Propositional Logic. Modern Logic. Boolean Logic. AKA Modern Logic AKA Symbolic Logic. AKA Boolean Logic. Chrysippos (3 rd Head of Stoic Academy). Main early logician
Stoic Philosophers Zeno ff301bc. Taught Philosophy under the Stoia Poikile (Painted Porch). Zeno Famous for logical paradoxes: This statement is false This school of philosophy persisted until 529 AD when the Christian Emperor Justinian ordered all philosophy schools closed. We know that several hundred Texts were written by Stoic Logicians. NONE SURVIVE.
Stoic Philosophers Stoic Philosophers were adamant that reason, rather then passion, was the correct attitude for someone seeking wisdom (and happiness). freedom is secured not by the fulfilling of one's desires, but by the removal of desire (Epictus). Thus in modern English, we call someone Stoic who is dispassionate, especially in the face of hardship.
Propositional VS Predicate Logic Propositional Stoics. More variety in allowable expressions. Arguments have variable forms. Allows more then one operator (if then, and, or, not). Less simple proofs. Scorned and Burned by Early Christians. Rediscovered in late 19 th Century. Predicate Aristotle. Very limited expressions. Arguments have fixed form (3 line syllogisms) Only operator set membership Very simple proofs for validity. Scorned and Burned by the Early Christians. Rediscovered in Early Medieval Period.
Atomic Propositions The simplest possible propositions. I am walking = Atomic I am walking and texting. Not atomic. I am walking and I am texting. Two atomic propositions conjoined. A statement is Atomic if no logical connectives are in the statement: words like and, or, if then, but, not.
Propositional operators. And Or conjunction. disjunction If then conditional Not negation Other logical relationships can be reduced to combinations of these.
Truth Values Every statement is either true or false. In ordinary language we represent a statement as true by merely writing, speaking etc. By convention, it is raining means: It is true that it is raining. To deny, we negate. it is not raining.
Propositional Notation Any atomic statement can be represented by a variable letter (usually uppercase). Ex. Phil is a Philosophy prof. P. To say that it is true that Phil is a Philosophy Prof. we merely write P. read as P is true. To deny that Phil is a philosophy prof. we write not-p or ~P, read as P is false
Examples of Notation There are 11 minors released from the Chilean mine = M Two junior and two senior faculty were present at the appeal hearing. = J and S. J = two junior faculty were present. S = two senior faculty were present. Either I will pass the course or drop out of the U. P or D.
Truth Tables We can represent all the possible truth values for any combination of statements using a truth table. One row represents one possible permutation of truth values. For one statement, there would be 2 rows. For two statements, this would double to 4 rows. For 3 statements this would double to 8 rows. Reasonable limit to usefulness.
Uses for truth tables Define operators. This is what we will be doing today. Analyze statements. Statements have different possible truth values. Analyze arguments. Valid arguments never have a false conclusion with all true premises.
Truth table for negation. Every statement is either true or false. A negated statement has the opposite truth value to the un-negated statement. Symbol for negation is the tilde: ~ P (possible values) T F ~P (result) F T
Truth table for Conjunction A conjunction is true when both conjuncts are true, otherwise false. Symbol for conjuction is the dot: P Q P Q (result) T T T T F F F T F F F F
Truth table for Disjunction A disjunction is true when either, or both, disjuncts are true, otherwise false. Symbol for disjunction is the wedge: v P Q P v Q (result) T T T T F T F T T F F F
Possible Equivocation on or Sometimes when we say or we mean: One or the other, or both. Eg. You can have cake or ice cream. This is the inclusive sense of or. Othertimes we mean: One or the other but not both: eg. you can have apple or pumpkin pie. This is the exclusive sense of or. In Symbolic logic we presume the inclusive sense. The exclusive sense can be represented by a combination of symbols.
Conditional Statements Statement form: If ANTECEDENT then CONSEQUENT. Interpretation = material conditional No implied relationship. Eg. If you have money then you ll get a car. Is this true or false when you have no money but you still get a car?
Categorical Equivalents Recall A form statements like: All A s are B s. These can be converted to Conditional statements All A s are B s = If A (member of A) then B (member of B). E form Statements: No A s are B s Converts to not (A then B) ~(A B)
Truth table for Conditional A Conditional is true in every case except when the antecedent is true and the consequent is false. Symbol for conditional is the horseshoe P Q P Q (result) T T T T F F F T T F F T
WHAT!!!!!! Consider a Bet on a Horse. My bet can be expressed as a conditional If Speedy wins then my bookie pays me $5.00 When is the deal broken? If my bookie pays me $5.00 for no reason, or for some other bet, the deal is still good. If speedy loses and I don t get $5.00 does it make sense to say the deal was not kept? Only when speedy wins and I don t get the $5.00 would we say the deal has been broken. (false).
If and only if Bi-conditional -> two conditionals A if and only if B means: (If A then B) and (If B then A) Recall how only worked in Categorical Logic: Only A s are B s = All B s are A s. The bi-conditional is the same as equivalence. Symbolized by the tri-bar:
Truth Table for Equivalence P Q P Q T T T T F F F T F F F T So if the university stipulates: you can graduate if and only if you have a GPA of C or better, they mean: the conditions for graduating are equivalent to the conditions of a C or better GPA
Compound propositions. You can take Law 2240 if you have credit for law 1140 and law 1240 or if you have written consent of the dean. Symbolized as: ((1140 1240) v W) 2240 Brackets work just like in math (they borrowed them from us).
Master truth table This table defines the logical operators. P Q ~P P Q P v Q P Q P Q T T F T T T T T F F F T F F F T T F T T F F F T F F T T http://editthis.info/logic/main_page
Truth Tables for Compound Statements Compound statements can be analyzed for their possible truth values. Some statements can never be true: Contradictions (P ~P) Some statements can never be false: Tautologies (P v ~P) Some statements can be either true or false: Contingent (P v Q)
Truth table for statements Nothing can be both red all over and green all over. ~(R G) R G ~(R *G) T T F T F T F T T F F T This table shows this statement is contingent. It can be either true or false depending on the input conditions of R and G.
Contradiction Nothing can be true and false at the same time. P ~P This table shows that the statement (P ~P) can never be true. P T F P ~ P F F It is internally contradictory.
Tautology Its not over till its over (Yoggi Berra) P if and only P P P P P P T F P P P P T T This table shows a statement that can never be false (a tautology).
Evaluate these statements 1. (P v Q) (P Q) 2. (P Q) (P v Q) 3. Q (P v Q) 4. (~P ~Q) ~(P v Q)
Propositional Arguments Arguments are any number of statements intended to support a conclusion. Notational Convention: each premise gets its own line. Each premise is numbered. Conclusion added to end of last line after /
A familiar example The syllogism Barbara (3 A forms) All A s are B s All B s are C s Therefore All A s are C s. If A then B IF B then C Therefore if A then C. 1.A B 2.B C / A C Propositional form AKA Hypothetical syllogism.
A familiar fallacy All A s are B s All C s Are B s Therefore All A s are C s 1. A B 2. C B / A C
Example Propositional argument I m going to drop critical thinking if my quiz score and my prose score are both low or if Prof Phil continues to lull me to sleep. Since my quiz score and my prose score are both low, I m headed over to the faculty office to VW. Let D = Drop Critical thinking Let Q = Low quiz score Let P = Low prose score Let S = Prof Phil Lulls me to sleep.
Example continued I m going to drop critical thinking if my quiz score and my prose score are both low or if Prof Phil continues to lull me to sleep. Since my quiz score and my prose score are both low, I m headed over to the faculty office to VW. 1. ((Q P) v S) D. 2. Q P / D Later we ll see how to add a proof to this argument by deriving further lines. For now, we ll prove it using the truth table. (see whiteboard).
Statements vs Arguments Statements and arguments are inter-changable. Any argument is a statement formed by a conjunction of the premises being the antecedent of a conditional with the conclusion being the consequent.
Statement vs Argument eg. Statement: since I agreed that If my dog bites then I ll have to keep him leashed and he bite me, so I ll have to keep him tied up. ((B L) B) L Argument form: If my dog bites then I ll have to keep him leashed. My dog bite me. Therefore I ll have to keep him tied up. B L B / L
Test for validity A Valid argument presented as a single statement will be a tautology. (final column all T) A valid argument with separate premise(s) and conclusion will satisfy the definition of valid. If the premises are true the conclusion must be true. Rows with a false conclusion must have at least one false premise. (Parallel truth table for previous example)
Test these arguments for validity Either employment levels will go up or there will be a revolt. Employment levels are not going up. Therefore, there will be a revolt.
Classic argument forms Hypothetical Syllogism (equivalent to Barbara 3Aform syllogism) does reduce to SCS form. Stoic philosophers identified several argument forms that did not obviously reduce to Aristotelian syllogisms. They worked hard to prove them. Eg. Disjunctive syllogism P or Q, not P; therefore Q. These argument forms can be proven using truth tables. These arguments can be used as intermediate inference rules in more extended arguments.
Using inference rules. 1. If A then B 2. If B then C 3. If C then D. Therefore A then D. lines 1 and 2 are a valid argument that implies A then C. A and C and line 3 implies the conclusion.
Copi s rules of inference. Total of 9 inference rules. Modus ponens Modus tollens. Hypothetical syllogism. Disjunctive syllogism. Constructive dilemma. Absorption. Simplification. Conjunction. Addition.
Modus Ponens AKA affirming the antecedent. If A then B A therefore B If speedy wins then $5.00 Speedy wins! $5.00. Proof on whiteboard. A
Modus Tollens AKA Denying the consequent. Recall IF antecedent then CONSEQUENT. IF P then Q Not Q Therefore not P Eg. If speedy wins then $5.00 No $5.00, so speedy didn t win.
Hypothetical Syllogism IF A then B If B then C Therefore If A then C Ex. If my car starts then I ll get to class. If I get to class then I can sleep in class Therefore If my car starts then I can sleep. Proof via truth table.
Disjunctive Syllogism AKA denying the disjunct. P or Q Not P Therefore Q You can have either pie or cake. No pie please. Cake it is! Proof on whiteboard.
Constructive Dilemma Di (2) lemma (theorm) Form has 3 premises 2 conditionals 1 disjunction Either A or B If A then C If B then D Therefore C or D.
Absorbtion P then Q Therefore P then (P and Q).
Simplification P and Q Therefor P
Conjunction P Q Therefore P Q
Addition P Therefore P or Q. Where does the Q come from? Anywhere If P is true then P or ANYTHING is true. Remember this when hearing answers to disjunctive questions.
Some Classic Fallacies Affirming the Consequent. Denying the Antecedent. Affirming the Disjunct.
Affirming the Consequent 1. A B 2. B / A Eg. If I m in SJ s College, then I m at the U of M. I m at the U of M. Therefore I m in St. John s College.
Denying the Antecedent 1. A B 2. ~A / ~B Eg. If I m convicted then I must be guilty. I wasn t convicted so I must not be guilty.
1. A v B 2. A / ~B Affirming the Disjunct. Eg. I can have either cake or ice cream. I had cake, so I can t have ice cream. This would be valid with exclusive sense of or.
Substitution rules We have seen that A if and only if B (A B) means the same thing as A equivalent to B (A B) Using truth tables we can prove other equivalencies. We can substitute these equivalencies to simplify reasoning.
Copi s logical equivalencies 10 logical equivalencies: De Morgan s Theorem(s). Commutation. Association. Distribution. Double negation. Transposition. Material Implication. Material Equivalence. Exportation. Tautology.
De Morgan s Theorems Explain logical relationship between addition and conjunction. ~(P Q) (~P v ~Q) ~(P v Q) (~P ~Q) Eg: U2 isn t playing in Edmonton and Calgary. ~(E C) U2 isn t playing in Edmonton or in Calgary. (~E v ~C)
Natural Deduction Any argument can be proven valid by a process of deduction. Deducing the conclusion from the premises using a series of steps, each of which is a valid argument form. How does this work? Remember the definition of Valid: If the premises are true, the conclusion must be true. Assume the premises are true, and derive the conclusion.
Natural Deduction example 1. (P v Q) (P Q) 2. P / Q 3. P v Q 2, add (from step 2 by Addition) 4. P Q 1, 3, MP (Steps 1 and 2 by Modus Ponens) 5. Q 4 Simp. (from step 4 by simplification)
Natural deduction cont. Fair game? Using rules of inference (copi 7.2) Not fair. Using rules of substitution. (copi 7.3)
Natural deduction problems 1. P (~Q ~P), P, ~Q / ~P 2. P v Q, ~P, Q R / R 3. P, P R, Q S / R v S (2 ways to solve) 4. (B v ~C), C v D, ~B / D
Comparing Proof Systems Counter-examples (intuitive) Venn Diagrams (visual) Aristotle s rules (simple) Truth Tables (simple but laborious) Natural Deduction (elegant but difficult)
Completeness of proof systems A proof system is complete if it can prove every invalid argument invalid, AND every valid argument valid.
Comparing Proof Systems Counter-examples Both predicate and propositional logic. Proves only invalid arguments. Cannot prove an argument valid (fallacy of ignorance.) Venn Diagrams. Only predicate logic. Complete (both valid and invalid) Aristotle s rules. Complete for all arguments that can be expressed as 3 term syllogisms. Truth Tables. Complete for all propositional arguments.
Is natural deduction complete? Recall, you prove an argument valid by deducing its conclusion from its premises using only valid inferences (and substitutions). This proves that if the premises are true, the conclusion cannot be false. Problem: suppose you cannot find the proof? Invalid or just really hard? Fallacy of ignorance?
Natural deduction for invalidity If you can deduce the contradiction of the conclusion, THEN you have proven the argument invalid. Eg: 1. P R, 2. ~R / P 3. ~P 1,2 MT
More Cool Stuff with Prop Logic Conditional proof. Often when we reason, we accept statements conditionally ( for the sake of the argument ). We tentatively treat these statements as true to see what might come of it. In prop logic there is a notational form to deal with this. (see whiteboard demo).
More Cool Stuff with Prop Logic Reductio Ad Absurdum (indirect Proof) Reduce to absurdity. If you can deduce a logical absurdity, you show that something in the set of starting assumptions is false. Logical absurdity (something is both true and false at the same time) P ~P Indirect proofs proceed by using a conditional proof method and assuming the opposite of what you are trying to prove. (See whiteboard demo)