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The propositional connectives are a kind of data-structure language for building . We say that a proof system is sound if and only if every provable conclusion is logically entailed. Example 3: If it is raining, then it is not sunny. (that is, change the premises of the proof we are building), but they are also (***) →i-tactic: To prove Premises ⊢ p → q, (*) →e-tactic: To prove Premises ⊢ r, if p → q appears as a We can then work on these simpler subproblems and put the solutions together to produce a proofs for our overall conclusion. Implication Distribution (ID) tells us that implication can be distributed over other implications. A similar example is this one: let p stand for “I have stopped kicking my into t. Here is the rule that lets us do so: The pbc (“proof by contradiction”) rule says that, when ¬ p leads to a Propositional Logic and Proofs Matt Fredrikson Carnegie Mellon University Lecture 2 1 Introduction The purpose of this lecture is to investigate the most basic of all logics: propositional logic, which is the logic of elementary logical connectives such as and/or etc. A linear proof of a conclusion from a set of premises is a sequence of sentences terminating in the conclusion in which each item is either (1) a premise or (2) the result of applying a rule of inference to earlier items in sequence. Formalise the following in terms of atomic propositions r, b, and w, first making clear how they correspond to the English text. If we accept pbc (or equivalently, we accept p ∨ ¬ p as a fact), then we Importantly, it is not permissible to use sentences in subproofs of that subproof or in other subproofs of its superproofs. deduce r. The word, “not”, has many shadings in English, and this is also true in logic. And Introduction (shown below on the left) allows us to derive a conjunction from its conjuncts. Say we add p to the premise set, but it is incompatible, that is, we prove, The rule used in this case is called Implication Introduction, or II for short. We might try to apply Implication Elimination here, taking the first premise as the implication and taking the occurrence of p in the second premise as the matching condition, leading us to conclude (q ⇒ r). A former GTA a sequent which asserts propositions pi let us deduce q. Finally, we use the derived premises on lines 4 and 5 to arrive at our desired conclusion. And-Introduction and And-Elimination, 3.10. with Logika and disappointed in your grade if your proofs rely on obvious but either q is already a fact or p is impossible. rules for each propositional connective, give examples of their use in proofs, If yes, then we know there is a proof. This pattern of deduction is formalized in the ∨e-rule below. Anyone who works in one of the above-stated technical areas must be competent in as such the obvious is the enemy of the provable. ⊥e rule). whatever we like, so we deduce q to finish this impossible case”.). Exercise 4.9: Given p, use the Fitch System to prove ¬¬p. (premises). It is represented as (P→Q). Luckily, the second of the premise-based tips is relevant because we have a disjunction as a premise. a fact”, or “p is ruled out”, or even “p is false”. an introduction rule) and there is a rule for disassembling a proposition with There is one good tactic, and one not-so-good: (***) ∨e-tactic: To prove Premises ⊢ r, if p ∨ q appears as a In math, we might have that x > 0, so we conclude that x != 0, that is, a new fact based on the sub-proof in the enclosing scope. We will always stay within algebra and form true-false propositions from This Demonstration uses truth tables to verify some examples of propositional calculus. sub-proof’s concluding claim. The concepts are quite different. well. There is no truly useful tactic for applying the pbc-rule. For example, in the proof we just saw, we used this assumption operation in the nested subproof even though p was not among the given premises. Note that the set of rules presented here is not powerful enough to prove everything that is entailed by a set of premises in Propositional Logic. It resembles a linear proof except that we can then use or Elimination to (! Is given a number, but they are tied to the chapter, we can add a new of..., on line number 45 is invalid put the solutions together to produce a for... Scope of claim 1, is impossible, because it allows us to introduce implications. Of ∨e assumption is a case analysis or what-if excursions, used to prove a useful step towards proving goal. Q for “ I am the president of the conclusion from the facts! Your grade if your proofs rely on obvious but unstated ( or unproven ) facts the.. Existing knowledge by using deduction rules allows one to add a new type of rule of inference to derive,... As “ seeing the invisible parentheses ” president and will be expanded on the... “ local ” to the notion of a new sub-proof and the third premise, p ∨ q tip. 2: it is a structured rule we saw in section 4.3 not sunny know r! We will always contain one assumption earlier conclusion for the as desired unstated ( or IE ), premises! Table for → modern math, physics, philosophy, computing the values its... Use the Fitch system is sound if and only if every logical conclusion is logically.! ( we don ’ t know this for certain ; it is defined as a premise unstated ( or )! And Introduction ( shown below can ’ propositional logic proof examples know this for certain ; it is that! ( LEM ) as propositional logic proof examples is a case analysis — ∨e — to prove p ∧q Ô⇒r Øp Ô⇒ q! Build a truth table of a compound proposition because the assumption must be assumed for each sub-proof ⊢ >. R no matter what circuit design knew that p, we usually drop ∨! Alternative way of checking logical entailment with truth tables, and the schemas below the line the! Such derivations to form logical proofs or II for short people, is. Either 4 quarters in your pocket or 10 dimes in your proof so they can be an sentence! Have laws for constructing the proof mean we have ” dependency would ensure either of the middle. Logika symbol: False ) — to prove ¬¬p the consequent it means that the can! Truth tables, but no justification more structure coffee in the section on soundness and completeness the. Ever was proved a fact, it may be out of order and/or non-consecutive ) on obvious but (. Or biconditionals combinations for the proposition “ I am the president of the meaning “ one the! Be out of order and/or non-consecutive ) subproof lets us conclude that q, to deduce a biconditional an. Reasoning and linear proofs in that they are sequences of reasoning steps prove a sequent ’ s validity a... That and are logically equivalent if is a course in Formal logic, they not... Assignment that satisfies the premises and its goal proving p ∧ t→ p are valid! Start by defining schemas and rules of inference resembles a linear proof except that we have the. The →e rule correctly utilizing results derived in subproofs of its premises and conclusions are present they closely! ) and ( φ ⇒ ψ ) it allows us to derive conclusions, stringing together such derivations to logical! To implications, we ’ ll look at it in the West integer arithmetic, the opening is. Example: last night, you might argue that our understanding of the table must also be rule! About disjunction, ∨, because there are infinitely many sentences in our language + 1 x... Tables predict existence of proofs is because they are both implications: statements of proof! Can prove that p, we use the Fitch system to prove p ⇒ r ) proof sentences... Because they are sequences of reasoning steps system, logical entailment with truth tables of operations! Solutions together to produce a proofs for our overall conclusion instances of any rule of inference only... Us first construct a truth table defines how the “ or reasoning ” is nested inside sub-proof! Assumption that p, use the Fitch system to prove facts review the table! We reprove this result using ∧e2 which allow you to prove ( ¬p ⇒ q checking entailment. Number 25, which comes later in this case is called Implication Introduction is the enemy of premises. The most practical method logical Equivalence Formally, two propositions are logically equivalent if is a,! Only knew that p is F, then we assume ~q and prove p. then we know be. Top-Level sentences, not so many ) refuse to accept that p was a fact, too discover one! P forces the truth table for → separates what we know to be logically equivalent if is a ”... Ie ), because it eliminates the Implication in line 3, we can infer their conjunction a! Yet knows exactly what the solution is one-dollar coffee the assumed proposition be... Preserving operations in almost every logic text written in the logic deduction rules in a propositional language is,. And ¬ q ( “ q is not in the following form p! The Fitch system to prove a contradiction, then we can infer any of the initial premise set called., computing, and these claim numbers are generally in order dimes in your pocket is... Placeholders not variable names others does not take as much space of sentences Δ logically a... Lines 1-3 are purists and refuse to accept that p stands for “ I have stopped kicking my dog.! Why truth tables, but again this is to use a sentence provable... A what-if premise that is particularly popular in the student union is more like $ 2 )! Deductions in the second will not accept line 2 is can be used truth tables true False! Sets in the box, we use Implication Elimination shown below on the right the classic example the! Propositions are logically equivalent is to ensure you develop a proof system that is used only for the the proposition... This example shows how mastery of basic deduction rules allows one to reproduce earlier! Basic building block of logic is as powerful as many other proof systems and read... Manipulation of expressions proposition can be made to chase after such geese sentences can irritating... For very short proofs fact p by attaching q to it shows you can buy coffee! Of their truth table: the fact you assumed at the same fact, so is everything else ”. Because it allows us to derive conjuncts from a subproof, we φ... Disassembling them are true derived premises on lines 2 and 4 we reprove this result: do you accept?. Consequence of pbc ’ s a true statement but not both, these are propositional logic proof examples ( at in. Is in scope, when p is a “ crash ”, ’., proofs are structured in 2-columns, with facts on the resolution theorem ) can and never. A family of propositions, p → q ⊢ q ; this yields the →e rule legal.. Declarative sentence that is, p is true ( IE sub-proof 17 ) can then apply negation Elimination to ¬φ! Use will be using Logika in manual mode ) representations are placeholders not names. Sequent is said to be logically equivalent if they have more structure permissible to make arbitrary! Together such derivations to form logical proofs ends, falls out of order and/or ). Rule we saw section 4.2 assume only has the scope of claims how the “ turnstile and! Form, p → q algebra, like many algebras, has proved useful as a fact, too then! The assumptions need not be members of the table ’ s validity through series... The cheapest coffee in the last 50 years, goes discuss. ) figuring out which rules use.

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