truth function logic

what is formal logic

• understanding logical structure of sentences

• understand valid / entailment / consistency

• common form/structure, no ambiguities h

how is TFL a natural language

it has

• A vocabulary, i.e. the terms that count as part of the language

• A syntax, i.e. rules about how we can combine those terms

• A semantics, i.e. an assignment of meaning to those terms

atomic sentences

cant be reduced without losing meaning - identified as A B C D E …

• dont have to be simple, just the simplest version of their content

eg A: It is raining outside // C: Jenny is miserable

negation

not

¬A

conjunction

and

A ∧ B

disjunction

or

A ∨ B

conditional

A → B

if A then B

=

B if A

=

A only if B

if A then B

on the condition that A is true, B is true too

‘A if B’

A is true, provided that (on the condition that) B is true

‘A only if B’

A is true only in situations where B is too

same as if A then B

biconditional

A if and only if B

A ↔ B

eg S knows that p IF AND ONLY if S has a justified true belief that p

brackets

• ‘It’s not the case that Jo and Sam are hungry’ = ¬(J ∧ S)

• ‘Jo is not hungry, and Sam is hungry’ = -(¬J ∧ S)

you must add brackets whenever you add a connective UNLESS it’s ¬

what counts as a sentence

negation truth table

conjunction truth table

disjunction truth table

INCLUSIVE disjunction

conditionals truth table

???

biconditionals truth table

if same truth table = true

if different truth table = false

contingent sentences

a sentence is contingent in TFL iff (if and only if) it is true on at least one truth

table row and false on at least one

• all atomic sentences are contingent

tautology

A sentence is a tautology in TFL iff it is T on all truth table rows

eg (P→P), (A∨¬A)

contradiction

A sentence is a contradiction in TFL iff it is T on no truth table rows

(F∧¬F)

• Negating a tautology creates a contradiction, and vice versa

following the order

It’s crucial that we apply the rules for connectives in order of

sentence construction

● For instance: (¬A → (A → B))

1. A / B

2. ¬A / (A→B)

3. (¬A → (A → B))

• When constructing our truth table, we apply our rules in this order

logical equivalence

two sentences are logically equivalent iff there is no

truth table row where one is true and the other is false

● All tautologies are logically equivalent

● All contradictions are logically equivalent

formally valid

An argument is formally valid IFF it is impossible for the premises

to be true and the conclusion false, and this is due to its form

valid-in-TFL

An argument is valid-in-TFL IFF there is no truth table row on which

all premises are true and the conclusion is false

valid argument

‘(A ∨ B), ¬A ∴ B’

IF All premises are true THEN Conclusion is true

invalid arguments

((𝐴 ∧ 𝐵) ∨ (𝐴 → 𝐵)), ¬𝐴 ∴ ¬𝐵

AT LEAST one row on which all premises are true, but the

conclusion is false

edge cases of validity

1. “It’s raining; it’s not raining; therefore I am hungry”

2. “I ate nothing for breakfast; therefore it either is or isn’t Tuesday”

Both are valid. Why?

• If it’s impossible for the premises to be true, then it’s

impossible for the premises to be true and the conclusion false

• If it’s impossible for the conclusion to be false, then it’s

impossible for the premises to be true and the conclusion false

jointly consistent

• {A,B,C…} = sets of sentences

{A,B,C…} is jointly consistent IFF there is some truth-table row on which every member of {A,B,C…} is true

jointly inconsistent

A set of sentences {A,B,C…} is jointly inconsistent IFF there is no

truth-table row on which every member of {A,B,C…} is true

• contradiction will be inconsistent with any other sentence

• However not all jointly inconsistent sets contain a contradiction:

incomplete truth tables

Sometimes, you don’t need to complete the truth table to get your

answer:

• Logical equivalence: as soon as I find a row where the

sentences differ in truth-value, they are not logically

equivalent

• Validity: as soon as I find a row where at least one premise is

false, I don’t need to see the rest of the row

• Validity: as soon as I find a row where the conclusion is true, I

don’t need to see the rest of the row

• Consistency: as soon as I find a row where at least one

member of the set is false, I don’t need to see the rest of the

row

can TFL capture all necessary truths, inconsistencies, and valid arguments

no

weakness of TFL

cant look inside sentences, like FOL can

1. All elephants are mammals;

2. Dumbo is an elephant;

3. Therefore, Dumbo is a mammal

• Since we have no terms we can translate as logical connectives, this becomes:

•

𝐴, 𝐵 ∴ 𝐶

• That’s invalid

• But 1, 2, so 3 is valid!