Modern logicians, in almost unanimous disagreement with Boole, take this expression to be a misnomer; none of the above propositions classed under "laws of thought" are explicitly about thought per se, a mental phenomenon studied by psychology, nor do they involve explicit reference to a thinker or knower as would be the case in pragmatics or in epistemology.
But this permission is limited by two indispensable conditions, first, that from the sense once conventionally established we never, in the same process of reasoning, depart; secondly, that the laws by which the process is conducted be founded exclusively upon the above fixed sense or meaning of the symbols employed.The law of non-contradiction (alternately the 'law of contradiction'[4]): 'Nothing can both be and not be.
Regarding the law of excluded middle, Aristotle wrote: But on the other hand there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate.
In one of Plato's Socratic dialogues, Socrates described three principles derived from introspection: First, that nothing can become greater or less, either in number or magnitude, while remaining equal to itself ... Secondly, that without addition or subtraction there is no increase or diminution of anything, but only equality ... Thirdly, that what was not before cannot be afterwards, without becoming and having become.The law of non-contradiction is found in ancient Indian logic as a meta-rule in the Shrauta Sutras, the grammar of Pāṇini,[6] and the Brahma Sutras attributed to Vyasa.
(Thomas Hughes, The Ideal Theory of Berkeley and the Real World, Part II, Section XV, Footnote, p. 38) Arthur Schopenhauer discussed the laws of thought and tried to demonstrate that they are the basis of reason.
He listed them in the following way in his On the Fourfold Root of the Principle of Sufficient Reason, §33: Also: The laws of thought can be most intelligibly expressed thus:
There would then have to be added only the fact that once for all in logic the question is about what is thought and hence about concepts and not about real things.To show that they are the foundation of reason, he gave the following explanation: Through a reflection, which I might call a self-examination of the faculty of reason, we know that these judgments are the expression of the conditions of all thought and therefore have these as their ground.
Thus by making vain attempts to think in opposition to these laws, the faculty of reason recognizes them as the conditions of the possibility of all thought.
[9]The title of George Boole's 1854 treatise on logic, An Investigation on the Laws of Thought, indicates an alternate path.
So we have an example of the "Law of Contradiction": This notion is found throughout Boole's "Laws of Thought" e.g. 1854:28, where the symbol "1" (the integer 1) is used to represent "Universe" and "0" to represent "Nothing", and in far more detail later (pages 42ff): In his chapter "The Predicate Calculus" Kleene observes that the specification of the "domain" of discourse is "not a trivial assumption, since it is not always clearly satisfied in ordinary discourse ... in mathematics likewise, logic can become pretty slippery when no D [domain] has been specified explicitly or implicitly, or the specification of a D [domain] is too vague (Kleene 1967:84).
They are necessary, for no one ever does, or can, conceive them reversed, or really violate them, because no one ever accepts a contradiction which presents itself to his mind as such.The sequel to Bertrand Russell's 1903 "The Principles of Mathematics" became the three-volume work named Principia Mathematica (hereafter PM), written jointly with Alfred North Whitehead.
He asserts that "Symbolic Logic is essentially concerned with inference in general" (Russell 1903:12) and with a footnote indicates that he does not distinguish between inference and deduction; moreover he considers induction "to be either disguised deduction or a mere method of making plausible guesses" (Russell 1903:11).
In his Part I "The Indefinables of Mathematics" Chapter II "Symbolic Logic" Part A "The Propositional Calculus" Russell reduces deduction ("propositional calculus") to 2 "indefinables" and 10 axioms: From these he claims to be able to derive the law of excluded middle and the law of contradiction but does not exhibit his derivations (Russell 1903:17).
Subsequently, he and Whitehead honed these "primitive principles" and axioms into the nine found in PM, and here Russell actually exhibits these two derivations at ❋1.71 and ❋3.24, respectively.
"Thus we must either accept the inductive principle on the ground of its intrinsic evidence, or forgo all justification of our expectations about the future".
Of these various "laws" he asserts that "for no very good reason, three of these principles have been singled out by tradition under the name of 'Laws of Thought'.
"[25] But he rates this a "large question" and expands it in two following chapters where he begins with an investigation of the notion of "a priori" (innate, built-in) knowledge, and ultimately arrives at his acceptance of the Platonic "world of universals".
For Russell the matter of "self-evident"[27] merely introduces the larger question of how we derive our knowledge of the world.
He cites the "historic controversy ... between the two schools called respectively 'empiricists' [ Locke, Berkeley, and Hume ] and 'rationalists' [ Descartes and Leibniz]" (these philosophers are his examples).
[30] This question of how such a priori knowledge can exist directs Russell to an investigation into the philosophy of Immanuel Kant, which after careful consideration he rejects as follows: His objections to Kant then leads Russell to accept the 'theory of ideas' of Plato, "in my opinion ... one of the most successful attempts hitherto made.
[32] Unfortunately, Russell's "Problems" does not offer an example of a "minimum set" of principles that would apply to human reasoning, both inductive and deductive.
But PM does at least provide an example set (but not the minimum; see Post below) that is sufficient for deductive reasoning by means of the propositional calculus (as opposed to reasoning by means of the more-complicated predicate calculus)—a total of 8 principles at the start of "Part I: Mathematical Logic".
At about the same time (1912) that Russell and Whitehead were finishing the last volume of their Principia Mathematica, and the publishing of Russell's "The Problems of Philosophy" at least two logicians (Louis Couturat, Christine Ladd-Franklin) were asserting that two "laws" (principles) of contradiction" and "excluded middle" are necessary to specify "contradictories"; Ladd-Franklin renamed these the principles of exclusion and exhaustion.
The definition of "consistent" is this: that by means of the deductive "system" at hand (its stated axioms, laws, rules) it is impossible to derive (display) both a formula S and its contradictory ~S (i.e. its logical negation) (Nagel and Newman 1958:50).
[36] Given PM's tiny set of "primitive propositions" and the proof of their consistency, Post then proves that this system ("propositional calculus" of PM) is complete, meaning every possible truth table can be generated in the "system": Then there is the matter of "independence" of the axioms.
In his commentary before Post 1921, van Heijenoort states that Paul Bernays solved the matter in 1918 (but published in 1926) – the formula ❋1.5 Associative Principle: p ⋁ (q ⋁ r) ⊃ q ⋁ (p ⋁ r) can be proved with the other four.
In his introduction to Post 1921, van Heijenoort observes that both the "truth-table and the axiomatic approaches are clearly presented".
The "dictum" appears in Boole 1854 a couple places: But later he seems to argue against it:[42] But the first half of this "dictum" (dictum de omni) is taken up by Russell and Whitehead in PM, and by Hilbert in his version (1927) of the "first order predicate logic"; his (system) includes a principle that Hilbert calls "Aristotle's dictum" [43] This axiom also appears in the modern axiom set offered by Kleene (Kleene 1967:387), as his "∀-schema", one of two axioms (he calls them "postulates") required for the predicate calculus; the other being the "∃-schema" f(y) ⊃ ∃xf(x) that reasons from the particular f(y) to the existence of at least one subject x that satisfies the predicate f(x); both of these requires adherence to a defined domain (universe) of discourse.
This extends the domain (universe) of discourse and the types of functions to numbers and mathematical formulas (Kleene 1967:148ff, Tarski 1946:54ff).
George Spencer-Brown in his 1969 "Laws of Form" (LoF) begins by first taking as given that "we cannot make an indication without drawing a distinction".