Deductive Reasoning

Inductive: premise offers support and evidenceInductive: premise offers support and evidence Deductive: premises offers proof that the conclusion is trueDeductive: premises offers proof that the conclusion is true DeductiveDeductive  Absolutely follows  Necessarily follows  Defiantly the case Deductive logic = If premises are true it proves the conclusion is true Inductive Likely follows Probably follows Best possible result

Two types of deductive logic Classical Logic(Socratic) – –Allows for simple arguments – –Tries to place subjects in/out of categories (predicate) – –Written in a very specific fashion (standard form) – –Inference = Form Modern Logic (Symbolic) – –Allows for more flexibility – –Shows connections between premises and conclusion – –Inference = Flow P v ~P Therefore, Q

Classical Deductive Logic Socratic Symbolism

Math Standard Form Equation – –Two things are equal – –X – 7 = 0 Linear Equation – –Equation for a line on graph – –3x – y = -2 Algebraic Polynomial – –Expression of mathematical terms – –4x + 3x – 7 (depends on variables) Logic tries to put language into expressions and equations as well.

Any statement made must be in a proper form (format) = Standard Form All statements must have a relationship: – –Subject: The what/whom the statement is about – –Predicate: Tells us something about the subject All apple are red All you is the best student ever All Wisconsin is in the United States What category does the subject fit into? Has to be ALL, None or Some This means you may have to rewrite statement to fit this format English to Standard Form

Putting into standard form " Ships are beautiful" translates to "All ships are beautiful things” "The whale is a mammal" translates to "All whales are mammals." "Whoever is a child is silly" translates to "All children are silly creatures." "Snakes coil" translates to "All snakes are coiling things."

Standard Form- Structure Can only have 3 lines or (statements) – –Two lines are premises & one is a conclusion Can only have 3 terms (subject/predicate relationship) – –Cannot switch meaning of words – –1 term must be in both premises – –2 terms in conclusion must be in one of the premise If either premise is negative than the conclusion must be negative Why do this? By doing this we guarantee an inference in our argument (Glue)

Categorical proposition (statement) – –Statement that is asserting an inclusion or exclusion into a catagory Class – –Objects that have same characteristics in common Quantity (All/None/Some) – –Subject is all, none or some of a certain class Quality (Affirmative or negative) – –“Copula” – –is or is not – –The verb in the sentence Standard Form – Statement Rules

Standard Form- Categorical Statements Only 4 types of statements you can have given standard format rules. – –A: All S are P – –E: No S are P – –I: Some S are P – –O: Some S are not P Quantifier + Subject Term + Copula (quality)+ Predicate Term

Standard Form- Examples Universal Affirmative- A – –All S are P (All Oaks are Trees) Universal Negative- E – –No S are P (No Oaks are fish) Particular affirmative- I – –Some S are P (Some Oaks are big) Particular negative- O – –Some S are not P (Some oaks are not big)

Standard Form- Argument All arguments will have a – –Major term: predicate of the conclusion – –Minor term: subject of the conclusion Each one of these premises will share a part of the overall conclusion Conclusion: All mortals are Greeks – –Middle term: term in both premises

Middle Term All human are Greeks (A) All mortal are humans (A) Conclusion: All mortal are Greeks (A) Middle Term= The inference in the premises Notice that the connecting term does not appear in the conclusion. It is the connection between the two premises. It is the subject in one and the predicate in the other

Standard Form- Inferences All humans are Greeks (A) All mortals are human (A) ∴ All mortals are Greeks (A) – –Major term – –Minor term – –Middle term

Valid argument- Clear line of inference Greeks Humans Mortals All human are Greeks (A) All mortals are humans (A) Conclusion: All mortals are Greeks (A)

Invalid Argument Has no inference to connect argument terms and statements together All dogs are mortal All cats are strange ∴ ∴ All dogs are cats Invalid = No middle term

Deductive Argument Valid Conclusion must be true if both premises are true Conclusion can only be false when one of the premises are false The form must be valid (Standard Form) Fill in any variable term and the structure is valid All X are Y All Z are X Therefore, all Z are Y Then talk about soundness (truth of premises)

Standard Form- Valid Only 15 valid standard form arguments – –That are in valid form – –If both premises are true than it forces you to accept the conclusion as true – –Because of the structure…the set up…the form…the inferences – –True premises?

Standard Form- Sound If the set up of argument is in standard form (validity) All premises are true (truth) Forces a true conclusion (truth) t If an argument is valid and has all true statement then it is a sound argument

Modern Deductive logic

Rules of modern logic Allows for many lines of logic – –Only one line is the conclusion Inferences – –Is shown through 9 rules of deductive logic – –Rules are valid inferences between premises or between premises and conclusion – –Rules will be valid 100% of the time Shows the flow of the argument -or- reasoning – –How do the reasons presented “flow” to the conclusion presented These offer proofs – like math (geometry) – –Not based on form of argument

Disjunctive Either you are red or you are green. You are not Red. Therefore, you are green P v Q ~P/Q Also deny the second term (~q)Also deny the second term (~q) Be careful for Fallacy of False DilemmaBe careful for Fallacy of False Dilemma

Modus Ponens If you repent, then you will go to heaven. You have repented. So you will go to heaven. P Q P P  Q P/Q If the IF part is true then the THEN part must be true as well

Modus Tollen If there is smoke, then there is fire. There is not fire, so there is no smoke. P Q ~Q P  Q ~Q~P If the THEN part is false then the IF part must be FALSE as well

Hypothetical Syllogism If something is a tree (P), then it is green (Q). If it is green (Q), then it is a plant (R). Therefore, if something is a tree (P), then it is a plant (R). P Q Q R. P  Q Q  R. /P R. /P  R.

Hypothetical Syllogism- Chain If something is a tree (P), then it is green (Q). If it is green (Q), then it is a plant (R). If it is a plant (R), then it needs the sun (Z). Therefore, if something is a tree (P), then it needs the sun (Z). P Q. Q R. P  Q. Q  R. R R  Z /P Z. /P  Z.

1. I  (C v K) 2. I 3. ~C 4. K  (~S E) 5. M  S / Therefore ~M 6. C v K (1)(2) MP 7. K (6)(3) DS 8. ~S E (4)(7) MP 9. ~S (8) Simp. 10. ~M (5)(9) MT If your reasoning has this “flow” than it is valid. You can put in any term and it will flow- validly- to the conclusion. Then argue if sound or not (truth)