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Standing to definitions :

A fact is a sentence that states that a relation holds between individuals. A rule is a relation that holds between individuals provided that some other relations hold.

According to this, why it is not possible to construct contradictory descriptions using facts and rules ?

For example, i can use as facts the fact that a = b and a = not(b) and i have a contradiction....

Qwerto
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  • A fact is a fact (a part of teh world) and not a sentence. A rule in logic is not "something that holds between individuals": this is a relation. – Mauro ALLEGRANZA Feb 12 '18 at 12:25
  • Tird account of the day ? – Mauro ALLEGRANZA Feb 12 '18 at 12:26
  • Im sorry if it's a problem. But im obsessed with knowledge :) – Qwerto Feb 12 '18 at 12:29
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    I do not think. You have made many similar questions in the last few days: at least twelve. Most of them received answers: no vote (up or down), no accept; no requests for clarification; no feed-back at all. If you are not interested in answers, why ask questions ? – Mauro ALLEGRANZA Feb 12 '18 at 12:32
  • A sentence expresses a fact, i.e. something that holds or not. In FOL, an atomic sentence expresses a relation between individual: e.g. $x=x, P(a), R(x,y)$, etc. – Mauro ALLEGRANZA Feb 12 '18 at 12:37
  • Two sentences $\varphi$ and $\lnot \varphi$ are said to be contradictory. Thus, e.g., $a=b$ and $\lnot (a = b)$ are contradictory sentences. – Mauro ALLEGRANZA Feb 12 '18 at 12:38
  • In FOL we negate formulas and not "objects", i.e. (individual) variables and constants. – Mauro ALLEGRANZA Feb 12 '18 at 12:39
  • If i write the rule a(x) <-- not(a(x)) expressed as a definite sentence, this is unsatisfiable. So why using the language of definite programs, it is not possible to construct contradictory descriptions ? – Qwerto Feb 12 '18 at 21:47
  • The rules of logic must be sound, i.e. they must "transfer the truth", i.e. they must produce true conclusion from true premises. Thus, if the rule is sound and the conclusion is false, there must be some false premise. – Mauro ALLEGRANZA Feb 13 '18 at 10:08

2 Answers2

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Who on earth says "A fact is a sentence that states that a relation holds between individuals"?

At least in my neck of the woods, there are two common views about facts -- namely that facts are true propositions (true declarative sentences), or that facts are truth-makers, the sort of thing that makes a true proposition true. But on either view, truth comes into it! A sentence that states that a relation between individuals can be false, and so is neither a fact-qua-true-sentence, nor does it correspond to a fact-qua-truth-maker.

Peter Smith
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You define a fact as an expression that says that some relation holds ... as such you can only express positive claims, and not any negations, i.e. you cannot express that some relation does not hold. The same is true for the rules: they apparently can only say that some relation holds when other relations holds, and so you cannot make any negations part of rules either. It is therefore impossible to use that $a \not = b$ as a fact, or make that part of a rule. And indeed, without negations you cannot get any contradictions at all.

Bram28
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  • This property of rules , is it connected to the fact that we use definite clauses to express rules ? – Qwerto Feb 12 '18 at 12:28
  • @Qwerto Well, it's because by only using positive claims you obtain a system for which you can define very efficient algorithms for checking logical consequence ... I assume this came up in the context of a logic programming course? – Bram28 Feb 12 '18 at 12:32
  • If a rule can say that some relation holds when other relations holds, why cant a rule say something as "if a(fact) so not(a)" ? – Qwerto Feb 12 '18 at 14:45
  • @qwerto if you say $a \not = b$ then you are not saying that some relation holds,but rather that some relation does not hold. – Bram28 Feb 12 '18 at 14:48
  • Maybe i've finally understood: everything we just said about facts and rules rest on the implicit principle that facts and rules are true ? It is that right ? – Qwerto Feb 13 '18 at 08:44