful concept.
Here’s a little guide on how to appreciate it using Propositional Logic (PL) and Truth Tables.

The law of non-contradiction simply says that a state of affairs cannot be the case and not the case at the same time. This might seem obvious, but unlike alot of other seemingly obvious things (like causality or induction) with PL you can prove it must always hold.

In PL we can write the law as “~(p&~p)”.

For starters a truth value is simply whether the statement is “true” or “false”. Whether it happens to be the case, or not the case.

“~” is negation or “not”. It inverts the truth value of the statement it applies to. Eg. “I am not French” is the opposite truth value of “I am French”

Brackets “()” are used in the same way as brackets in maths; they show what the preceeding operator applies to (in this case, the first “~” applies to the rest of the statement as opposed to just the first “p”, while the second “~” only applies to the second “p”

“&” is conjunction or “and”. It is true only when both the parts on either side of it are true. Eg. the statement “Pierre and Ivan are French” is only true if (and only if) both Pierre and Ivan are French.

“p” is a variable which can be substituted to any statement you like. Non-contradiction happens to apply to EVERY statement as we shall see, but I’ll carry on using the French thing.

So, “~(p&~p)” is “It is not the case that I am French and I am not French”.

You got that people? After testing we should be able to write this in its “absolute” form of “It cannot be the case that I am French and I am not French”.

Lets test this theory in the real world then.
Since we only have one variable (p) this vastly simplifies things. Either “p” is the case, or “p” is not the case, ie. “I am French” or “I am not French”.

So, we write:

p
-
T
F

and this summarises all the possible combinations of the truth value of the statement “p”, ie. it can be True (T) or it can be False (F). Pretty damn simple, no?

Lets add the law in then, and work out how those truth values would effect the truth value of the law.

The left hand bit represents the possible states of affairs in the world then, and the right hand bit shows how those possible states of affairs will affect the truth value of the law.

p | ~(p&~p)
-----------
T | 
F |

First things first, lets copy over the truth value for “p”.

p | ~(p&~p)
-----------
T |   T  T
F |   F  F

Second, there is a negation for the second “p” so lets invert those truth values (swap T for F, and F for T).
I’ll put these values next to the original ones, under the “~” symbol.

p | ~(p&~p)
-----------
T |   T FT
F |   F TF

We now have the values either side of the “&”. So, if they are both true, we can mark the “&” as true, if not it is false.
As this is the value of the whole area in brackets, I’ll place these truth values in brackets.

p | ~(p&~p)
-----------
T |   T(F)FT
F |   F(F)TF

So, the statement “p&~p” is always false regardless of the truth value of “p”. If we add the negation that applies to all the stuff in brackets however… (ive put the final result in square brackets)

p | ~(p&~p)
-----------
T | [T]T(F)FT
F | [T]F(F)TF

We see that the whole statement is always true. Regardless of whether “p” is true or false, the law of non-contradiction is true.

This is what makes the law of non-contradiction a law, it is always true, and although this statement doesnt tell me alot about “p”, it does tell me what the structure of the world is like.

In other words, although the statement “It cannot be the case that I am French and I am not French” doesnt tell me whether I am French or not, it does tell that I cant be both at once.

That is one sexy bit of logic. :sunglasses:

Try it yourself and find your own laws; ie. PL statements that are always true regardless of the truth value of the constituent statements.

Cheers!

2

Very interesting.

I’d be interested in seeing “I think therefore I am” proven in this method, as I am sure Descartes would admire this sort of logic.

3

Certainly. There is, in the predicate calculus, a rule of inference called “the rule of existential generalization” which says that if there is a property, then something must have that property. [ In symbols: f entails, E! X (f).} Thus, for instance, if there is walking going on, then someone or something must be doing the walking. Similarly, if thinking is going on, then someone, or something, must be doing the thinking.

4

Bah, you Russelians and your metaphysics. Wittgenstein wouldnt approve you know. :slight_smile: You cant use your logic to make up a new metaphysics.

In any case, you havent actually done more than what our basic grammar already does; predicates have subjects (either explicit or implied).

5

Of course such logic is beautiful in its own way but it makes no difference to the tabloid journalist who writes ‘Sampras didn’t win the match, Rafter lost it!’ and such things…

6

now you are just being unfair. :slight_smile: What would tabloid journalism be if they had to say what they mean rather than play on words? Treat tabloids as poetry not a bunch of essays on current affairs.

Btw, anyone want to argue against non-contradiction? Any resident Hegelians for example?

Cheers!

7

You missed my point entirely. Language evades the law of non-contradiction almost continually…

8

Not at all. We just dont mean what we say (but what we mean is still implied).

Grammatically simple sentances can be logically complex. In your instance, the tabloid journalist is relating that the cause of Sampras’ win was due to Rafter’s poor performance (compared to some expectation) rather than good performance by Sampras himself. What is meant makes perfect logical sense, it is just expressed in grammatical shorthand for poetic effect, it is a completely legit method of using language and does not evade any logical laws.

9

On the contrary, my example completely evades the law of non-contradiction. Either Sampras won or lost the game of tennis. ‘Sampras didn’t win, Rafter lost’. Can you see it?

We do mean what we say, just often not in anything like a literal fashion. Logic simply has no say over such operations of language…

10

The doctrine of the included middle reveals the limitations of the law of non-contradiction. However, reading between the lines, it suggests the possibility of a truly scandelous unification of science and religion. Now we can’t have that, can we.

nicol.club.fr/ciret/bulletin/b12/b12c3.htm

11

How is this ‘evading’ a logical law? It is simply playing with different senses of the term “winning”; ie. the connotation that you have to play well to win which didnt apply in this case.

Of course it does, or we couldnt mean anything, the language is just not usually transparent.

12

there are no real laws. there are so many laws they are all touchong each other., we are in a biodome of reality… … . .

13

Dear oreso

Precisely, it is playing with connotation, something for which logic cannot account.

Logic cannot deal with connotation. For Logic A=A. In actual languahe A can mean something different in two adjacent sentences.

14

Dear someonisatthedoor (that felt wierd, but good),

Then those two different meanings for A are two different signs in logic (A1 and A2 perhaps). This is no more tricky than the existence of homophones. No problem.

Yours, sincerely,
oreso

15

In the absence of a meta-language there is only the interplay of language which can never be held to account. One can never be sure if the two are one or two…

p.s. you almost got my name right. It is an absolute bitch to spell (one of the reasons I chose it)…

edit - from one of Imp’s posts:

Some triangles are obtuse
Whatever is obtuse is ignorant
.:Some triangles are ignorant

16

I understood well enough. Sure, i could have interpreted it differently if i was feeling contrary, and im sure when current spoken english is a dead language some people would be confused, but those arent the issue. You dont need a meta-language, you just have to accept you cant ignore context. Language isnt transcendental, big deal. Surely?

I’ll just spell it phonetically then. Which is obviously sohmeyohneysat-tehdooer. You know, with a pause between the t’s and the second t is aspirated.
Apparantly my name is spookily similar to an american biscuit. Coincidence? I think not.

Well obviously, no one is better than Imp, but fortunately, I think we can all agree, I am better than no one, thus at least i can be comforted that i am better than Imp.

:stuck_out_tongue:

17

Dear biscuit oreso,

One cannot ignore context but nor can one put context into language. One cannot use language (and therefore logic) to determine, finally and completely, ‘context’. As such one can never be sure because one lacks a meta-language…

Context is not all…

There you go…

18

Dear Shat (or “siatd”)

Why would context need to be defined? This is just question begging:
Shat: “You cant get a contextless definition”
Chocolate Oreo: “So why not use context to define stuff?”
Shat: “Because you cant use context in a contextless definition! n00b!”

Language’s job is not to define context. Only lang and context together have meaning, so obviously you cant have a meaningful definition of the context in isolation. You would be building the ladder out of the rungs you are stood on. So i say, whats wrong with using context and lang together? Why does context require definition?

aside from lang then, what else is there?

19

Sorry I missed your post earlier, that essay is pretty cool.

I’ll have to look into it closely, but it seems to rely on a specific interpretation of quantum mechanics. I dont think we are in a position to state any interpretation with certainty (or uncertainty, har har) or if we can interpret QM at all.

A law of “included middle” would seem to sort things, if that interpretation turns out to be the case. I disagree with (very fudged) stuff about nature and god, etc though.

20

Dear Oreso

philosophypages.com/dy/i.htm#illo
en.wikipedia.org/wiki/J.L._Austin
hydra.umn.edu/derrida/sec.html