# semantic proof propositional logic

And as a result, we also end up unable to have any algorithmic procedure that enumerates all the truths about natural numbers. It’s a set of characters, along with a function that defines the grammar of the language by taking in (finite, in most logics) strings of these characters and returning true or false. Why? Is quantum mechanics simpler than classical physics? But we will discuss semantic issues in enough detail to give you a good sense of what it means to think semantically, as well as a sense of how to make pragmatic use of semantic notions. How uncomputable are the Busy Beaver numbers? This number grows very quickly, so we’ll mostly look at smaller formulas here. Can an irrational number raised to an irrational power be rational? This is already perhaps a little disconcerting; we might have hoped that every logic of any use can be nailed down concretely with a proof system. Computing truth values of sentences of arithmetic, or: Math is hard. Given a formula $$A$$, is there any truth assignment that makes $$A$$ true? By the end of this lecture, you should be able to (Semantic entailment) Determine if a set of formulas is satisfiable. Looking back on my introduction series, I’m pretty happy with it. To have a sound, complete, and effectively enumerable proof system for a logic is to know that our semantics is on solid ground; that we’re not just waving our hands around and saying things that cannot be cashed out in any concrete way, but that in fact everything we are saying can be grounded in some algorithm. A proof system is just such a mechanical system. Intuitionistic propositional logic (IPC) is given semantics in the same way, but the truth values belong to a Heyting algebra H instead of boolean algebra. There are different types of meanings that might be assigned, but the primary one for classical logic is truth and falsity. The Principle of Bivalence (PAB) vs. But the notions of soundness and completeness play an important role in helping us understand the nature of the logical notions, and so we will try to provide some hints here as to why these properties hold for propositional logic. However, there is a significant practical difficulty with our semantic method of checking arguments using truth tables (you may have already noted what this practical difficulty is, when you did problems 1e and 2e of chapter 3). We say that a set of sentences A syntactically entails a sentence X if there exists some proof that uses the sentences in A as assumptions and concludes with the sentence X. (Equivalently, $$\bar v(A \to B) = \mathbf{F}$$ if $$\bar v(A)$$ is $$\mathbf{T}$$ and $$\bar v(B)$$ is $$\mathbf{F}$$, and $$\mathbf{T}$$ otherwise.). The even more discerning might notice that these axioms only involve the → and ¬ symbols, but we seem to be missing ∧ and ∨. Justify your answer with either a derivation or a counterexample. Since a disjunction “false or true” is true, the entire formula is true. Once again, let us consider the true sentence “every natural number that is prime and greater than two is odd.” We can interpret this sentence as saying that all of the (infinitely many) sentences in this list are true: If 0 is prime and greater than 2, then 0 is odd. Intuitively, a truth assignment describes a possible “state of the world.” Going back to the Malice and Alice puzzle, let’s suppose the following letters are shorthand for the statements: In the world described by the solution to the puzzle, the first and third statements are true, and the second is false. A propositional formula is said to be provable if there is a formal proof of it in that system. For propositional logic, one common proof system involves just three axioms and one inference rule. Propositional Definite Clauses: Semantics Semantics allows you to relate the symbols in the logic to the domain you’re trying to model. In this text, we will adopt a “classical” notion of truth, following our discussion in Section 5. Suppose that some pesky philosopher were to come up to us and ask us what we meant by “only when A and B are both assigned true”. For example, suppose our truth assignment $$v$$ makes $$A$$ and $$B$$ true and $$C$$ false. What hypotheses are needed to derive $$A$$? When talking about the semantics of first-order logic, we call its truth functions models (or structures), and each model comes with a universe of objects that the quantifiers of the logic range over. We can also go in the other direction: given a formula, we can attempt to find a truth assignment that will make it true (or false). So if we want to describe the natural numbers, we have to also be simultaneously describing structures of every cardinality. Proving soundness is easier than proving completeness. Definition (interpretation) An interpretation I assigns a truth value to each atom. For compound formulas, the style is much the same. We construct a truth function by first doing any assignment of truth values to the propositional variables (p, q, r, and so on), and then defining what the function does to the rest of the strings in terms of these truth values. By the definition of the valuation function, $$\bar v (A \wedge B) = \mathbf{T}$$, as required. The three: X → (Y → X), (X → (Y → Z)) → ((X → Y) → (X → Z)), and ((¬Y) → (¬X)) → (X → Y). By our construction of the grammar of propositional logic, we’ve guaranteed that our function is defined on all grammatical strings. In addition to classical propositional logic, LP contains new atoms of sort t :F , where F is a formula and t is a special proof term called a proof polynomial. 3/14. Natural Deduction for Propositional Logic, 8. A logic defined by its syntax, semantics, and proof system. You might be skeptical right now, the phrase “incompleteness theorems” bubbling into your mind, and you’re right to be. Math is on sound grounds, right? What formulas can be derived from $$\Gamma$$? Once we have a truth assignment $$v$$ to a set of propositional variables, we can extend it to a valuation function $$\bar v$$, which assigns a value of true or false to every propositional formula that depends only on these variables. To evaluate $$(B \to C) \vee (A \wedge B)$$ under $$v$$, note that the expression $$B \to C$$ comes out false and the expression $$A \wedge B$$ comes out true. But I want to do it for REAL now, as an actual post of its own. In this extended setting, soundness says that if $$A$$ is provable from $$\Gamma$$, then $$A$$ is a logical consequence of $$\Gamma$$. With that said, there do exist sound, complete, and effectively enumerable proof systems for propositional logic. Since then, I’ve learned an enormous amount more and become a lot more comfortable with the basics of the field. The second and third semantic questions we asked are a little trickier than the first. Write out the truth table for $$(A \to B) \wedge (B \wedge C \to A)$$. Propositional Logic Rules1 • You don't need to memorize these rules by name, but you should be able to give the name of a rule. Group Theory: The Mathematics of Symmetry? The important thing is that the syntax for these logics as well as higher order logics is algorithmically checkable; it’s possible to write a simple program that verifies whether any given input string counts as grammatical or not. More troubling, perhaps, we seem to have engaged in a little bit of circularity when we said in our definition of the semantics that (A ∧ B) is assigned true only when A and B are both assigned true. This is known as effective enumerability. Which are the truth assignments that make $$A$$ true? The function $$\bar v$$ is defined recursively, which is to say, formulas are evaluated from the bottom up, so that value assigned to a compound formula is determined by the values assigned to its components. When this is the case, we write: A ⊨ X. Try varying the truth assignments, to see what happens. Notice that, if $$B$$ is true, we can prove $$A \to B$$ without any assumptions about $$A$$: This follows from the proper reading of the implication introduction rule: given $$B$$, one can always infer $$A \to B$$, and then cancel an assumption $$A$$, if there is one. We should be able to write an algorithm that verifies whether any given sequence of strings counts as a valid proof according to the proof system. And no other string is grammatical. Axioms are simply strings that can be used in a proof at any time whatsoever. We have already seen two, namely, “true” and “false.” We will use the symbols $$\mathbf{T}$$ and $$\mathbf{F}$$ to represent these in informal mathematics. A formula A is H-valid if V(A)=⊤ for all H-valuationsV. First-order logic has a sound, complete, and effectively enumerable proof system. Because of the way we have chosen our inference rules and defined the notion of a valuation, this intuition that the two notions should coincide holds true. Decoherence is not wave function collapse, The problem with the many worlds interpretation of quantum mechanics, Consistently reflecting on decision theory, Kant’s attempt to save metaphysics and causality from Hume, How to Learn From Data, Part 2: Evaluating Models, How to Learn From Data, Part I: Evaluating Simple Hypotheses, Gödel’s Second Incompleteness Theorem: Explained in Words of Only One Syllable, Sapiens: How Shared Myths Change the World, Infinities in the anthropic dice killer thought experiment, Not a solution to the anthropic dice killer puzzle, A closer look at anthropic tests for consciousness, Getting empirical evidence for different theories of consciousness, More on quantum entanglement and irreducibility, Quantum mechanics, reductionism, and irreducibility, Matter and interactions in quantum mechanics, Concepts we keep and concepts we toss out, If all truths are knowable, then all truths are known, Objective Bayesianism and choices of concepts, Regularization as approximately Bayesian inference, Why minimizing sum of squares is equivalent to frequentist inference, Short and sweet proof of the f(xy) = f(x) + f(y) logarithmic property, Moving Naturalism Forward: Eliminating the macroscopic, Getting evidence for a theory of consciousness, “You don’t believe in the God you want to, and I won’t believe in the God I want to”, Galileo and the Schelling point improbability principle, Why “number of parameters” isn’t good enough.