In keeping with the concept of vacuous truth, when disjunction is defined as an operator or function of arbitrary arity, the empty disjunction ( OR-ing over an empty set of operands ) is often defined as having the result 0.

In keeping with the concept of vacuous truth, when conjunction is defined as an operator or function of arbitrary arity, the empty conjunction ( AND-ing over an empty set of operands ) is often defined as having the result 1.

However, vacuous truth also appears in, for example, intuitionistic logic in the same situations given above.

Indeed, the first two forms above will yield vacuous truth in any logic that uses material conditional, but there are other logics which do not.

This is a complex question and, for simplicity of exposition, we will here consider only vacuous truth as concerns logical implication, i. e., the case when has the form, and is false.

By convention, the formula is always true, regardless of the formula P ( x ); see vacuous truth.

See also vacuous truth.

For a related problem, see vacuous truth.

This statement is an example of a vacuous truth since there is no x in the domain.

It should also be noted that the definition of makes vacuously true certain sentences, since when it speaks of " every world that is accessible to w " it takes for granted the usual mathematical interpretation of the word " every " ( see vacuous truth ).

The first problem is that the definition is not rigorous – all classes without instances would pass the definition, so that they become metaclasses by vacuous truth.

" vacuous truth.

# REDIRECT vacuous truth

Because that goal is significantly less vacuous than " maximizing utility " and the like, rational choice theory is apt.

Amelia begins as a warm-hearted and friendly girl, though sentimental and naive, but by story's end she is portrayed as vacuous and shallow.

Note that a field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way.

When the right-hand side is greater than one, so the inequality becomes vacuous, as the probability of any event cannot be greater than one.

All of her identity has been centered upon doing well academically ; she is unsure of what to make of her life once she leaves school, and the choices presented to her ( motherhood, as exemplified by the prolific child-bearer and vacuous Dodo Conway, or stereotypical female careers such as stenography ) do not appeal to her.

In accordance with its relationship to the sounds W and F it is sometimes weak and flustured ( German verwirrt ), as in the words vain, vacuous, vapid, vague, vacillate, vagrant, vaporous, vertigo, veer, and vary.

This quip is not vacuous: the study of embeddings in codimension 2 is knot theory, and difficult, while the study of embeddings in codimension 3 or more is amenable to the tools of high dimensional geometric topology, and hence considerably easier.

It was frequently criticized in the Toronto media, particularly for flashy but vacuous and repetitive local content, newscasts that had a tabloid feel and n uninspired prime time schedule based heavily on movies, much like CHUM's longstanding Citytv.

Such statements are considered vacuous because the falsity of the antecedent prevents one from using the conditional to infer the consequent.

Seven major software developers — including Ashton-Tate, Hewlett-Packard and Sybase — formed a council in 1990, and issued a report condemning the " vacuous product announcement dubbed vaporware and other misrepresentations of product availability " because they felt it had hurt the industry's credibility.

Aerogel is mostly air because its structure is like that of a highly vacuous sponge.

At the same time, they will have to, because otherwise Hegel's system concepts would say nothing about something that is not itself a concept and the system would come down to being only an intricate game involving vacuous concepts.

Subsequent nationalist movements, writers and politicians wrote about the Ottoman presence in very hostile and negative terms, with many works being vacuous, based on suspect sources and heavily biased.

Since rococo music tends to be pretty and elegant above all, it can seem rather vacuous to twentieth-century ears that have grown accustomed to the stress and dissonances of composers from Beethoven to Boulez.

Professor Larsen, however, while agreeing that Weatherhead was " a brilliant preacher ", judges his sermons to be theologically " vacuous and empty ".

* A 2007 study conducted on rats showed a complete remission of Haloperidol-induced tardive vacuous chewing symptoms when Busiprone was co-administered for five weeks.

Turtle does not seem hurt, but she has stopped speaking and has the same vacuous look in her eyes that she had when Taylor first saw her.

The Register has accused Gladwell of making arguments by weak analogy and commented that Gladwell has an " aversion for fact ", adding that, " Gladwell has made a career out of handing simple, vacuous truths to people and dressing them up with flowery language and an impressionistic take on the scientific method.

* Boubou Flaring: glamorous but vacuous actress.

He holds the position that the alternative seems to be the elaborate philosophical reduction of the word " moral " into a vacuous, useless term.

Scholars have described Ukraine's political system as " weak, fractured, highly personal and ideologically vacuous while the judiciary and media fail to hold politicians to account " ( Dr. Taras Kuzio in 2009 ).

Mrs. Allen: Somewhat vacuous, she sees everything in terms of her obsession with clothing and fashion, and has a tendency to utter repetitions of remarks made by others in place of original conversation.

A controversial critic of the blithely functionalist and symbolically vacuous architecture of corporate modernism during the 1950s, Venturi has been considered a counterrevolutionary.

NME magazine's Paul Du Noyer called the album " colossally smug and cosmically dull ; an interminable, vacuous and drearily egotistical stab at spirituality: Into the muzak.

Dawn French and Jennifer Saunders parodied Zeta-Jones as a vacuous über-celebrity named Catherine Spartacus-Zeta-Douglas-Jones, ( variously described as a princess of the Spartacus dynasty or the queen of Wales ) on their show French & Saunders in the series Back With a Vengeance.

For several years Sickert had already been painting lugubrious female nudes on beds, and continued to do so, deliberately challenging the conventional approach to life painting —" The modern flood of representations of vacuous images dignified by the name of ' the nude ' represents an artistic and intellectual bankruptcy "— giving four of them, which included a male figure, the title, The Camden Town Murder, and causing a controversy, which ensured attention for his work.

There is much truth in both these charges, and not many Bourbons deny them.

Poetry for a Persian is nothing less than truth and beauty.

To my knowledge, Lincoln remains the only Head of State and Commander-in-Chief who, while fighting a fearful war whose issue was in doubt, proved man enough to say this publicly -- to give his foe the benefit of the fact that in all human truth there is some error, and in all our error, some truth.

So great a man could not but understand, too, that the thing that moves men to sacrifice their lives is not the error of their thought, which their opponents see and attack, but the truth which the latter do not see -- any more than they see the error which mars the truth they themselves defend.

I suggested that one must let it in because it is the truth, but Beckett did not take to the word truth.

The central concern of Erich Auerbach's impressive volume called Mimesis is to describe the shift from a classic theory of imitation ( based upon a recognition of levels of truth ) to a Christian theory of imitation in which the levels are dissolved.

For this change is not a change from one positive position to another, but a change from order and truth to disorder and negation.

This, however, cannot be done by a community whose very experience of truth is confused and incoherent: it has no absolute standard, and consequently cannot distinguish the absolute from the contingent.

Its ontological status is itself most tenuous because apart from individual men, who are its `` matter '', tradition, the `` form '' of society exists only as a shared perception of truth.

And the direction of that movement is determined by his perception of the truth about himself.

And it would seem that history is a witness to this truth.

The basic truth in the reactionary response is to be found in its realistic assumption of the primacy of the real over the ideational.

But this truth is distorted by its extreme application: the assumption of the separate existence of tradition.

Whatever the psychological truth in the Oedipus myth, an Oedipus who is drawn to his fate by irresistible external forces can carry the symbol of humanity and its archaic crime, and the incest that is unknowing renews the mystery of the eternal dream of childhood and absorbs us in the secret.

But a modern Oedipus who is doomed because he cannot oppose his own childhood is only pathetic, and for renouncing the mystery in favor of psychological truth he gives up the claim on our sympathies.

Mann understood better than most men the incest comedy at the center of the myth and the psychological truth in which dread is shown as the other face as longing was for him just the kind of deep and complicated joke he liked to tell.

But however we come, finally, to explain and account for the present, the truth we are trying to expose, right now, is that the makers of constitutions and the designers of institutions find it difficult if not impossible to anticipate the behavior of the host of all their enterprises.

Its truth is illustrated by the skill, sensitivity, and general expertise of the English professor with whom one attends the theatre.

This truth that the moral law is natural has other important corollaries.

Nogaret is hardly an impartial witness, and even he did not make his charges against Boniface until the latter was dead, but there is some truth in what he said and more in what he did not say.