The world is the totality of facts, not of things.
To begin: a distinction. A thing is a discrete object — a stone, a number, a name. A fact is a relation that obtains between things. The world we reason about is composed not of inert items but of states of affairs: this is the case; that is not.
Logic concerns itself with facts. It does not ask what exists; it asks what follows.
A proposition is a picture of reality.
To say something true is to construct a model that shares its form with the world. The proposition p is true when reality is arranged as p describes; otherwise p is false. Truth is correspondence; falsity is its absence.
Atomic propositions are the bricks of thought.
An atomic proposition asserts a single state of affairs and cannot be decomposed further. From atoms, compound propositions are built using the connectives ∧ (and), ∨ (or), ¬ (not), and → (implies).
A picture has the same logical multiplicity as what it pictures.
A diagram of a circuit must contain as many distinguishable parts as the circuit itself. So too with thought: any sentence whose structure does not mirror the structure of the world it claims to describe is, however eloquent, not yet a proposition.
Form is what cannot be said but only shown.
The relation between a picture and what it pictures is itself a fact — but it is not a fact within the picture. The grammar of clear thinking is exhibited; it is never spoken.
A thought is a logical picture of the facts.
To think is to model. The mind which reasons well is one whose internal structure tracks the structure of the world it inhabits. Where mind and world diverge in form, error arises — and the work of logic is to detect that divergence and repair it.
In a proposition a thought finds an expression that can be perceived by the senses.
A thought, kept private, is mute. Written or spoken, it becomes available for inspection — and so for refutation. Public reasoning is the discipline by which thought is forced to make itself accountable.
Therefore: write your premises down. The page is a more honest reasoner than the mind.
A proposition has one and only one complete analysis.
Every well-formed assertion can, in principle, be reduced to a unique combination of atomic propositions and logical connectives. This is not a claim about psychology — it is a constraint on what counts as a proposition at all. Ambiguity is not a richness of meaning but a failure of analysis.
Most questions and propositions of the philosophers arise from failure to understand the logic of our language.
When a question resists answer for centuries, suspect the question. The remedy for confusion is rarely a new theory; usually it is a clearer notation. Ask: what would count as evidence here? If nothing would, the dispute is verbal.
A proposition is a truth-function of elementary propositions.
Given the truth-values of the atoms, the truth-values of every compound are determined. Logic does not discover new facts; it makes explicit what was already entailed. Validity is conservation: nothing is in the conclusion that was not, in some form, in the premises.
The limits of my language mean the limits of my world.
What we can say clearly, we can think clearly. What we cannot articulate, we cannot examine; and what we cannot examine, we cannot trust ourselves to know. To enlarge the world available to thought, enlarge the language available to expression. Read the careful writers. Translate. Define your terms.
This is the daily practice: take one belief you hold and try to write it as a proposition. If you cannot, you do not yet hold it — you only feel it.
Whereof one cannot speak, thereof one must be silent.
Logic ends where it ends. Beyond the boundary of what can be stated lies a region not of falsity but of formlessness. The honest reasoner does not pretend to map it. The honest reasoner stops, marks the edge, and waits.