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The Principles

Leibniz variously invoked one or another of seven fundamental philosophical Principles (Mates 1986: chpts. 7.3, 9):

• Identity / Contradiction. If a proposition is true, then its negation is false and vice versa.
• Identity of indiscernibles. Two things are identical if and only if they share the same properties. Frequently invoked in modern logic and philosophy.
• Sufficient reason. "There must be a sufficient reason [often known only to God] for anything to exist, for any event to occur, for any truth to obtain." (LL 717).
• Pre-established harmony. See Jolley (1995: 129–31), Woolhouse and Francks (1998), and Mercer (2001). "[T]he appropriate nature of each substance brings it about that what happens to one corresponds to what happens to all the others, without, however, their acting upon one another directly." (Discourse on Metaphysics, XIV) A dropped glass shatters because it "knows" it has hit the ground, and not because the impact with the ground "compels" the glass to split.
• Continuity. Natura non saltum facit. A mathematical analog to this principle would go as follows. If a function describes a transformation of something to which continuity applies, then its domain and range are both dense sets.
• Optimism. "God assuredly always chooses the best." (LL 311).
• Plenitude. "Leibniz believed that the best of all possible worlds would actualize every genuine possibility, and argued in Théodicée that this best of all possible worlds will contain all possibilities, with our finite experience of eternity giving no reason to dispute nature's perfection." (From Plenitude.)

The second principle here is often referred to as Leibniz's Law [2]. The Identity of Indiscernibles has attracted the most controversy and criticism, especially from corpuscular philosophy and quantum mechanics.

Leibniz would on occasion give a speech for a specific principle, but more often took them for granted. For a precis of what Leibniz meant by these and other Principles, see Mercer (2001: 473–84). For a classic discussion of Sufficient Reason and Plenitude, see Lovejoy (1957).