Propositional Logic (5A) Young W. Lim
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Based on Contemporary Artificial Intelligence, R.E. Neapolitan & X. Jiang Logic and Its Applications, Burkey & Foxley 3
An argument consists of a set of propositions : The premises propositions The conclusion proposition List of premises followed by the conclusion A 1 propositions A 2 A n -------- B premises conclusion 4
Entail The premises is said to entail the conclusion If in every model in which all the premises are true, the conclusion is also true List of premises followed by the conclusion A 1 A 2 A n -------- B whenever all the premises are true the conclusion must be true for the entailment 5
A Model A model or a possible world: Every atomic proposition is assigned a value T or F The set of all these assignments constitutes A model or a possible world All possible worlds (assignments) are permissiable models A B A B A B A T T T T T F F T F T F T F F F T T T T F F T F F T T T F T T T F F T T T T F F T F F Every atomic proposition : A, B 6
Interpretation An interpretation of a formal system is the assignment of meanings to the symbols, and truth values to the sentences of a formal system. The study of interpretations is called formal semantics Giving an interpretation is synonymous with constructing a model. An interpretation is expressed in a metalanguage, which may itself be a formal language, and as such itself is a syntactic entity. https://en.wikipedia.org/wiki/syntax_(logic)#syntactic_consequence_within_a_formal_system 7
Entailment Notation Suppose we have an argument whose premises are A 1, A 2,, A n whose conclusion is B Then A 1, A 2,, A n B if and only if A 1 A 2 A n B (logical implication) logical implication: if A 1 A 2 A n B is tautology (always true) The premises is said to entail the conclusion If in every model in which all the premises are true, the conclusion is also true 8
Entailment and Logical Implication A 1, A 2,, A n B A 1 A 2 A n B A 1 A 2 A n B is a tautology (logical implication) If all the premises are true, then the conclusion must be true T T T T T T T F F X X T 9
Sound Argument and Fallacy A sound argument A 1, A 2,, A n B If the premises entails the conclusion A fallacy A 1, A 2,, A n B If the premises does not entail the conclusion 10
Modal Accounts Modal accounts of logical consequence are variations on the following basic idea: Γ A is true if and only if it is necessary that if all of the elements of Γ are true, then A is true. Alternatively : Γ A is true if and only if it is impossible for all of the elements of Γ to be true and A false. Such accounts are called "modal" because they appeal to the modal notions of logical necessity and logical possibility. 'It is necessary that' is often expressed as a universal quantifier over possible worlds, so that the accounts above translate as: Γ A is true if and only if there is no possible world at which all of the elements of Γ are true and A is false (untrue). https://en.wikipedia.org/wiki/syntax_(logic)#syntactic_consequence_within_a_formal_system 11
Validity and Soundness (1) An argument form is valid if and only if whenever the premises are all true, then conclusion is true. An argument is valid if its argument form is valid. If If premises : true false false true then conclusion : true true false then never false An argument is sound if and only if it is valid and all its premises are true. Always premises : true therefore conclusion : false http://math.stackexchange.com/questions/281208/what-is-the-difference-between-a-sound-argument-and-a-valid-argument 12
Validity and Soundness (2) A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. If true then never false Otherwise, a deductive argument is said to be invalid. for the premises to be true and the conclusion is false. A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound. Always premises : true therefore conclusion : false http://www.iep.utm.edu/val-snd/ 13
Validity and Soundness (3) A B A B A (A B) A (A B) B T T T T T T F F F T F T T F T F F T F T valid If premises : true then never conclusion : false A B A B A (A B) A (A B) B T T T T T T F F F T F T T F T F F T F T sound Always premises : true therefore conclusion : true http://www.iep.utm.edu/val-snd/ 14
Validity and Soundness (4) the author of a deductive argument always intends that the premises provide the sort of justification for the conclusion whereby if the premises are true, the conclusion is guaranteed to be true as well. if the author's process of reasoning is a good one, if the premises actually do provide this sort of justification for the conclusion, then the argument is valid. an argument is valid if the truth of the premises logically guarantees the truth of the conclusion. it is impossible for the premises to be true and the conclusion to nevertheless be false: http://www.iep.utm.edu/val-snd/ 15
Entailment Examples A, B A A, B A A B A A, (A B) B A, (A B) B A (A B) B 16
Entailment Examples and Truth Tables A B A B A B A T T T T T F F T F T F T F F F T A B A The premises is said to entail the conclusion If in every model in which all the premises are true, the conclusion is also true any of the premises are false, still premises conclusion is true (F T and F F always T) Tautology A B A B A (A B) A (A B) B T T T T T T F F F T F T T F T F F T F T A (A B) B 17
Deduction System Propositional logic Given propositions (statements) : T or F Deductive inference of T or F of other propositions Deductive Inference A process by which the truth of the conclusion is shown to necessarily follow from the truth of the premises A B A B A (A B) A (A B) B T T T T T T F F F T F T T F T F F T F T A (A B) B Deductive Inference Entailment (logical implication) 18
Deduction System Deduction System : a set of inference rules Inference rules are used to reason deductively Sound Deduction System : if it derives only sound arguments Each of the inference rules is sound A sound argument: If the premises entails the conclusion A fallacy: If the premises does not entail the conclusion Complete Deduction System : It can drive every sound argument Must contain deduction theorem rule 19
Inference Rules Combination Rule A, B A B Simplification Rule A B A Addition Rule A A B Modus Pones A, A B B Modus Tolens B, A B A Hypothetical Syllogism A B, B C A C Disjunctive Syllogism A B, A B Rule of Cases A B, A B B Equivalence Elimination A B A B Equivalence Introduction A B, B A A B Inconsistency Rule A, A B AND Commutivity Rule A B B A OR Commutivity Rule A B B A Deduction Theorem If A 1, A 2,, A n,b C then A 1, A 2,, A n, B C 20
Deduction Theorem A 1, A 2,, A n, B C A 1, A 2,, A n, B C The premises is said to entail the conclusion If in every model in which all the premises are true, the conclusion is also true A 1 A 2 A n B C A 1 A 2 A n B C A 1, A 2,, A n B if and only if A 1 A 2 A n B (A 1 A 2 A n B is a tautology) A,B C B C A B C B C If A is T, then B C is always T (for the tautology) Even if A,B is T, then B C is always T And if A,B is T, then B is T By modus ponens in the RHS, A,B is T, then C is true 21
Deduction Theorem Δ,A B iff Δ A B where Δ : a set of formulas, if the formula B is deducible from a set Δ of assumptions, together with the assumption A, then the formula A B is deducible from Δ alone. Δ,A B Δ A B Conversely, if we can deduce A B from Δ, and if in addition we assume A, then B can be deduced. Δ A B Δ,A B http://planetmath.org/deductiontheorem 22
Deduction Theorem The deduction theorem conforms with our intuitive understanding of how mathematical proofs work: if we want to prove the statement A implies B, then by assuming A, if we can prove B, we have established A implies B. http://planetmath.org/deductiontheorem 23
Deduction Theorem The converse statement of the deduction theorem turns out to be a trivial consequence of modus ponens: A, A B B if Δ A B, then certainly Δ,A A B Since Δ,A A, we get, via modus ponens, Δ,A B as a result. Δ,A A Δ,A A B Δ,A B http://planetmath.org/deductiontheorem 24
Deduction Theorem Deduction theorem is needed to derive arguments that has no premises An argument without premises is simply a tautology A A no premises appear before the symbol an argument without premises Tautology if it is sound 25
Argument Example 1. Q 2. P Q 3. P R S 4. R T 5. U 6. U T 7. P Q, P Q P (1, 2) 8. R S P R S, P R S (3, 7) 9. T U, U T T (5, 6) 10. R T, R T R (9, 4) 11. S R S, R S (8, 10) 26
Argument without premises A A A assume A X X Y A A A A A A A A A discharge A A assume A A A X X Y A A X Y Y X A A A A A A A A A A A discharge A X Y, X Y Y A A A, A A A A A A A 27
Argument without premises A A assume A A A A A A A A A A A A discharge A A A assume A A A A A A A A A A A A A A A A A discharge A A A A A 28
To prove a sound argument Prove using truth tables Whether an argument is sound or fallacy 1. time complexity (2^n) 2. not the way which humans do Prove using inference rules To reason deductively 29
Logical Consequences A S B Syntactic Consequences there is a derivation, in the proof-system S, from the premise A to the conclusion B. [If context fixes the relevant system S, we suppress the subscript.] A L B Semantic Consequences Logical Implication on every possible interpretation of the non-logical vocabulary of language L, if A comes out true, so does B. [If context fixes the relevant language L we suppress the subscript.] A B Material Implication on the truth-functional interpretation, if the atomic wff A happens to be false and the atomic wff B happens to the false too, then A B evaluates as true. But we don't have A B (q isn't true on every valuation which makes p true). A B A B True True True True False False False True True False False True http://math.stackexchange.com/questions/365569/whats-the-difference-between-syntactic-consequence-%e2%8a%a2-and-semanticconsequence-%e2%8a%a8 30
Double Turnstile 1. semantic consequence: a set of sentences on the left a single sentence on the right to denote that if every sentence on the left is true, the sentence on the right must be true, e.g. Γ φ. This usage is closely related to the single-barred turnstile symbol which denotes syntactic consequence. 2. satisfaction: a model (or truth-structure) on the left a set of sentences on the right to denote that the structure is a model for (or satisfies) the set of sentences, e.g. A 3. a tautology: φ to say that the expression φ is a semantic consequence of the empty set. Γ. https://en.wikipedia.org/wiki/double_turnstile 31
Syntactic Consequences A formula A is a syntactic consequence within some formal system FS of a set Γ of formulas if there is a formal proof in FS of A from the set Γ. Γ FS A Syntactic consequence does not depend on any interpretation of the formal system. A formal proof or derivation is a finite sequence of sentences (called wwf), each of which is an axiom, an assumption, or which follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. Sound argument Fallacy https://en.wikipedia.org/wiki/double_turnstile 32
Semantic Consequences A formula A is a semantic consequence within some formal system FS of a set of statements Γ Γ FS A if and only if there is no model I in which all members of Γ are true and A is false. the set of the interpretations that make all members of Γ true is a subset of the set of the interpretations that make A true. https://en.wikipedia.org/wiki/double_turnstile 33
Summary Syntactic consequence Γ φ sentence φ is provable from the set of assumptions Γ. Semantic consequence Γ φ sentence φ is true in all models of Γ. Soundness If [Γ φ] then [Γ φ]. Completeness If [Γ φ] then [Γ φ]. The propositional logic has a proof system (propositional calculus) Syntactic consequences a semantics (truth-tables) Semantic consequences http://philosophy.stackexchange.com/questions/10785/semantic-vs-syntactic-consequence 34
Syntactic and Semantic Consequences (1) Syntactic consequence Γ φ sentence φ is provable from the set of assumptions Γ. A A B B Semantic consequence Γ φ sentence φ is true in all models of Γ. A B A B A (A B) A (A B) B T T T T T T F F F T F T T F T F F T F T http://philosophy.stackexchange.com/questions/10785/semantic-vs-syntactic-consequence 35
Syntactic and Semantic Consequences (2) A, A B B Syntactic consequence Γ φ if we take the assumptions A and A B as given, by modus ponens we can deduce B. A, A B B Semantic consequence Γ φ in any model for which it is the case that A is true and also A B is true, then, in that model, B is also true. talks about the propostions themselves as syntactic objects, talks about what the propositions mean i.e. semantics. https://www.quora.com/what-are-the-differences-between-semantic-consequence-and-syntactic-consequence-in-logic 36
Soundness and Completeness (1) Sound Deduction System : if it derives only sound arguments each of the inference rules is sound Soundness. If [Γ φ] then [Γ φ]. A sound argument: If the premises entails the conclusion A fallacy: If the premises does not entail the conclusion Complete Deduction System : It can drive every sound argument must contain deduction theorem rule Completeness. If [Γ φ] then [Γ φ]. 37
Soundness and Completeness (2) Soundness is the property of only being able to prove "true" things. Completeness is the property of being able to prove all true things. So a given logical system is sound if and only if the inference rules of the system admit only valid formulas. Or another way, if we start with valid premises, the inference rules do not allow an invalid conclusion to be drawn. A system is complete if and only if all valid formula can be derived from the axioms and the inference rules. So there are no valid formula that we can't prove. http://philosophy.stackexchange.com/questions/6992/the-difference-between-soundness-and-completeness 38
Invalid Argument Examples P1 P2 C Grass is green Paris is the capital of France A poodle is a dog P1, P2 and C all true, but argument not deductively valid. If you object that this doesn t count as an argument because there is no connection between the Ps or between the Ps and the C, try P1 P2 C All atoms are tiny The smallest particle of hydrogen gas is tiny The smallest particle of hydrogen gas is an atom All true, but not deductively valid. To see this, substitute oxygen for hydrogen (the smallest part of oxygen gas is a molecule not an atom, so C false) or pollen for hydrogen gas (the smallest particle of pollen is a grain, C false) https://askaphilosopher.wordpress.com/2012/02/08/invalid-argument-with-true-premisses-and-true-conclusion/ 39
Unsound Argument Examples P1 P2 C Craig is a Scot All Scots are drunks Craig is a drunk Here, P1 can be true, C follows from the Ps (validity), C can be true, but the argument is unsound because P2 is false. So, although the C is true we can t rely on the argument to establish it. It is an unsound argument. P1, P2 C T, T = T T, F = T https://askaphilosopher.wordpress.com/2012/02/08/invalid-argument-with-true-premisses-and-true-conclusion/ 40
Invalid Argument Examples Either Elizabeth owns a Honda or she owns a Saturn. (True / False) Elizabeth does not own a Honda. Therefore, Elizabeth owns a Saturn. A valid argument even if one of the premises is actually false, that if they had been true the conclusion would have been true as well All toasters are items made of gold. (False) All items made of gold are time-travel devices. Therefore, all toasters are time-travel devices. A valid and unsound argument even if one of the premises is actually false, that if they had been true the conclusion would have been true as well No felons are eligible voters.(true) Some professional athletes are felons. (True) Therefore, some professional athletes are not eligible voters. (True) A valid and sound argument http://www.iep.utm.edu/val-snd/ 41
Valid Argument Examples All tigers are mammals. No mammals are creatures with scales. Therefore, no tigers are creatures with scales. A valid and sound argument [P1,P2 C], [P1, P2 C] All spider monkeys are elephants. (False) No elephants are animals. (False) Therefore, no spider monkeys are animals. (False) A valid but unsound argument [P1,P2 C], [P1, P2 C] These arguments share the same form: A valid arguments, Syntactic Consequences All A are B; No B are C; Therefore, No A are C. http://www.iep.utm.edu/val-snd/ 42
Invalid Argument Examples All basketballs are round. The Earth is round. Therefore, the Earth is a basketball. An invalid and unsound argument [P1,P2 C], [P1, P2 C] All popes reside at the Vatican. (True) John Paul II resides at the Vatican. (True) Therefore, John Paul II is a pope. (True) An invalid and unsound argument [P1,P2 C], [P1, P2 C] These arguments also have the same form: an invalid arguments All A's are F; X is F; Therefore, X is an A. http://www.iep.utm.edu/val-snd/ 43
Syntactic and Material Consequences https://en.wikipedia.org/wiki/logical_consequence 44
Logical Equivalences,,,, 45
References [1] en.wikipedia.org [2] en.wiktionary.org [3] U. Endriss, Lecture Notes : Introduction to Prolog Programming [4] http://www.learnprolognow.org/ Learn Prolog Now! [5] http://www.csupomona.edu/~jrfisher/www/prolog_tutorial [6] www.cse.unsw.edu.au/~billw/cs9414/notes/prolog/intro.html [7] www.cse.unsw.edu.au/~billw/dictionaries/prolog/negation.html [8] http://ilppp.cs.lth.se/, P. Nugues,` An Intro to Lang Processing with Perl and Prolog 46