In Square One, Steve Patterson argues that the three laws of logic—the law of identity, the law of non-contradiction, and the law of the excluded middle—demonstrate that absolute, certain, and objective truths exist. He shows how all arguments against the absoluteness of these laws ultimately fail. Patterson believes that all apparent paradoxes or contradictions are simply linguistic errors. Given their nature, Patterson posits that the laws of logic should be the foundation of any accurate worldview.
Chapter One: Foundations
Chapter Summary: “Certain truths exist and they should be used as the foundation for any accurate worldview.”
The Roots of a Tree
Foundations determine the accuracy of a worldview.
The Foundation of a House
It is difficult to change the most fundamental ideas of a worldview.
Use skepticism to critically examine and revise your worldview.
Challenge and doubt every idea you encounter but be open to the possibility that certain truths might exist.
From Top to Bottom
Use “Why?” to get from premises to foundations, where “Why?” no longer applies.
Chapter Two: Implausible and Impossible
Chapter Summary: “The central argument for implausibility claims that certainty is beyond the reach of humans.”
Two Types of Argument
There are two kinds of arguments against certain truths: implausibility (humans can’t access certain truths) and impossibility (certain truth can’t be accessed in principle).
Insisting that all knowledge is uncertain is itself dogmatic.
No Land Across the Sea
People who argue that we will never know if certain truths exist have not ventured far enough.
Faith In Something
A worldview can be rational down to its foundations; faith is unnecessary.
The Evolved Brain
The implausibility of accessing certain truths because of the human brain’s limitation does not rule out the possibility that certain truths exist.
A Senseless Universe
If we can make sense of some parts of the universe, then the universe is comprehensible.
Language Is Imprecise
Communication might be imprecise, but the understanding of concepts can be precise.
We can have knowledge that is true regardless of sensory input.
Chapter Three: Logic and Existence
Chapter Summary: “The central argument for implausibility fails because the existence of certain truths can be demonstrated easily.”
To understand certain truths, it helps to first analyze the less-than-certain ideas.
Logic is the rules of existence: logic is inseparable from existence.
There are two kinds of existents: one dependent on our conception (ideas) and another independent of it (things).
Existents follow the law of identity: “A is A.”
Existents follow the law of non-contradiction: “It is not the case that A and not-A.”
Logic and existence are inseparable: existence cannot occur without identity and non-contradiction.
True or False
Existence is inseparable from logic, and logic is inseparable from truth and falsehood.
The law of non-contradiction preserves the meaning of “negation,” which in turn, creates “mutually exclusive” things.
The laws of logic are necessary, universal, and inescapable. Doubting them is proof that they are correct.
The Law of the Excluded Middle
Claims about things either correspond to reality or they do not. There is no third option.
Logic is the self-identity of everything even if things are different.
There are certain truths that can be known without logical necessity (ie, awareness).
Chapter Four: Implication and Application
Chapter Summary: “Tools are available to create logical structures in our minds (theories) that correspond with the logical structure of the world.”
Theories are constructed out of concepts, which are referenced with words within a language. Conceptual reasoning is unpacking the meaning and implication of our concepts to ensure that they are not contradictory.
Analyzing the implicit concepts bundled with an idea.
Theory Versus Data
Theoretical critiques are more powerful than empirical ones because a theory is inescapably prior to interpretation of data.
Deduction and Validity
Deduction is a technique for conceptual reasoning where you try to demonstrate that conclusions of an argument necessarily follow from the premises; those that do are “valid arguments” while those that don’t are “invalid arguments.”
Propositional logic breaks down the structure of complex arguments into their individual components and uses “rules of inference” to analyze the logical relationship between propositions. Sound arguments are those with true premises and are valid in structure.
Using axioms (self-evidently true statements) in deductive reasoning could create logically airtight theories about the world.
Certainty and the Mind
The mind is simultaneously able to state subjective truths and objective truths.
Logic can be used to construct systems (i.e. chess).
Poker in Your Mind
Rationalism, which ensures internal consistency, should be coupled with the real-world feedback advocated by empiricism to create external consistency.
Hierarchy of Knowledge
The existence of foundational truth implies the existence of a hierarchy of knowledge because some ideas are presupposed by others.
Mathematics is simply an extension of logic. Therefore, the conclusions are necessary.
Paradoxes and Puzzles
There are no actual paradoxes because the Universe could not contradict itself. Apparent paradoxes involve conceptual or linguistic errors that could be resolved through logical analysis.
Chapter Five: Objections and Paradoxes
Chapter Summary: “The central argument for impossibility, which claims that logic could not be universal because contradictions exist, is flawed because apparent contradictions are simply linguistic errors.”
The Liar’s Paradox
The liar’s paradox is not a meaningful proposition; it is a linguistic error.
The Bittersweet Paradox
The bittersweet paradox is inconsistent because the meanings of “opposite” and “mutually exclusive” that it adopts are imprecise.
The Logic of Nothing
Nothing is the concept of universal negation. Nothing is a concept without a referent, it is not an actual existent thing, therefore we shouldn’t talk about “nothing” as if it is “something.”
The argument that reality isn’t so binary and therefore logic couldn’t accurately represent it is flawed. A proposition cannot meaningfully be “half true” therefore something cannot be half-existent. Language could be ambiguous but the world cannot.
The argument that the law of identity does not apply to anything because everything is constantly changing is flawed. Constant change presupposes identity because before something could change it has to have an “identity” where it started. Even if a “thing” constantly changes, it is in a particular way at a particular time; this still corresponds to the law of identity.
Mystical experiences do not prove that real paradoxes exist. The apparent contradiction between reality and the mystic’s description of her experience is a reflection of the limitations of language and not of logic. Some knowledge is ineffable. Therefore, grant the importance of mystical experiences but recognize that language might not be able to communicate them precisely.
The Copenhagen interpretation of quantum mechanics, which is used to prove the existence of contradictions, is flawed and there are far more superior alternatives.
The Unknown World
The argument that we cannot know if the world operates in a fundamentally logical way because we haven’t experienced everything about it, is flawed. Even if all our conceptions about physics are wrong, the world would still be bound by the law of identity (it is what it is).
That’s Just Your Logic
The argument that people can have different logic is flawed because the laws of logic are not agreed to.
The argument that the laws of logic are mere tautologies—unnecessary propositions that do not teach us anything new about the world—is flawed. Tautologies are self-evidently true and therefore should be the foundations of any accurate worldview and any scientific exploration.
The Ultimate Resolution
Contradictions cannot exist in the world because of logic. Logic is the ultimate foundation of all forms of knowledge.