Vector logic

Key takeaways

Vector logic belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Vector logic to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Vector logic from memory before moving on to harder problems.

Reference excerpt

Vector logic is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the monadic and dyadic operations are executed by matrix operators. "Vector logic" has also been used to refer to the representation of classical propositional logic as a vector space, in which the unit vectors are propositional variables. Predicate logic can be represented as a vector space of the same type in which the axes represent the predicate letters S {\displaystyle S} and P {\displaystyle P} . In the vector space for propositional logic the origin represents the false, F, and the infinite periphery represents the true, T, whereas in the space for predicate logic the origin represents "nothing" and the periphery represents the flight from nothing, or "something".

Overview Classic binary logic is represented by a small set of mathematical functions depending on one (monadic) or two (dyadic) variables. In the binary set, the value 1 corresponds to true and the value 0 to false. A two-valued vector logic requires a correspondence between the truth-values true (t) and false (f), and two q-dimensional normalized real-valued column vectors s and n, hence:

t ↦ s {\displaystyle t\mapsto s} and f ↦ n {\displaystyle f\mapsto n}

(where q ≥ 2 {\displaystyle q\geq 2} is an arbitrary natural number, and "normalized" means that the length of the vector is 1; usually s and n are orthogonal vectors). This correspondence generates a space of vector truth-values: V2 = {s,n}. The basic logical operations defined using this set of vectors lead to matrix operators. The operations of vector logic are based on the scalar product between q-dimensional column vectors: u T v = ⟨ u , v ⟩ {\displaystyle u^{T}v=\langle u,v\rangle } : the orthonormality between vectors s and n implies that ⟨ u , v ⟩ = 1 {\displaystyle \langle u,v\rangle =1} if u = v {\displaystyle u=v} , and ⟨ u , v ⟩ = 0 {\displaystyle \langle u,v\rangle =0} if u ≠ v {\displaystyle u\neq v} , where u , v ∈ { s , n } {\displaystyle u,v\in \{s,n\}} .

Monadic operators The monadic operators result from the application M o n : V 2 → V 2 {\displaystyle Mon:V_{2}\to V_{2}} , and the associated matrices have q rows and q columns. The two basic monadic operators for this two-valued vector logic are the identity and the negation:

Identity: A logical identity ID(p) is represented by matrix I = s s T + n n T {\displaystyle I=ss^{T}+nn^{T}} . This matrix operates as follows: Ip = p, p ∈ V2; due to the orthogonality of s with respect to n, we have I s = s s T s + n n T s = s ⟨ s , s ⟩ + n ⟨ n , s ⟩ = s {\displaystyle Is=ss^{T}s+nn^{T}s=s\langle s,s\rangle +n\langle n,s\rangle =s} , and similarly I n = n {\displaystyle In=n} . This vector logic identity matrix is not generally an identity matrix in the sense of matrix algebra. Negation: A logical negation ¬p is represented by matrix N = n s T + s n T {\displaystyle N=ns^{T}+sn^{T}} Consequently, Ns = n and Nn = s. The involutory behavior of the logical negation, namely that ¬(¬p) equals p, corresponds with the fact that N2 = I.

Dyadic operators The 16 two-valued dyadic operators correspond to functions of the type D y a d : V 2 ⊗ V 2 → V 2 {\displaystyle Dyad:V_{2}\otimes V_{2}\to V_{2}} ; the dyadic matrices have q2 rows and q columns. The matrices that execute these dyadic operations are based on the properties of the Kronecker product. Two properties of this product are essential for the formalism of vector logic:

Using these properties, expressions for dyadic logic functions can be obtained:

Conjunction. The conjunction (p∧q) is executed by a matrix that acts on two vector truth-values: C ( u ⊗ v ) {\displaystyle C(u\otimes v)} .This matrix reproduces the features of the classical conjunction truth-table in its formulation:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Vector logic

Start with the simplest possible case. Write down what Vector logic claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Vector logic before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Vector logic ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Vector logic

In research: Vector logic appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Vector logic in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Vector logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Boolean algebra, Logic, so understanding it makes those chapters shorter.
In everyday life: Look for Vector logic outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Vector logic in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Vector logic means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Vector logic out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Vector logic in simple terms?

Vector logic is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the monadic and dyadic operations are executed by matrix operators. "Vector logic" has also been used to refer to the representation of classical propo…

Why does Vector logic matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Vector logic?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Vector logic.