* Axiomatic proof: Proofs are deductive derivations of propositions from primitive premises that are ‘ true ’ in some sense.

Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects ( for example, numbers and functions ) from all the traditional areas of mathematics ( such as algebra, analysis and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them.

Ethical subjectivism stands in opposition to moral realism, which claims that moral propositions refer to objective facts, independent of human opinion ; to error theory, which denies that any moral propositions are true in any sense ; and to non-cognitivism, which denies that moral sentences express propositions at all.

It states that contradictory statements cannot both be true in the same sense at the same time, e. g. the two propositions " A is B " and " A is not B " are mutually exclusive.

*** Moral realism ( in the robust sense ; see moral universalism for the minimalist sense ) holds that such propositions are about robust or mind-independent facts, that is, not facts about any person or group's subjective opinion, but about objective features of the world.

There are propositions in his writings which can be understood in a purely rationalistic sense.

The pope responded with the bull Ad Sanctam Beati Petri Sedem ( dated October 16, 1656 ) in which he stated " We declare and define that the five propositions have been drawn from the book of Jansenius entitled Augustinus, and that they have been condemned in the sense of the same Jansenius and we once more condemn them as such.

Clement IX ordered his nuncio to conduct a new investigation ' reporting back, the nuncio declared: " they have condemned and caused to be condemned the five propositions with all manner of sincerity, without any exception or restriction whatever, in every sense in which the Church has condemned them ".

The existence of propositions in sense ( a ) above, as well as the existence of " meanings ," is disputed by some philosophers.

In this sense, propositions are " statements " that are truth bearers.

That is, propositions in this sense are meaningless, formal, abstract objects.

Bertrand Russell famously rejected Frege's sense-reference distinction, though there is some possibility that the two were misinterpreting and arguing past one another: Frege talks about ( for example ) sentences, which have both a sense ( a proposition ) and a reference ( a truth value ); Russell on the other hand deals directly with propositions, but construes these not as abstract para-linguistic items but as tuples, or sets, of objects and concepts.

The propositions listed had been condemned as erroneous opinions in the sense and context in which they originally occurred ; without the original context, the document appeared to condemn a larger range of ideas than it actually did.

The fourth section considers how common sense propositions like " Here is my hand " are to be analysed.

Moore concludes that we are absolutely certain about the common sense belief, but that no analysis of the propositions has been offered that is even close to being certain.

Ethical subjectivism stands in opposition to moral realism, which claims that moral propositions refer to objective facts, independent of human opinion ; to error theory, which denies that any moral propositions are true in any sense ; and to non-cognitivism, which denies that moral sentences express propositions at all.

As Volf later recalled about his childhood, he did not have the luxury of “ entertaining faith merely as a set of propositions that you do or don ’ t assent to .” In school, especially in his early teens, the faith of his parents and their community was a heavy burden ; Volf ’ s sense of being different from his peers and from the larger culture around him caused him “ almost unbearable shame ” and he rebelled against faith.

Note: This includes propositions which are necessarily true, in the sense below.

The word amphibian became restricted in the taxonomical sense to what we now use around 1600, with the taxon " Amphibia " first published in scientific classification circa 1819.

Still, even when used in this second sense, the word is often restricted to apply only to societies that have attained a particular level of advancement — especially the founding of cities.

While most international law has a broader interpretation of the term treaty, the U. S. sense of the term is more restricted.

This term, it is true, has sometimes been used, and is still sometimes used, in almost as restricted a sense as that of Tamil itself, so that though on the whole it is the best term I can find, I admit it is not perfectly free from ambiguity.

Although the modern term " gun " is often used as a synonym for firearm, in specialist or military use it has a restricted sense referring only to an artillery piece with a relatively high muzzle velocity, such as a field gun, a tank gun, or a gun used in the delivery of naval gunfire ; or in sporting use for a shotgun.

Hand-held firearms such as rifles, carbines, pistols and other small firearms, are rarely called " guns " in the restricted sense among specialists.

Often, the term glass is used in a restricted sense to refer to this specific use.

… British hackers mostly learned / kluhj / orally, use it in a restricted negative sense and are at least consistent.

Descriptive epithets are a common literary device in many parts of the world, whereas kennings in this restricted sense are a distinctive feature of Old Norse and, to a lesser extent, Old English poetry.

In ancient times, literacy in its restricted sense was always confined to a small elite.

Thus, not all sound deductive systems are complete in this special sense of completeness, in which the class of models ( up to isomorphism ) is restricted to the intended one.

In a more restricted sense, a scientist is an individual who uses the scientific method.

In the United States, the term " treaty " has a different, more restricted legal sense than exists in international law.

It includes, but is not restricted to, sexuality in the sense of eroticism.

The separation axioms are axioms only in the sense that, when defining the notion of topological space, one could add these conditions as extra axioms to get a more restricted notion of what a topological space is.

In technical and development circles the old restricted sense of an OS persists because of the continued active development of embedded operating systems for all kinds of devices with a data-processing component, from hand-held gadgets up to industrial robots and real-time control-systems, which do not run user applications at the front-end.

Tabu itself has been derived from alleged Tongan morphemes ta (" mark ") and bu (" especially "), but this may be a folk etymology ( note that Tongan does not actually have a phoneme / b /), and tapu is usually treated as a unitary, non-compound word inherited from Proto-Polynesian * tapu, in turn inherited from Proto-Oceanic * tabu, with the reconstructed meaning " sacred, forbidden ".< ref name = POLLEX > In its current use on Tonga, the word tapu means " sacred " or " holy ", often in the sense of being restricted or protected by custom or law.

However, when used in the second ( more restricted ) sense, iteration describes the style of programming used in imperative programming languages.

In flowering plants, the term " apomixis " is commonly used in a restricted sense to mean agamospermy, i. e. asexual reproduction through seeds.

Though the literal meaning of the words could include anyone who fights for the cause of freedom, in common use it may be restricted to those who are actively involved in an armed rebellion, rather than those who campaign for freedom by peaceful means ( though they may use the title in its literal sense ).

* In a less restricted sense, internationalism is a word describing the impetus and motivation for the creation of any international organizations.

Some similar results about restricted partitions can be obtained by the aid of a visual tool, a Ferrers graph ( also called Ferrers diagram, since it is not a graph in the graph-theoretical sense, or sometimes Young diagram, alluding to the Young tableau ).

In common parlance, all members of male religious institutes are often termed " monks " and those of female religious institutes " nuns ", although in a more restricted sense, a monk is one who lives in a monastery under a monastic rule such as that of Saint Benedict and the term " nun " was in the 1917 Code of Canon Law officially reserved for members of a women's religious institute of solemn vows, and is sometimes applied only to those who devote themselves wholly to the contemplative life and belong to one of the enclosed religious orders living and working within the confines of a monastery and reciting the Liturgy of the Hours in community.

Smilacaceae appears to be the family most closely related to Liliaceae in its modern restricted sense.