Wilkins has had 1 FULL season now

[Uncle Buck]

We must have lost a couple of games then...

 

[Les Biehn]

Too right. I was not trying to persude him not to post or was I?

I didn't say you were. You were suggesting he was oppressing Spielbergs opinion, however I would suggest he was merely commenting on how boring it is to have it rammed down people's throats every 5 minutes.

 

[Perry Milkins]

No - your point about it being an open forum bears no relevance as to the tedium of the anti-Wilkins posts or not. I'm not going to but if I posted a poll thread asking who found all these U.S. anti-Wilkins threads boring there'd be a fair few who'd agree.

What's childish about that, then?

Until such time that the mods declare that US should not start threads about Wilkins he is free to do so, as are you to counter his points. I felt you where pasting my previous response and trying to bring it contextually into a reply to a further point I made. hence ths comment I made.

 

[Perry Milkins]

I didn't say you were. You were suggesting he was oppressing Spielbergs opinion, however I would suggest he was merely commenting on how boring it is to have it rammed down people's throats every 5 minutes.

It is topical is it not?

 

[Buzzer]

Until such time that the mods declare that US should not start threads about Wilkins he is free to do so, as are you to counter his points. I felt you where pasting my previous response and trying to bring it contextually into a reply to a further point I made. hence ths comment I made.

which makes your point about it being an open forum a tautology then. Hence, it didn't need saying.

 

[Les Biehn]

It is topical is it not?

Yes, but more importantly is it needed?

 

[Perry Milkins]

We must have lost a couple of games then...

Just the one on Saturday.

Here is my agenda. I was hiding it from you.

Dean spoilt my weekend.

Sussex won the Campiones 'ship.

Both my sons won their respective footie games on Sunday.

But hey the 'albion' screw it up by losing to Yeovil Nathan Jones has the license to do as he pleases whilst some french bloke acclimatises himself to English football whilst Jake warms the bench.

There..you see my point and my heartache?

 

[Perry Milkins]

Yes, but more importantly is it needed?

That is relevant but we will need the wider kinfolk of NSC to answer the question? Anyone for a poll?

 

[Perry Milkins]

which makes your point about it being an open forum a tautology then. Hence, it didn't need saying.

If you say so Buzzer, if you say so. Did I need to say that?

 

[Uncle Buck]

Just the one on Saturday.

Here is my agenda. I was hiding it from you.

Dean spoilt my weekend.

Sussex won the Campiones 'ship.

Both my sons won their respective footie games on Sunday.

But hey the 'albion' screw it up by losing to Yeovil Nathan Jones has the license to do as he pleases whilst some french bloke acclimatises himself to English football whilst Jake warms the bench.

There..you see my point and my heartache?

Saturday's game is a bit of a blur, was more interested in goings on else where.

They key from here is how Wilkins can turn it around from here, he has to avoid any more runs like the ones we had last season.

 

[Barrel of Fun]

BOF

I am bored quite frankly of the " they are inexperienced finding their feet " line now, how long before we stop using that excuse for Cox, Robinson & Co, 19, 25 , 35, 75

Until they have gained enough experience?

Someone like Lynch has been fantastic and very rarely had a bad game for us, but I don't think it is unreasonable to accept that many of the first/second year pros are going to blow hot and cold.

For goodness sake, plenty of teams are able to use a decent mix of players as opposed to our team that was at different ends of the spectrum, which is gradually being redressed.



Carwash, I am not saying that Deano is perfect. There have been some strange decisions, but then I am not a manager and I am not privvy to the performances in training. Some things work and somethings do not. If anything, he needs to keep things simple and decide on his team. The real test will be to see how we bounce back from this latest setback. Shame we have a potentially tricky trip to Wales!

 

[Uncle Spielberg]

Until they have gained enough experience?

Someone like Lynch has been fantastic and very rarely had a bad game for us, but I don't think it is unreasonable to accept that many of the first/second year pros are going to blow hot and cold.

For goodness sake, plenty of teams are able to use a decent mix of players as opposed to our team that was at different ends of the spectrum, which is gradually being redressed.



Carwash, I am not saying that Deano is perfect. There have been some strange decisions, but then I am not a manager and I am not privvy to the performances in training. Some things work and somethings do not. If anything, he needs to keep things simple and decide on his team. The real test will be to see how we bounce back from this latest setback. Shame we have a potentially tricky trip to Wales!

to talk far too much sense for a young man BOF, can't you just have a rant or say something outrageous or silly just once, go on

 

[Uncle Spielberg]

I didn't say you were. You were suggesting he was oppressing Spielbergs opinion, however I would suggest he was merely commenting on how boring it is to have it rammed down people's throats every 5 minutes.

I do not love you anymore Les

 

[Les Biehn]

I do not love you anymore Les

It is part of my charm that I don't take sides.

 

[Perry Milkins]

It is part of my charm that I don't take sides.

What other charms do you possess...

 

[Perry Milkins]

I do not love you anymore Les

US am I on your side?

 

[Woodchip]

to talk far too much sense for a young man BOF, can't you just have a rant or say something outrageous or silly just once, go on

No, BoF leaves it to the older generation to post stupid Wilkins threads every 5 minutes, even if we haven't played for 2 days.

Oh, and CarWash - I presume you have no knowledge of the meaning of tautology going by your response. Bless ya.

 

[Uncle Spielberg]

No, BoF leaves it to the older generation to post stupid Wilkins threads every 5 minutes, even if we haven't played for 2 days.

Oh, and CarWash - I presume you have no knowledge of the meaning of tautology going by your response. Bless ya.

is there a time limit after a game to post Wilkins comments I did not realise that

 

[Perry Milkins]

No, BoF leaves it to the older generation to post stupid Wilkins threads every 5 minutes, even if we haven't played for 2 days.

Oh, and CarWash - I presume you have no knowledge of the meaning of tautology going by your response. Bless ya.

Thank you.

 

[Uncle Spielberg]

No, BoF leaves it to the older generation to post stupid Wilkins threads every 5 minutes, even if we haven't played for 2 days.

Oh, and CarWash - I presume you have no knowledge of the meaning of tautology going by your response. Bless ya.

Tautology (logic)
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In propositional logic, a tautology (from the Greek word ταυτολογία) is a sentence that is true in every valuation (also called interpretation) of its propositional variables, independent of the truth values assigned to these variables. For example, is a tautology, because any valuation either makes A and B both true, or makes one or the other false. According to Kleene (1967, p. 12), the term was introduced by Ludwig Wittgenstein (1921).

The negation of a tautology is a contradiction, a sentence that is false regardless of the truth values of its propositional variables, and the negation of a contradiction is a tautology. A sentence that is neither a tautology nor a contradiction is logically contingent. Such a sentence can be made either true or false by choosing an appropriate interpretation of its propositional variables.

Contents [hide]
1 Examples
2 Notation
3 Tautological implication
4 Substitution
5 Verifying tautologies
6 Tautologies versus validities in first-order logic
7 Tautology and its application in Logic Synthesis
8 References
9 External links
10 See also
10.1 Normal forms
10.2 Related logical topics
10.3 Related topics



[edit] Examples
There are infinitely many tautologies. One example is


the law of the excluded middle. A truth valuation for this formula must, by definition, assign A one of the truth value true or false, and assign the other truth value. Thus the disjunction in this law is satisfied by every valuation.

Another tautology is


One method of verifying that every valuation causes this sentence to be true is to make a truth table that examines every possible valuation. There are 8 possible valuations for the propositional variables A, B, C, represented by the first three columns of the following table. The remaining columns show the truth of subsentences of the sentence above, culminating in a column showing the truth value of this sentence under each interpretation.

A B C
T T T T T T T T
T T F T F F F T
T F T F T T T T
T F F F T T T T
F T T F T T T T
F T F F T F T T
F F T F T T T T
F F F F T T T T

Because each row of the final column shows T, the sentence in question is verified to be a tautology.


[edit] Notation
The notation is used to indicate that S is a tautology. The symbol is sometimes used to denote an arbitrary tautology, with the dual symbol (falsum) representing an arbitrary contradiction.


[edit] Tautological implication
A sentence S is said to tautologically imply a sentence T if every truth valuation that causes S to be true also causes T to be true. This situation is denoted . It is equivalent to the sentence being a tautology (Kleene 1967 p. 27).

It follows from the definition that if S is a contradiction then S tautologically implies every sentence, because there is no truth valuation that causes S to be true and so the definition of tautological implication is trivially satisfied.


[edit] Substitution
There is a general procedure, the substitution rule, that allows additional tautologies to be constructed from a given tautology (Kleene 1967 sec. 3). Suppose that S is a tautology and for each propositional variable A in S a fixed sentence SA is chosen. Then the sentence obtain by replacing each variable A in S with the corresponding sentence SA is also a tautology.

For example, let S be , a tautology. Let SA be and let SB be . It follows from the substitution rule that the sentence


is a tautology.


[edit] Verifying tautologies
The problem of constructing practical algorithms to determine whether sentences with large numbers of propositional variables are tautologies is an area of contemporary research in the area of automated theorem proving.

The method of truth tables illustrated above is provably correct – the truth table for a tautology will end in a column with only T, while the truth table for a sentence that is not a tautology will contain a row whose final column is F, and the valuation corresponding to that row is a valuation that does not satisfy the sentence being tested. This method for verifying tautologies is an effective procedure, which means that given unlimited computational resources it can always be used to mechanistically determine whether a sentence is a tautology.

As an efficient procedure, however, truth tables are constrained by the fact that the number of valuations that must be checked increases as 2k, where k is the number of variables in the formula. This exponential growth in the computation length renders the truth table method useless for formulas with thousands of propositional variables, as contemporary computing hardware cannot execute the algorithm in a feasible time period.

The problem of determining whether there is any valuation that makes a formula true is the Boolean satisfiability problem; the problem of checking tautologies is equivalent to this problem, because verifying that a sentence S is a tautology is equivalent to verifying that there is no valuation satisfying . It is known that the Boolean satisfiability problem is NP complete, and widely believed that there is no polynomial-time algorithm that can perform it. Current research focuses on finding algorithms that perform well on special classes of formulas, or terminate quickly on average even though some inputs may cause them to take much longer.


[edit] Tautologies versus validities in first-order logic
The fundamental definition of a tautology is in the context of propositional logic. The definition can be extended, however, to sentences in first-order logic (see Enderton (2002, p. 114) and Kleene (1967 secs. 17–18)). These sentences may contain quantifiers, unlike sentences of propositional logic. In the context of first-order logic, a distinction is maintained between logical validities, sentences that are true in every model, and tautologies, which are a proper subset of the first-order logical validities. In the context of propositional logic, these two terms coincide.

A tautology in first-order logic is a sentence that can be obtain by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). For example, because is a tautology of propositional logic, is a tautology in first order logic. Similarly, in a first-order language with a unary relation symbols R,S,T, the following sentence is a tautology:


It is obtained by replacing A with , B with , and C with in the propositional tautology considered above.

Not all logical validities are tautologies in first-order logic. For example, the sentence


is true in any first-order interpretation, but it corresponds to the propositional sentence which is not a tautology of propositional logic.


[edit] Tautology and its application in Logic Synthesis
In Logic Synthesis tautology plays an important role especially for Logic Optimization. Though the problem is intractable, whether or not a function is a tautology can be efficiently answered using the Recursive Paradigm. Any binary-valued function F is a tautology if and only if its cofactors with respect to any variable and its complement are both tautologies. Hence it can be easily concluded whether or not a function F is reducible to a tautology by recursive Shannon Expansion and the application of the above theorem.


[edit] References
Enderton, H. B. (2002). A Mathematical Introduction to Logic. Harcourt/Academic Press. ISBN 0-12-238452-0
Kleene, S. C. (1967). Mathematical Logic. Reprinted 2002, Dover. ISBN 0-486-42533-9
Rechenbach, H. (1947). Elements of Symbolic Logic. Reprinted 1980, Dover. ISBN 0-486-24004-5
Wittgenstein, L. (1921). "Logisch-philosophiche Abhandlung," Annalen der Naturphilosophie (Leipzig), v. 14, pp. 185–262. Reprinted in English translation as Tractatus logico-philosophicus, New York and London, 1922.

[edit] External links
Eric W. Weisstein, Tautology at MathWorld.

[edit] See also

[edit] Normal forms
Algebraic normal form
Conjunctive normal form
Disjunctive normal form
Logic Optimization

[edit] Related logical topics
Boolean algebra (logic)
Boolean domain
Boolean function
First-order logic
Logical consequence
Logical graph
Propositional logic
Table of logic symbols
Truth table
Vacuous truth
Zeroth order logic



[edit] Related topics
Tautology (rhetoric), use of redundant language that adds no information.