What Is the Lowest Oxidation Number of Mn

In the IUPAC Recommendations 2016 the definition of oxidation state underwent a significant and comprehensive change. It does now resemble the version wich was proposed be Hans-Peter Loock and is quoted in the earlier version of this answer, parts of which are included below.

The electronegativity battle scheme is most helpful for all kinds of compounds since it is one of the most generic ways to derive oxidation states. The table represents just a cheat sheet that might be very helpful in the beginning. If you are spending most of your time with chemistry, this table will be present as some sort of muscle memory - usually referred to as chemical intuition.

The IUPAC gold book now defines oxidation state as follows:

oxidation state
gives the degree of oxidation of an atom in terms of counting electrons. The higher the oxidation state (OS) of a given atom, the greater is its degree of oxidation. Definition:
OS of an atom is the charge of this atom after ionic approximation of its heteronuclear bonds.

The underlying principle is that the ionic sign in an $\ce{AB}$ molecule is deduced from the electron allegiance in a LCAO-MO model: The bond's electrons are assigned to its main atomic contributor. Homonuclear $\ce{AA}$ bonds are divided equally. In practical use, the ionic-approximation sign follows Allen electronegativities (see Source). There are two general algorithms to calculate OS:

1. Algorithm of assigning bonds, which works on a Lewis formula showing all valence electrons in a molecule: OS equals the charge of an atom after its heteronuclear bonds have been assigned to the more electronegative partner (except when that partner is a reversibly bonded Lewis-acid ligand) and homonuclear bonds have been divided equally:
   

2. Algorithm of summing bond orders: Heteronuclear-bond orders are summed at the atom as positive if that atom is the electropositive partner in a particular bond and as negative if not, and the atom's formal charge (if any) is added to that sum, yielding the OS. This algorithm works on Lewis formulas and on bond graphs of atom connectivities for an extended solid:
   

NOTE:

1. Specific uses may require modified OS values: Electrochemical OS is nominally adjusted to represent a redox-active molecule or ion in Latimer or Frost diagrams. Nominal OS values may also be chosen from close alternatives for systematic-chemistry descriptions.
2. Some OS may be ambiguous, typically when two or more redox-prone atoms enter bonding compromises and nearest integer values have to be chosen for the OS.
3. The caveat of reversibly bonded Lewis-acid ligands originates from the simplifying use of electronegativity instead of the MO-based electron allegiance to decide the ionic sign. Typical examples are the transition-metal complexes with so called Z ligands in the CBC ligand-classification scheme (see Source).
4. When used in chemical nomenclature as a symbol, the OS value is in roman numerals.
5. At the introductory teaching level, prior to the bonding-based definition and algorithms: OS for an element in a chemical formula is calculated from the overall charge and postulated OS values for all the other atoms. For example, postulating $\text{OS} = +1$ for $\ce{H}$ and $−2$ for $\ce{O}$ yields correct OS in oxides, hydroxides, and acids like $\ce{H2SO4}$; with coverage extended to $\ce{H2O2}$ via decreasing priority along the sequence of the two postulates. Additional postulates may expand the range of compounds to fit a textbook's scope.

Source:

1. Karen, P.; Mcardle, P.; Takats, J. Toward a comprehensive definition of oxidation state (IUPAC Technical Report). Pure Appl. Chem. 2014, 86 (6), 1017–1081 DOI: 10.1515/pac-2013-0505.
2. Karen, P.; Mcardle, P.; Takats, J. Comprehensive definition of oxidation state (IUPAC Recommendations 2016). Pure Appl. Chem. 2016, 88 (8), 831–839 DOI: 10.1515/pac-2015-1204.
3. Nomenclature of Inorganic Chemistry. IUPAC Recommendations 2005. p. 34. Available as pdf from old.iupac.org

I recommend having a look at the sources cited above, they contain a lot more information. (Note that the link from the gold book to the red book on the IUPAC website is broken.)

Prior to 2016 the IUPAC defined oxidation states in their gold book (via the Internet Archive) differently. I am including this definition, since it will be still present in prominent text books and it is what lead to the postulates given in the table of the original question.

oxidation state (deprecated, pre-2016 definition) A measure of the degree of oxidation of an atom in a substance. It is defined as the charge an atom might be imagined to have when electrons are counted according to an agreed-upon set of rules:

1. the oxidation state of a free element (uncombined element) is zero;
2. for a simple (monatomic) ion, the oxidation state is equal to the net charge on the ion;
3. hydrogen has an oxidation state of $1$ and oxygen has an oxidation state of $-2$ when they are present in most compounds. (Exceptions to this are that hydrogen has an oxidation state of $-1$ in hydrides of active metals, e.g. $\ce{LiH}$, and oxygen has an oxidation state of $-1$ in peroxides, e.g. $\ce{H2O2}$;
4. the algebraic sum of oxidation states of all atoms in a neutral molecule must be zero, while in ions the algebraic sum of the oxidation states of the constituent atoms must be equal to the charge on the ion. For example, the oxidation states of sulfur in $\ce{H2S}$, $\ce{S8}$ (elementary sulfur), $\ce{SO2}$, $\ce{SO3}$, and $\ce{H2SO4}$ are, respectively: $-2$, $04, $+4$, $+6$ and $+6$. The higher the oxidation state of a given atom, the greater is its degree of oxidation; the lower the oxidation state, the greater is its degree of reduction.

What we see is, that the table in the original question actually reflects some of these rules. However, this set is not generic at all and it lacks a definition for compounds that do not contain oxygen or hydrogen, are elemental or monoatomic ions. With only these rules it is impossible do determine the oxidation states for $\ce{BF3}$ and many, if not most, compounds.

The lack of the definition is very well known, but it took the IUPAC until 2016 to actually change the official set. Hans-Peter Loock proposed a much simpler concept in Expanded Definition of the Oxidation State, which resembles the currently used definition quite well.

The oxidation state of an atom in a compound is given by the hypothetical charge of the corresponding atomic ion that is obtained by heterolytically cleaving its bonds such that the atom with the higher electronegativity in a bond is allocated all electrons in this bond. Bonds between like atoms (having the same formal charge) are cleaved homolytically.

So he basically came to the same conclusion as Dissenter in the question. Another statement from this article is

This is not a new definition, but it predates the IUPAC rules by several decades. For example, Linus Pauling provided a similar definition of the oxidation state in his 1947 edition of General Chemistry (3).

(3):Pauling, L. General Chemistry; Freeman: San Francisco, CA, 1947. Republished by Courier Dover Publications, 2012.

What Is the Lowest Oxidation Number of Mn

Source: https://chemistry.stackexchange.com/questions/10944/electronegativity-considerations-in-assigning-oxidation-states

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