Sentential logic

Lecture based on: Graeme Forbes, Modern Logic Katarzyna Paprzycka, online lectures

Sentential logic or propositional logic?

Logic is the most boring class Jane has ever taken. The most boring class Jane has ever taken is logic.

A sentence in the logical sense is only an unambiguous declarative sentence. it is either true or false. (a) No exclamations are sentences L. “Heyah!” “Come here!” (b) No questions are sentences L Are you bored already?

(c) No ambiguous declarative sentences are sentences L (d) No declarative sentences with indexical expressions are sentences L. There is a class of sentences containing the so- called “indexical” expressions (like ‘I’, ‘he’, ‘she’, ‘there’, ‘now’, etc.) that are notoriously ambiguous because the indexical expressions change their referent depending on the context in which they are uttered. I am hungry now.

NEGATION Connective: It is not the case that Symbol: ~ [the tilde] Logical form: ~p Component: p – negated sentence

CONJUNCTION Connective: and Symbol:  Logical form: p  q Components: p, q – conjuncts

both... and as well as but however though although even though nevertheless …... still …... but still …... also …... while …... despite the fact that …... moreover …... in addition …

DISJUNCTION Connective: or Symbol: ∨ [the wedge] Logical form: p ∨ q Components: p, q – disjuncts

either... or... … or else …

CONDITIONAL Connective: if … then … Symbol: → [the arrow] Logical form: p → q Components: p – antecedent q – consequent

... if provided that given that in case in the event that as long as assuming that supposing that on the condition that on the assumption that...

BICONDITIONAL Connective: if and only if Symbol:  Logical form: p  q Components: p, q – terms of the biconditional; right and left hand sides

... if but only if... … when and only when … … just in case … … iff … … exactly if …

Symbolization Our currency will lose value. Our currency will lose value and our trade deficit will narrow. Either our currency will lose value, or exports will decrease and inflation will rise.

John will attend the meeting but Mary will not. Insider trading is unethical and illegal. At least one of our two deficits, budget and trade, will rise. Neither John nor Mary will attend the meeting. John and Mary won’t both attend the meeting.

If Smith bribes the instructor, then Smith will get an A.

Smith will get an A if Smith bribes the instructor.

Smith’s bribing the instructor is a sufficient condition for Smith to get an A.

Smith will get an A only if Smith bribes the instructor. Smith bribing the instructor is a necessary condition for Smith to get an A.

Smith will get an A provided Smith bribes the instructor. Smith will get an A whenever Smith bribes the instructor.

Smith will fail unless he bribes the instructor.

Your car will not start unless it has fuel in its tank.

Smith will get an A if and only if Smith bribes the instructor. Smith will get an A just in case Smith bribes the instructor.

John will study hard and also bribe the instructor, and if he does both then he’ll get an A, provided the instructor likes him.

If Smith bribes the instructor then he’ll get an A. And if he gets an A potential employers will be impressed and will make him offers. But Smith will receive no offers. Therefore Smith will not bribe the instructor.

If logic is difficult, then few students will take logic courses unless such courses are obligatory. If logic is not difficult then logic courses will not be obligatory. So if a student can choose whether or not to take logic, then either logic is not difficult or few students take logic courses.

If God exists, there will be no evil in the world unless God is unjust, or not omnipotent, or not omniscient. But if God exists the He is none of these, and there is evil in the world. So we have to conclude that God does not exist.

SYNTAX uninterpreted: the basic elements (apart from the connectives) do not have their own meaning the basic elements are sentence-letters (not words) The lexicon of LSL contains infinitely many sentence-letters

A sentence-letter symbolizes an atomic sentence; stands for a fixed sentence p, q – sentential variables, metavariables; do not stand for fixed sentences

The lexicon of LSL: All sentence-letters “A”, “B”, “C”, Connectives Punctuation marks “(“, “)” Well-formed formula = grammatical formula (wff)

Formation rules: conditionals: if it is given a wff or wffs as input, then such and such a wff results as an output

Formation rules Every sentence letter is a well formed formula (wff). If p is a wff, so is  p  If p and q are wffs, so is  p  q  If p and q are wffs, so is  p  q  If p and q are wffs, so is  p  q  If p and q are wffs, so is  p  q  Closure condition: nothing is a wff unless certified as such by the above rules

The object language – the metalanguage mouse ”mouse”

Use/mention distinction Warsaw is the capital of Poland. use “Warsaw” contains 6 letters.Mention “Mouse” is an animal. Mouse is a noun.

Corner quotes = selective quotes – corner quotes around a whole expression put ordinary quotes selectively around parts of the expression, those parts which are object language symbols. If p and q are wffs, so is “p  q” – incorrect If p and q are wffs, the result of writing p followed by “  ” followed by q is a wff.

SEMANTICS If a sentence is true then it is said to have the truth-value true The Principle of Bivalence: there are exactly two truth-values, truth and false. Every meaningful sentence, simple or compound, has one or the other, but not both, of these truth-values.

Any sentential connective whose meaning can be captured in a truth-table is called a truth-functional connective and is said to express a truth-function.