are all abbreviations found in Muslim writings and represent the same Arabic phrase, salla Allahu alaihi wa-sallam. In English, Muslims often use the acronym "PBUH" instead.

The meaning of this phrase is given on Muslim websites as:

"SAW" and "PBUH" represent the Arabic phrase salla Allah alaihi wa sallam, meaning "may Allah's peace and blessings be upon him." It is a standard Muslim expression of love and respect for the Prophet. (Source)

When the name of Prophet Muhammad (saw) is mentioned or written, a Muslim is to respect him and invoke this statement of peace upon him. The meaning of it is: "May the blessings and the peace of Allah be upon him (Muhammad)". (Source)

Sallallahu 'Alaihe wa Sallam (S.A.W.)
"May the peace and blessings of Allah be upon him." This is said whenever the name of prophet Muhammad (S.A.W.) is mentioned or read. The equivalent English phrase is usually abbreviated as S.A.W. (peace be upon him). (Source)

However, a number of important aspects get lost in these inaccurate translations of the phrase. Salla and sallam are both verbs not nouns as the above rendering "blessings" and "peace" may suggest. Salla and sallam are verbs in the perfective aspect. This form of the verb can be used for both past tense (something accomplished), future perfective (like English "he will have done" etc.) and wishes that something may be accomplished.

The main meaning for salla, as every dictionary will show, is "to pray" and salla ala means "to pray for". Ibn 'Abbas states:

"The people of Israel said to Moses (peace be upon him): ‘Does your Lord pray?’ His Lord (az wa gal) called him [saying]: O Moses, they asked thee if your Lord prays. Say [to them] ‘Yes, I do pray, and my angels [pray] upon my prophets and my messengers’, and Allah (az wa gal) then sent down on his messenger (prayer and peace be upon him): ‘Allah and His angels pray ...’" [quoted by Ibn Kathir on Surat Al-Ahzaab 33:56; translated from the online Arabic edition]

[Az wa gal = exalted and majesty, a phrase of praise spoken after God was mentioned.]

The verb sallam can convey various concepts. One aspect is "safety and security". Hans Wehr in his dictionary renders it as "grant him salvation!" Another meaning is to greet or salute. Nearly all Muslim and non-Muslim translations of the Qur'an render the final phrase of Sura 33:56, wasallimoo tasleeman, as "and salute him with a worthy salutation" (Pickthall), "and salute him with all respect" (Yusuf Ali), "and salute him with a salutation!" (Palmer).

Thus, at a literal level, the following translations for salla Allahu alaihi wa-sallam are all possible:

"Allah prayed for him and saluted him" (past tense, a statement of fact).
"(May) Allah pray for him and salute him!" (a wish).
"(May) Allah pray for him and save him!" (a wish).

Context/usage will have to determine what the actual meaning of it is (in each instance).

For a detailed discussion of various issues around this eulogy of Muhammad, particularly the issue of Allah praying for Muhammad, see the articles Does Allah Pray? and "The "Mystery" of PBUH revealed" [Part 1], [Part 2].

It is noteworthy that "PBUH" is used by western Muslims for all prophets in Islam, but in Arabic, "SAW" is reserved for Muhammad only. The consequences of this are discussed in the entry on "PBUH".

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