Syntax and Semantics/Truth tables

Syntax

Has to do with the rules for combining words or phrases within a natural or formal language (English)

Semantics

Has to do with the meanings of words or phrases

Truth functional connective

a word that joins statements where the truth of the whole statement depends only on the truth of the parts

Ex: ‘and’ it is raining and it is cold (true only if both parts are true)

Non truth functional connective

a word that joins statements where the truth of the whole depends on more than just the truth of the parts

Ex: ‘because’ She is happy because she won the prize (truth depends on whether winning caused the happiness)

Truth table - Negations (~)

When P is true, ~P is false

When P is false, ~P is true

Truth table - Conjunctions (&)

Whenever both conjuncts (P and Q) are true, the conjunction is true

In all other cases, the conjunction is false

Truth table - Disjunctions (v)

Whenever both disjuncts (P and Q) are false, the disjunction is false

In all other cases, the disjunction is true

Truth table - Conditionals (⊃)

Whenever the antecedent is true (P), but the consequent (Q) is false, the conditional is false

In all other cases, the conditional is true

Truth table - Bioconditionals

Whenever both sides of the biconditional have the same truth value. The bioconditional is true (FF) or (TT)

When there is a mismatch, the bioconditional is false

How do you construct truth tables for complex statements

1) List all of the distinct sentence letters to the left of the vertical line

2) Determine the number of rows (2n where n is the number of variables in the statement)

3) List all the truth value combinations

4) Determine the truth value of the statements whose connectives have the smallest scope and work your way out

5) work out the truth value of the entire sentence

Truth value equivalence

two statements always have the same truth value - they are either both true or both false in every possible situation

Ex: It is either raining or not raining

Logical truths

a statement that is always true no matter what

Both always true

Ex: It is raining or it is not raining

Contingencies

a statement that is sometimes true and sometimes false

true and false in exactly the same situation

Ex: it is raining

Contradictions

a statement that is always false

They are both false

Ex: It is raining and it is not raining at the same time

How to use truth tables to test for validity

1) Translate each premise and the conclusion into the language of sentential logic

2) Construct a truth table and determine the truth-value of each

3) Put an asterisk next to every row in which all premises are true (regardless of whether the conclusion is true or false

4) For each row with the asterisk check if conclusion is true or false

• If false, write CE (counterexample)

• If there is even one row with CE, the argument is invalid

Conditionalizing an argument

combine all the premises using (&), then make a sentence that says “if all premises are true, then the conclusion is true.”

If an argument is valid, it’s conditionalized expression will be a logical truth