Then, once this claim ( expressed in the previous sentence ) is proved, it will suffice to prove " φ is either refutable or satisfiable " only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ ( i. e., ( φ ≡ ψ ) is provable ), then it is indeed the case that " ψ is either refutable or satisfiable " → " φ is either refutable or satisfiable " ( the soundness theorem is needed to show this ).

We see that we can restrict φ to be a sentence, that is, a formula with no free variables.

Our generic formula φ now is a sentence, in normal form, and its prefix starts with a universal quantifier and ends with an existential quantifier.

The compactness theorem implies Robinson's principle: If a first-order sentence holds in every field of characteristic zero, then there exists a constant p such that the sentence holds for every field of characteristic larger than p. This can be seen as follows: suppose φ is a sentence that holds in every field of characteristic zero.

The diagonal lemma states that there is a sentence φ such that φ ↔ ψ (< u >#( φ )</ u >) is provable in T.

Intuitively, φ is a self-referential sentence saying that φ has the property ψ.

The sentence φ can also be viewed as a fixed point of the operation assigning to each formula θ the sentence ψ (< u >#( θ )</ u >).

Let φ be the sentence β (< u >#( β )</ u >).

He used these to give an explicit definition of a formula Bew ( x ) that is true if and only if x is the Gödel number of a sentence φ and there exists a natural number that is the Gödel number of a proof of φ ( The German word for proof is Beweis ).

It is characteristic of early literature that the evolution of the thought, or the grammatical form of the sentence, is guided by the structure of the verse ; and the correspondence which consequently obtains between the rhythm and the syntax — the thought being given out in lengths, as it were, and these again divided by tolerably uniform pauses — produces a swift flowing movement such as is rarely found when periods are constructed without direct reference to the metre.

This sentence was constructed by Noam Chomsky as an illustration that phrase structure rules are capable of generating syntactically correct but semantically incorrect sentences.

* Naffarin was the first language Tolkien constructed by himself during his adolescence ; only one sentence remains which shows a Spanish affinity.

Finally, the bottom of the structure is composed of the words out of which the sentence is constructed.

In both LexRank and TextRank, a graph is constructed by creating a vertex for each sentence in the document.

In English, a cleft sentence can be constructed as follows:

The typical United Nations resolution is constructed as a single, very long sentence.

The means of carrying out the sentence of death used at Holman Prison was an electric chair constructed by an inmate in 1927.

Space builders are elements within a sentence that establish spaces distinct from, yet related to the base space constructed.

Although French poetry during the reign of Henri IV and Louis XIII was still largely inspired by the poets of the late Valois court, some of their excesses and poetic liberties found censure — especially in the work of François de Malherbe, who criticized La Pléiade's and Philippe Desportes's irregularities of meter or form ( the suppression of the cesura by a hiatus, sentence clauses spilling over into the next line — enjambement — neologisms constructed from Greek words, etc .).

Post gave a version of Gödel's Incompleteness Theorem using his creative sets, where originally Gödel had in some sense constructed a sentence that could be freely translated as saying " I am unprovable in this axiomatic theory.

It says that for any first-order theory T with a well-orderable language, and any sentence S in the language of the theory, there is a formal proof of S in T if and only if S is satisfied by every model of T ( S is a semantic consequence of T ).

The proof constructs a specific Gödel sentence for each effectively generated theory, but there are infinitely many statements in the language of the theory that share the property of being true but unprovable.

First, common natural-language proof techniques can be used to prove that the example sentence is true.

In this context, it shows that if we assume there is a formal sentence ( X → Y ), where X itself is equivalent to ( X → Y ), then we can prove Y with a formal proof.

A complete proof would show that the property displayed in quotes in the previous sentence is definable in the language of Peano arithmetic by a formula.

Aside from Section 1519 ’ s 20-year maximum prison sentence ( no small benefit to the government in big-dollar fraud loss cases such as Wolff ), its primary appeal is that it uniquely removes certain key proof burdens from prosecutors ’ collective shoulders.

The Talmud approves of the death penalty in principle but the standard of proof required for application of death penalty is extremely stringent, so that situations in which a death sentence could be passed are effectively impossible.

The proof is non-constructive ( a pure existence proof ): while it guarantees that if a sentence α follows ( semantically ) from a set of sentences Σ, then there is a proof of α from Σ, it gives no indication of the nature of that proof.

Rather than using the liar paradox sentence equivalent to " I am not provable ," he used a sentence that stated " For every proof of me, there is a shorter proof of my negation ".

The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists.

In this method, each sentence and formal proof in first-order arithmetic is assigned a particular natural number.

" La Razón " contends that the Wolfensons were convicted without sufficient proof and that their sentence is in effect political revenge.

If the system is suitably complex, like first-order arithmetic, then the set T of Gödel numbers of true sentences in the system will be a productive set, which means that whenever W is a recursively enumerable set of true sentences, there is at least one true sentence that is not in W. This can be used to give a rigorous proof of Gödel's first incompleteness theorem, because no recursively enumerable set is productive.

His explanations are all of a homiletical character ; his sole object being to find in each sentence or in each word of the Scriptures a moral lesson, a support for trust in God, encouragement to patient endurance, and a proof of the vanity of all earthly goods as compared with the everlasting bliss to be acquired in the future life.

This resolution technique uses proof by contradiction and is based on the fact that any sentence in propositional logic can be transformed into an equivalent sentence in conjunctive normal form.

Since attack serves to stimulate interest in broadcasts, I added to my opening statement a sentence in which I claimed that German youth seemed to lack the enthusiasm which is a necessary ingredient of anger, and might be classified as uninterested and bored rather than angry.

the athlete of by-gone years who wears a size 46 suit and puffs when he has finished a sentence of any length then, it is time to break it up and move on.

But the task is beyond me because I hold it impossible to compress in a sentence or two the complicated and prodigious contributions Sam Rayburn has made as an individual, as a legislator, as a statesman and as a leader and conciliator, to the majestic progress of this Nation.

In the second sentence if drinking water is a gerundial clause and without drinking water is roughly equivalent in meaning to unless I drink water, there will be stronger stress on water than on drinking ; ;

In the Steiners have busy lives without visiting relatives only context can indicate whether visiting relatives is equivalent in meaning to paying visits to relatives or to relatives who are visiting them, and in I looked up the number and I looked up the chimney only the meanings of number and chimney make it clear that up is syntactically a second complement in the first sentence and a preposition followed by its object in the second.

The jail sentence is to begin the day after Sarkees graduates from Eastern High School in June.

If the petition is granted the appellant could be released from incarceration, the sentence could be modified, or a new trial could be ordered.

Swift ’ s specific strategy is twofold, using a " trap " to create sympathy for the Irish and a dislike of the narrator who, in the span of one sentence, " details vividly and with rhetorical emphasis the grinding poverty " but feels emotion solely for members of his own class.

Allocution is sometimes required of a defendant who pleads guilty to a crime in a plea bargain in exchange for a reduced sentence.

In most United States jurisdictions a defendant is allowed the opportunity to allocute — that is, explain himself — before sentence is passed.

In his spare time, Grand polishes up his Latin, and he is also writing a book, but he is such a perfectionist that he continually rewrites the first sentence and can get no further.

This is because they were only carrying out the sentence of death that he would have faced had he not been acquitted on a technicality.

It is in the Boethius that the oft-quoted sentence occurs: " My will was to live worthily as long as I lived, and after my life to leave to them that should come after, my memory in good works.

The primary purpose of this text is to refine the literary concept dhvani or poetic suggestion, by arguing for the existence of rasa-dhvani, primarily in forms of Sanskrit including a word, sentence or whole work " suggests " a real-world emotional state or bhāva, but thanks to aesthetic distance, the sensitive spectator relishes the rasa, the aesthetic flavor of tragedy, heroism or romance.

The next sentence is often translated " Crantor adds, that this is testified by the prophets of the Egyptians, who assert that these particulars are narrated by Plato are written on pillars which are still preserved.

" But in the original, the sentence starts not with the name Crantor but with the ambiguous He, and whether this referred to Crantor or to Plato is the subject of considerable debate.

In the sentence I see the car, the noun phrase the car is the direct object of the verb " see ".

One can correctly use " the car " as the subject of a sentence also: " The car is parked here.

This is the form in the nominative case, used for the subject of a sentence.