I. Leibniz' Law: x = y, if, and only if, x has every property which y has, and y has every property which x has. Principia Mathematica defines the notion of equality as follows (in modern symbols); note that the generalization "for all" extends over predicate-functions f( ): Axiom 2. The law of crossing: The value of a (boundary) crossing made again is not the value of the crossing. Aristotle, "On Interpretation", Harold P. Cooke (trans.), pp.111–179 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

To supplement the four (down from five; see Post) axioms of the propositional calculus, Gödel 1930 adds the dictum de omni as the first of two additional axioms. Both this "dictum" and the second axiom, he claims in a footnote, derive from Principia Mathematica. Indeed, PM includes both as

Every idea starts with a distinction. Even the simplest thought involves drawing a boundary that distinguishes something from nothing or a thing from other things. Most of the time we communicate these ideas with words, yet words fail to communicate the hidden elements of our thoughts. This is true of all the terminology we use, but especially terms that are political or divisive, such as torture, terrorist, Muslim, conservative, liberal, us, or them. These words mean what we make them mean and include what we choose to include in their meaning. What these words mean is determined by what we load into them and what we don't. Induction principle: Russell devotes a chapter to his "induction principle". He describes it as coming in two parts: firstly, as a repeated collection of evidence (with no failures of association known) and therefore increasing probability that whenever A happens B follows; secondly, in a fresh instance when indeed A happens, B will indeed follow: i.e. "a sufficient number of cases of association will make the probability of a fresh association nearly a certainty, and will make it approach certainty without limit." [15]

Inference principle: Russell then offers an example that he calls a "logical" principle. Twice previously he has asserted this principle, first as the 4th axiom in his 1903 [20] and then as his first "primitive proposition" of PM: "❋1.1 Anything implied by a true elementary proposition is true". [21] Now he repeats it in his 1912 in a refined form: "Thus our principle states that if this implies that, and this is true, then that is true. In other words, 'anything implied by a true proposition is true', or 'whatever follows from a true proposition is true'. [22] This principle he places great stress upon, stating that "this principle is really involved – at least, concrete instances of it are involved – in all demonstrations". [4]All men (x) except Asiatics (y)" is represented by x − y. "All states (x) except monarchical states (y)" is represented by x − y

On the other hand, the knowledge of the laws of the mind does not require as its basis any extensive collection of observations. The general truth is seen in the particular instance, and it is not confirmed by the repetition of instances. ... we not only see in the particular example the general truth, but we see it also as a certain truth – a truth, our confidence in which will not continue to increase with increasing experience of its practical verification." (Boole 1854:4) Boole's signs and their laws [ edit ] The notion of a particular as opposed to a universal: To represent the notion of "some men", Boole writes the small letter "v" before the predicate-symbol "vx" some men. It seems to me that the doctrine of the laws of thought could be simplified if we were to set up only two, the law of excluded middle and that of sufficient reason. The former thus: "Every predicate can be either confirmed or denied of every subject." Here it is already contained in the "either, or" that both cannot occur simultaneously, and consequently just what is expressed by the laws of identity and contradiction. Thus these would be added as corollaries of that principle which really says that every two concept-spheres must be thought either as united or as separated, but never as both at once; and therefore, even although words are joined together which express the latter, these words assert a process of thought which cannot be carried out. The consciousness of this infeasibility is the feeling of contradiction. The second law of thought, the principle of sufficient reason, would affirm that the above attributing or refuting must be determined by something different from the judgment itself, which may be a (pure or empirical) perception, or merely another judgment. This other and different thing is then called the ground or reason of the judgment. So far as a judgment satisfies the first law of thought, it is thinkable; so far as it satisfies the second, it is true, or at least in the case in which the ground of a judgment is only another judgment it is logically or formally true. [9] Boole (1854): From his "laws of the mind" Boole derives Aristotle's "Law of contradiction" [ edit ] Theaetetus, by Plato". The University of Adelaide Library. November 10, 2012. Archived from the original on 16 January 2014 . Retrieved 14 January 2014. Aristotle, "The Categories", Harold P. Cooke (trans.), pp.1–109 in Aristotle, Vol. 1, Loeb Classical Library, William Heinemann, London, UK, 1938.By 1912 Russell in his "Problems" pays close attention to "induction" (inductive reasoning) as well as "deduction" (inference), both of which represent just two examples of "self-evident logical principles" that include the "Laws of Thought." [4] Hilbert 1927:467 adds only two axioms of equality, the first is x = x, the second is (x = y) → ((f(x) → f(y)); the "for all f" is missing (or implied). Gödel 1930 defines equality similarly to PM:❋13.01. Kleene 1967 adopts the two from Hilbert 1927 plus two more (Kleene 1967:387). x = y = def ∀f:(f(x) → f(y)) ("This definition states that x and y are to be called identical when every predicate function satisfied by x is satisfied by y" [47]