Logical Arguments

Strength  1. A useless argument is one in which the truth of the premisses has no effect at all on the truth of the conclusion. 2. A weak argument is one in which the likelihood of the

conclusion’s being true is not much affected by the truth or falsity of the premisses. 3. A moderate argument is one in which the likelihood of the conclusion being false if the premisses are true is

quite low. 4. A strong argument is one in which the likelihood of the conclusion being false if the premisses are true is very low. 5. A valid argument is one in which it is just impossible

for the conclusion to be false if the premisses are true.

Validity   D1. An argument is Valid if and only if it is impossible for the premisses to be true and the conclusion false   D2. An argument is Invalid if and only if it is possible for

the premisses to be true and the conclusion false                   All men are mortal                Socrates is a man                 Socrates is mortal                   All men are mortal                Socrates is mortal                 Socrates is a man

1. Valid arguments can have false premises.

  Socrates is a wombat All wombats are Greek Socrates is Greek

  2. Valid arguments can have false

conclusions. 

Socrates is a wombat All wombats are Spanish Socrates is Spanish

Soundness  D3. An argument is Sound if and only if it is valid and has true premisses.

Form   All men are mortal Socrates is a man Socrates is mortal   is valid: and any argument that has the form                   All A are B                 X is an A                 X is a B   is going to be valid   This sort of form is called the Logical Form of the

argument   compare                   Socrates is a bachelor Socrates is unmarried.   First kind of argument is Formally Valid.

Propositional Connectives 

 The standard words that we consider are:   1. ‘or’ for disjunction                            ‘Grass is green or snow is white’ is true if ‘grass is green’ is true or ‘snow is white is true’   Grass or leaves or mulch make a good compost.   Grass makes a good compost or leaves make a good compost or mulch makes a good compost.  

1.  exclusive (xor) We’ll have hamburgers or pizza for dinner (but not both.)

2.  inclusive There’s a pen or a pencil in that drawer (or both.)

               

Propositional Connectives  2. ‘and’ (‘but’, ‘as well as’, …) for conjunction  

‘Grass is green and snow is white’ is true if ‘grass is green’ is true and ‘snow is white is true’  

‘Grass and trees are plants’   3. ‘not’ (‘it is not the case that’, ‘no’, …) for negation  

‘Grass is not green’ is true if ‘grass is green’ is not true.  

‘It isn’t the case that grass is green’   4. ‘if … then ---’ for implication  

‘If grass is green then snow is white’ is true if in any case that ‘grass is green’ is true it is also the case that ‘snow is white is true’.

Disjunctive Syllogism:   P or Q Not Q     P   You are English or you are French. You are not French. So, you are English.   Hypothetical Syllogism:   If P then Q If Q then R If P then R   If you are English then you like fish and chips. If you like fish and chips then you are fat. So, if you are English then you are fat.

Truth Tables   Use the following definitions for the truth functions.   1.            A                B                A or B

T                T                  T T                F                   T F                T                  T F                F                   F

  2.            A                B                A and B

T                T                  T T                F                   F F                T                   F F                F                   F

    3.            A                not A

T                F F                T

  4.            A                B                if A then B

T                T                  T T                F                   F F                T                  T F                F                   T

We combine these functions in the truth tables.                   For the disjunctive syllogism  

A = You are English                 B = You are French                   A                B                A or B           not B                 T                T                  T                F                 T                T                  T                F

T                F                   T                T T                F                   F                 T

                  For the hypothetical syllogism  

D = You are English                 E = You like fish and chips                 F = You are fat                   D                E                F                if D then E              if E then F      if D then F                 T                T                T T                                 T            T                 T                T                F T                                  F                   F

T                F                T                F                                  T                       T T                F                F                  F                                  T                       F F                T                T                 T                                  T                       T

                F                T                F                  T                                  F                       T F                F                T                 T                                  T                      T F                F                F                  T                                  T                       T

Conditional Statements   D4. a. A statement of the form ‘if P then Q’ is a Conditional  

Statement.b. In a statement of the form ‘if P then Q’ the constituent statement P is

the Antecedent c. In a statement of the form ‘if P then Q’ the constituent statement Q is

the Consequent   The following are all equivalent                   If P then Q                 P only if Q                 If P, Q                 Q if P                 Unless Q, not P   The conditional ‘if P then Q’ can be drawn as:

Necessity and Sufficiency  A conditional statement claims that:   a situation in which P is true is a sufficient condition for Q to be true,   and   a situation in which Q is true is a necessary condition for P to be true.   If P is a necessary and sufficient condition for Q then Q is also a

sufficient and necessary condition for P.                   If P then Q and if Q then P.   This is a biconditional.   P if and only if Q

Arguments  Modus ponens:   If P then Q P              Q   If you are English then you like fish and chips. You are English. So, you like fish and chips.   Modus tollens:       If P then Q Not Q     Not P   If you are English then you like fish and chips. You do not like fish and chips. So, you are not English.

formal fallacies result from an error in the form of the argument.

  Affirming the consequent:   If P then Q Q             P   If you are English then you like fish and chips. You like fish and chips. So, you are English.   Denying the antecedent:   If P then Q Not P      Not Q   If you are English then you like fish and chips. You are not English. So, you do not like fish and chips.

Categorical Propositions  The categorical propositions are of four possible forms:     1.                   All S are P  Universal Affirmative                          A 2.                   No S are P     Universal Negative                              E 3.                   Some S are P               Particular Affirmative                          I 4.                   Some S are not P         Particular Negative                              O   where S and P are names of categories (standing for Subject and Predicate.)

Syllogistic Arguments  Given two categorical propositions as premisses we can sometimes draw a

conclusion in the form of another categorical proposition.   Ex 1.       No G are H                                                             All F are G                                                             No F are H                                                               No men are perfect                                 All Greeks are men                                 No Greeks are perfect   Ex 2.       No G are H                                                             Some F are G                                                         Some F are not H                                                    No philosophers are wicked                                 Some Greeks are philosophers                                 Some Greeks are not wicked

Showing Validity   We shall simply suppose that:   (i) there are some simple, uncontroversial cases of valid reasoning.   And then:   (ii) given any argument for consideration, we will consider it to

be valid just in case we can show that its conclusion can be inferred from the premises by use of these simple steps or rules.

An Example  Step 1     What is the argument to be analysed?   Ned Kelly was either an outlaw or a political activist. If he was an outlaw or a killer, he deserved what he got. Only if he was fairly tried and justly hung did he deserve what he got. But he was not fairly tried. Therefore, he was a political activist.

Step 2     How are we to evaluate the argument?                                 By attending to its structure or logical

form.   Using the following abbreviations:   O - Ned Kelly was an outlaw; P - Ned Kelly was a political activist; K - Ned Kelly was a killer; D - Ned Kelly deserved what he got; T - Ned Kelly was fairly tried; H - Ned Kelly was justly hung.   the argument can be formalized as follows:  

O or R                 if (O or K) then D                 if D then (T and H)                 not T                                      R

Step 3     Is the argument above valid or invalid?                   

   Valid. I can prove it as follows: 

Step 4:                                    By assumption, if (O or K) then D, and if D then (T and H), so (1)           if (O or K) then  (T and H)                   [since the basic rule: 'If A then B, If B then C; hence If A then C'

is valid].                 But, by assumption, not-T, so (2)           not-(T and H)                 [since the basic rule: 'not-A; hence not both (A and B)' is valid]. (3)           So, by (1) and (2), not-(O or K)                 [since the basic rule of modus tollens is valid]. (4)           So, neither O nor K                 [since the basic rule: 'not-(A or B); hence neither A nor B' is

valid]. (5)           So, by (4), not-O                 [since the basic rule: 'neither A nor B; hence not-A' is valid].                 Yet, by assumption, O or R. (6)           Hence, by (5), R                 [since the basic rule: 'not-O, O or P; hence P' is valid].

Showing Invalidity

Method 1: Find a Counterexample To show that an argument is invalid, you should: (i)       Determine the pattern of the argument to be criticised (ii)    Construct a new argument with:            (a)       the same pattern;            (b)       obviously true premises; and            (c)       an obviously false conclusion.  

       

     

If God created the universe then the theory of evolution is wrong The theory of evolution is wrong God created the universe   But this is just like arguing:   If Dominic is a wombat then Dominic is a mammal Dominic is a mammal Dominic is a wombat   same structure and is obviously invalid. Premises are true, conclusion is false! So it’s invalid.

Method 2: Invalidating Possible Situations

To show that an argument is invalid, you should:  

describe a possible situation in which the premises are obviously true and the conclusion is obviously false

E.g.                   Fallacy of affirming the consequent                             If my car is out of fuel, it won't start                           My car won't start                                                                                My car is out of fuel.   Consider the situation.   My car will indeed not start without fuel (it is a fuel-driven car) and the

electrical system needed to start the car has been taken out for repairs (so it won't start). Yet     the car has a full tank of petrol.

  The premises are true and the conclusion is false. The situation is not impossible (i.e. it is possible).                                        So, it is not impossible for the premises to be true and the conclusion false. So, the argument is invalid.

E.g.                   Fallacy of denying the antecedent   If the Committee addresses wilderness value then it must address naturalness. It will not address wilderness value.                 It need not address naturalness.             Consider the following possible situation.             Wilderness value involves, amongst other things, naturalness (Federal

legislation actually defines 'wilderness value' this way). Moreover, the Committee's terms of reference do not include consideration of wilderness value  (so it won't address it). Yet the Committee is explicitly formed to consider naturalness (to feed their findings into those of other committees, so that a joint finding can be made

regarding wilderness values).   The premises are obviously true in this situation and conclusion obviously false. The situation is not impossible (i.e. it is possible).                                        So, it is not impossible for the premises to be true and the conclusion false. So, the argument is invalid.